I have developed a way of thinking about how society progresses in the quest for scientific knowledge. Have you ever wondered about why mathematics and science are two separate subjects in school? Here is my version of the progression of knowledge.
Mathematics is for the things that we fully understand and science is for the things that we only partially understand. It is only when we fully understand the way something operates, such as a process in nature, that we describe it in terms of mathematics. Of course, science and mathematics fit closely together, the backbone of any of the physical sciences is mathematics. I define science as that which we cannot yet describe completely with mathematics because we do not have a complete understanding of it.
The difference between mathematics and science is the same as the difference between words and numbers. Words are science and numbers, of course, are mathematics. Words are less precise than numbers but have much greater flexibility and subjectivity that is necessary to describe that which is not completely understood. To make possible a description of something in terms of mathematics, all relevant factors must be thoroughly understood. When this is not the case, when it is only partially understood, it is the realm of science and words rather than mathematics.
Everything is really mathematics at it's most fundamental level, it is when we do not thoroughly understand it's workings that we classify it as science. The word "science" means "knowledge" but I think it could better be defined as "in the process of being understood". It is when we can apply definite numbers to a process that we really understand it.
Here is my definition of the knowledge progression: Information moves from the unknown or from superstition to the realm of science as it becomes understood and from there to the realm of mathematics as it becomes completely understood. Unknown - Science - Mathematics.
The chemical elements, for example, can be easily described in terms of mathematics because the very definition of a particular element is the number of protons in the nucleus. The calendar, the course of a year as the earth moves around the sun, was once the realm of science but is now that of mathematics because it is thoroughly understood. The measurement of the height of a tower by the length of it's shadow would also be considered as mathematics because there is no mystery to the process, it is well understood. We do not yet thoroughly understand why like electric charges repel and unlike charges attract so although this seems like it could be described easily mathematically, it is still considered as science.
All we have to do to tell how far along we are in understanding the nature of reality is to look inside a school. As long as science and mathematics are considered as separate subjects, it goes to show that we do not fully understand the operation of reality. When we do fully understand the operation of reality, it will be described primarily in mathematical terms and science will become one with mathematics.
Suppose there was a math book illustrating all mathematical processes and all relevant examples of each process and also a science book describing the sum of scientific knowledge. As knowledge advanced, the science book would decrease while the math book increased but the science book would decrease faster than the math book increased.
This also goes to validate my Theory of Primes, that all of reality consists of fundamental patterns. The way I see it, the categorization of the physical sciences into chemistry, physics, astronomy, etc. is also evidence of our incomplete knowledge. If we did have complete knowledge, these fields would merge together and into mathematics. When we get closer to complete understanding, specialization of topic will give way to categorization by pattern as is done in mathematics books. There will be no more chemistry, physics and, astronomy but only mathematical patterns.
On my patterns and complexity blog, http://www.markmeekpatterns.blogspot.com/ , I pointed out how coincidences that we observe can be a powerful tool to measure complexity in certain circumstances. The complexity section of that blog is about the tremendous benefits that we would gain if we could only quantify complexity. That is, put an actual numerical measurement on it instead of describing it in vague and subjective terms.
My definition of coincidence is a random reduction in apparent complexity, as seen by the observer. A reverse coincidence is a random increase in apparent complexity. Each coincidence must be balanced by a reverse coincidence that is equal in magnitude.
The ideal example of coincidence could be a call center. Suppose the call center has one hundred operators, but that you get the same few operators every time you call. This would be a coincidence that causes you to perceive the center as being less complex than it actually is. But if you called a hundred times and got a different operator every time, this would be a reverse coincidence that might make you think that the center was actually much more complex than it actually is.
There is actually no such thing as either a coincidence or a reverse coincidence. Both are only a matter of our perspective of incomplete understanding. Coincidences are somewhat like a mirage, the shimmering water mirage seen ahead in the distance on a hot day on flat ground. There appears to be water but when you arrive at where the water appears to be, it has moved further back.
Likewise, the more understanding we have of whatever system we are observing, the fewer coincidences and reverse coincidences we will observe. Neither actually exist and once we understand everything about a system of some kind, we will observe no more of either coincidences or reverse coincidences.
There is no such thing as a coincidence because any coincidence must be balanced by a reverse coincidence somewhere else in the system. If we do not see this balance, then we do not completely understand the system. Coincidences and reverse coincidences are perspective illusions which indicate that we have less than complete understanding of the system being observed.
A coincidence exists only when we do not fully understand it. Suppose we had four people walk in the same direction with equal space between them. Now suppose that someone happened to be walking by on the opposite side of the road. They might think "What a coincidence, those people walking just happen to have equal space between them".
If we could really develop an eye for coincidences, they would also be a powerful measurement tool in the progression of knowledge. We know what we know, but usually do not know what we don't know. When observing a system of some type, such as a galaxy or a bee hive, suppose we could keep a close count of every coincidence and reverse coincidence that we see and, if possible, put some value on it. When we got to the point where we saw none of either coincidences or reverse coincidences, we would know that we then knew about all that we could practically know about that system.
This concept makes partial values useful. If, when observing a system of some kind, we observed a certain value of coincidences and a certain value of reverse coincidences. First, we would know that we could assume the higher value of the two for both because coincidences and reverse coincidences must balance out.
Then, if we learned more about the system and then made another observation and this time logged only half the previous value for coincidences and reverse coincidences, we would know that we had progressed halfway in the progression of knowledge.
PART TWO
Since the universe is not infinite, the body of knowledge that we can possibly have cannot be infinite either. There must be a day in our future when, at least theoretically, we will reach the point where we know all that we can practically know. The question is: where do we, with our present accumulation of knowledge, stand with regard to that point?
In Part One, there are two possible methods described for answering this question. One, that more and more of our knowledge will be expressed as mathematics rather than as words and two, the use of apparent coincidences as a measurement of where we stand with regard to knowledge.
Today, I would like to describe another possible way to measure our progression of knowledge.
Humans began to learn and record knowledge. At first, every new fact would be something completely new. But as time went on, there would be an increasing proportion of newly-learned facts that are not totally new, but are a previously-unseen connection between facts that are already known. The more facts that we have already, the higher the likelyhood that a new fact will be a connection between two already-known facts, rather than something totally new.
We could call this first- and second-tier learning. But this is only an illusion of our perspective. There are no such tiers of facts in absolute reality.
As an example, the discovery of electrons was a totally new fact even though we already knew about the atoms of which electrons are a part. Their discovery was not a connection between previously-known facts, this was first-tier learning. Now, suppose that it was discovered that zebras like pumpkins. This would be second-tier learning, since it would be a connection between two entities that we already know about.
If we could get used to dividing newly-discovered facts into these two categories, this should give us a good idea about where we stand in the progression of knowledge. As we move along, an increasing proportion of newly-discovered facts should be second-tier, a connection between two previously-known facts. As we near the maximum potential of our practical knowledge, virtually every new fact should be a connection between two previously-known facts rather than a fact that is totally new to us.
Thus, this can be used as a measurement tool.
Thursday, June 18, 2009
The Floating Base System
Have you ever thought about how we are continually improving the performance of computers in such ways as processor speeds and hard drive capacity while we are still using the basic coding structure that has been in place since the beginning of the modern computer era and that it would be very beneficial if we would bring that into the Twenty-First Century?
The incredible progress in other areas of computing is overshadowed by our continued use of the primitive ASCII coding system for the characters of the alphabet, numbers and, punctuation. We now have the chance to make just as much progress in computer capacity and efficiency with simple arithmetic as we can with the usual chip and storage technology.
The way we have been making continuous upgrades in applications and operating systems while still using the old ASCII coding system is like developing entirely new ways of printing documents with the latest version of Microsoft Office but then delivering the documents to their destination by Pony Express. You can see an ASCII table at http://www.asciitable.com/ or read all about it on http://www.wikipedia.com/
If we could make the storage and transmission of data more efficient, it would have the same effect as improving processor and transmission speeds and hard drive capacity. I prefer the idea of Numbering Sentences that I described in the posting below but that will require some work and time to break all sentences down into numbers. There are other ways that I have noticed that could be implemented in the next generation of software.
TRIMMING BYTES
Suppose we could reduce the size of the byte used in ASCII from eight bits to six? That would immediately make storage and transmission of data more efficient because it would require only 75% of today's capacity for the same amount of data. It is true that six bits gives us only 64 possible combinations, as opposed to the 256 of eight-bit bytes. But those eight-bit bytes were implemented in the days before spellcheck technology in word processing. Why not eliminate capital letters from the coding for the purposes of storage and transmission and then have spellcheckers capitalize the proper letters later?
I believe that we could also eliminate quite a few of the other characters presently used in ASCII coding. We could even eliminate the number characters; 1234567890 by having numbers automatically spelled out for storage and transmission and then being reduced to numbers later on, if required. It is true that the word "five" takes up more space than "5" but it would reduce the number of different characters that must be encoded.
THE BASE SYSTEM
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Now, let's move on. Suppose we could use a flexible, or floating, number base for each block of data that we store or transmit? It would make it even more efficient. This is done by automatically scanning the data, detecting how many different characters along with invisible control keys it contains and using that number as the number base to encode the data as a single large number. This would be incredibly efficient.
How many documents or blocks of text contain characters such as ! # ^ & * +=( )? The answer is only a relative few. So, why is it necessary to have space to encode characters that are not there on each block of text? Also, a document may not contain letters such as q, x or, z, making it unnecessary to include space in the coding for them. The goal should be to make the number base for encoding the text as low as possible.
Back in my music days there was a song I liked, a line of which is "It don't come easy, you know it don't come easy." To encode this, the program would count the number of different characters, including spaces, punctuation and, control characters and use that number as the base by which it would be encoded. This is the type of calculation at which computers excel. If a block of text were too large for the computer processor to calculate the number representing the text, it would simply break it up into two or more blocks.
The incredible progress in other areas of computing is overshadowed by our continued use of the primitive ASCII coding system for the characters of the alphabet, numbers and, punctuation. We now have the chance to make just as much progress in computer capacity and efficiency with simple arithmetic as we can with the usual chip and storage technology.
The way we have been making continuous upgrades in applications and operating systems while still using the old ASCII coding system is like developing entirely new ways of printing documents with the latest version of Microsoft Office but then delivering the documents to their destination by Pony Express. You can see an ASCII table at http://www.asciitable.com/ or read all about it on http://www.wikipedia.com/
If we could make the storage and transmission of data more efficient, it would have the same effect as improving processor and transmission speeds and hard drive capacity. I prefer the idea of Numbering Sentences that I described in the posting below but that will require some work and time to break all sentences down into numbers. There are other ways that I have noticed that could be implemented in the next generation of software.
TRIMMING BYTES
Suppose we could reduce the size of the byte used in ASCII from eight bits to six? That would immediately make storage and transmission of data more efficient because it would require only 75% of today's capacity for the same amount of data. It is true that six bits gives us only 64 possible combinations, as opposed to the 256 of eight-bit bytes. But those eight-bit bytes were implemented in the days before spellcheck technology in word processing. Why not eliminate capital letters from the coding for the purposes of storage and transmission and then have spellcheckers capitalize the proper letters later?
I believe that we could also eliminate quite a few of the other characters presently used in ASCII coding. We could even eliminate the number characters; 1234567890 by having numbers automatically spelled out for storage and transmission and then being reduced to numbers later on, if required. It is true that the word "five" takes up more space than "5" but it would reduce the number of different characters that must be encoded.
THE BASE SYSTEM
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Now, on to another idea for increasing computer efficiency by reforming ASCII. I have a way to not only store and transmit computer data with fewer bits but to speed up processing by freeing the processor from deciphering the eight-bit bytes of ASCII. As you know, a number system can be of any base. Ours happens to be base-ten because ancient people counted things on their ten fingers. Computers are coded in binary, or base-two and the computer world also uses hexidecimal, or base-sixteen.
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Suppose we could consider any text that had to be stored or transmitted not as text, but as a number. If we could eliminate capital letters and numbers, our alphabet could be considered a base-twenty seven number system because there are twenty-six letters and a space between words.
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In fact ASCII is, in effect, a base-two hundred fifty six number system. If we could see any text as actually a number, all we would have to do is store and transmit that number in binary. Not only would it require far less space but would also be much easier on the computer processor because it would be just one long number and not composed of bytes.
Let's consider something simple like my name, Mark Meek. If encoded into ASCII it would require nine bytes, including the space. In other words, 72 bits. But if we devised a base-fifty number system, let the space = zero, a = 1, b = 2, z = 26 and so on, so that my name was considered by the computer to be a number, it would require only 49 bits, according to my calculations. Using a base-fifty number system should allow us to incorporate all the required punctuation and control keys since capital letters and possibly numerical characters can be eliminated until later.
Let's consider something simple like my name, Mark Meek. If encoded into ASCII it would require nine bytes, including the space. In other words, 72 bits. But if we devised a base-fifty number system, let the space = zero, a = 1, b = 2, z = 26 and so on, so that my name was considered by the computer to be a number, it would require only 49 bits, according to my calculations. Using a base-fifty number system should allow us to incorporate all the required punctuation and control keys since capital letters and possibly numerical characters can be eliminated until later.
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THE FLOATING BASE SYSTEM
Now, let's move on. Suppose we could use a flexible, or floating, number base for each block of data that we store or transmit? It would make it even more efficient. This is done by automatically scanning the data, detecting how many different characters along with invisible control keys it contains and using that number as the number base to encode the data as a single large number. This would be incredibly efficient.
How many documents or blocks of text contain characters such as ! # ^ & * +=( )? The answer is only a relative few. So, why is it necessary to have space to encode characters that are not there on each block of text? Also, a document may not contain letters such as q, x or, z, making it unnecessary to include space in the coding for them. The goal should be to make the number base for encoding the text as low as possible.
Back in my music days there was a song I liked, a line of which is "It don't come easy, you know it don't come easy." To encode this, the program would count the number of different characters, including spaces, punctuation and, control characters and use that number as the base by which it would be encoded. This is the type of calculation at which computers excel. If a block of text were too large for the computer processor to calculate the number representing the text, it would simply break it up into two or more blocks.
Numbered Sentences
It is really amazing how much rapid progress has been made in improving computer processor speeds as well as hard drive capacity. But while this progress is being made, our basic system of digital coding remains so inefficient. I find this to be yet another example of how we can be technically forward but system backward at the same time.
The best-known system of digital coding is ASCII. It uses eight digital bits, known as a byte, to encode information. In digital form, a byte would look like this: 01101001 since each bit can be either on or off, represented by a 1 or 0.
Each magnetic particle on a hard drive stores a bit. Eight bits of two possibilities each mean that a byte has 256 different possibilities. ASCII uses each bit to store a text character such a letter, number or, punctuation mark. Some of the 256 ASCII cominations in a byte are unprintable control codes. Besides ASCII, there are other systems such as EBCDIC, used in mainframe computers, and Unicode.
The problem with a coding system such as ASCII is that it's coding represents characters, such as numbers and the letters of the alphabet. No matter how we improve processor speeds and hard drive capacity, this simple coding system remains inefficient in the extreme and thus limits the potential of computers.
I find that our digital communications and data storage would be multiplied many times in efficiency if our primary unit of communication was not the character, not the word, but the sentence. If we can categorize DNA in the Genome Project, then why can't we categorize all things that all people say and write to each other while communicating?
People all across the world say pretty much the same things to each other. The sentences could be arranged in a logical order and each one assigned a number. This would greatly simplify and increase in efficiency all digital storage and communications. This should have been done long ago, actually in the early days of computing.
Microsoft Office has standardized and categorized office documents into word processing, spreadsheets, databases and, presentations. Why not categorize every sentence that is used in communications and assign it a number? Then we would only need to communicate and store that number instead of the sentence written out in characters. This would be immeasurably more efficient. Dictionary writers take great care to categorize words, we can expend the idea to entire sentences. The sentences could be arranged into a hundred or so logical categories and then selected from there.
It is all right, in most cases, if the sentences are somewhat generic. In most human communications, flowery prose is unnecessary. Several words may have the same meaning many sentences can be phrased in different ways, but for our purposes of efficient data storage and communications, only one choice of sentence would be necessary.
This concept of using sentences, rather than characters or words, as our main unit of digital communication and storage has another tremendous advantage besides the great increase in efficiency. Grammar and alphabet or character is not the same from one language to another so literal translation word by word from one language to another usually produces little more than gibberish. It is the sentence, not just the words, that must be translated.
This system of sentence numbering would also make quick and easy translation of data from one language to another possible. The entire world uses the same numbering system. Data could be stored and transmitted as numbers and each number would represent a sentence. The data numbers could be easily displayed in any language. If, for example, all communication was broken down into a million sentences, sentence 130461 might be "I went to the store today." All we would need to do would be to transfer and store the number.
In data transfers, computer systems make extensive use of codecs, compression and decompression so why not take the same approach to the basic coding of data? Numbering sentences would be far more efficient than today's coding of characters, which was developed long before the easy and widespread global communication of the internet. This sentence transmission and display could readily be included in future operating systems.
Next, let's move on to make this concept even more efficient with what I will call the "sentence package". Each sentence will be assigned a number, my guess is that we can expect to have a million or so sentences which will then be arranged into a logical sequence before being assigned numbers. The way to make this process more efficient is to use direct binary to encode each sentence, instead of the ASCII characters for the numbers assigned to the sentences.
A string of 20 of the digital bits that computers use to store and transmit data will give us 1,048,576 possible combinations. I believe that this will be enough to assign a number to all necessary sentence combinations, along with any control characters that will be needed. We will call this 20-bit string a "sentence package".
It will operate in the same way as the 8-bit bytes used to encode each character and number in ASCII. It might be more effective to enclose this 20-bit sentence package within a string of 24 bits because that would comprise 3 of the bytes that the computer world is accustomed to dealing with. This would also provide plenty of room for any specialized sub-sets of sentences that may be necessary such as one each for doctors, physicists, astronomers, etc. to include the sentences only used by these particular groups in communication. Names could still be spelled out in ASCII when necessary.
The thing that makes this great increase in efficiency possible is vast gaps in our written communication of potential words that are not used as words. We could call them non-words. For example, "Ncbda" could be a word but it isn't. The existence of such non-words means that our alphabetic system has much built-in spatial inefficiency. Quite a bit of this inefficiency is because the words we use revolve around positioning of vowels and consonants.
Our wording system, if printed as a graph would look something like a map of the South Pacific or the West Indies. The words we use would be islands but there would be vast gaps, represented by sea, of potential but non-words. The way to cut out this inefficiency is to use this idea of sentence packaging, it is the ultimate codec.
In ASCII coding, a simple sentence like "I went to the store today", requires 25 bytes, one for each character, including spaces. Since a byte consists of 8 bits, that means a total of 200 bits of data. In this new system of sentence packaging, it will require only the 20 bits of one sentence package. This is an increase in efficiency by a factor of 10. Even if we use 24 bits, 3 bytes, per sentence, it brings an increase in efficiency by more than a factor of 8. Plus the fact that text encoded in this way can be easily displayed in any language.
The best-known system of digital coding is ASCII. It uses eight digital bits, known as a byte, to encode information. In digital form, a byte would look like this: 01101001 since each bit can be either on or off, represented by a 1 or 0.
Each magnetic particle on a hard drive stores a bit. Eight bits of two possibilities each mean that a byte has 256 different possibilities. ASCII uses each bit to store a text character such a letter, number or, punctuation mark. Some of the 256 ASCII cominations in a byte are unprintable control codes. Besides ASCII, there are other systems such as EBCDIC, used in mainframe computers, and Unicode.
The problem with a coding system such as ASCII is that it's coding represents characters, such as numbers and the letters of the alphabet. No matter how we improve processor speeds and hard drive capacity, this simple coding system remains inefficient in the extreme and thus limits the potential of computers.
I find that our digital communications and data storage would be multiplied many times in efficiency if our primary unit of communication was not the character, not the word, but the sentence. If we can categorize DNA in the Genome Project, then why can't we categorize all things that all people say and write to each other while communicating?
People all across the world say pretty much the same things to each other. The sentences could be arranged in a logical order and each one assigned a number. This would greatly simplify and increase in efficiency all digital storage and communications. This should have been done long ago, actually in the early days of computing.
Microsoft Office has standardized and categorized office documents into word processing, spreadsheets, databases and, presentations. Why not categorize every sentence that is used in communications and assign it a number? Then we would only need to communicate and store that number instead of the sentence written out in characters. This would be immeasurably more efficient. Dictionary writers take great care to categorize words, we can expend the idea to entire sentences. The sentences could be arranged into a hundred or so logical categories and then selected from there.
It is all right, in most cases, if the sentences are somewhat generic. In most human communications, flowery prose is unnecessary. Several words may have the same meaning many sentences can be phrased in different ways, but for our purposes of efficient data storage and communications, only one choice of sentence would be necessary.
This concept of using sentences, rather than characters or words, as our main unit of digital communication and storage has another tremendous advantage besides the great increase in efficiency. Grammar and alphabet or character is not the same from one language to another so literal translation word by word from one language to another usually produces little more than gibberish. It is the sentence, not just the words, that must be translated.
This system of sentence numbering would also make quick and easy translation of data from one language to another possible. The entire world uses the same numbering system. Data could be stored and transmitted as numbers and each number would represent a sentence. The data numbers could be easily displayed in any language. If, for example, all communication was broken down into a million sentences, sentence 130461 might be "I went to the store today." All we would need to do would be to transfer and store the number.
In data transfers, computer systems make extensive use of codecs, compression and decompression so why not take the same approach to the basic coding of data? Numbering sentences would be far more efficient than today's coding of characters, which was developed long before the easy and widespread global communication of the internet. This sentence transmission and display could readily be included in future operating systems.
Next, let's move on to make this concept even more efficient with what I will call the "sentence package". Each sentence will be assigned a number, my guess is that we can expect to have a million or so sentences which will then be arranged into a logical sequence before being assigned numbers. The way to make this process more efficient is to use direct binary to encode each sentence, instead of the ASCII characters for the numbers assigned to the sentences.
A string of 20 of the digital bits that computers use to store and transmit data will give us 1,048,576 possible combinations. I believe that this will be enough to assign a number to all necessary sentence combinations, along with any control characters that will be needed. We will call this 20-bit string a "sentence package".
It will operate in the same way as the 8-bit bytes used to encode each character and number in ASCII. It might be more effective to enclose this 20-bit sentence package within a string of 24 bits because that would comprise 3 of the bytes that the computer world is accustomed to dealing with. This would also provide plenty of room for any specialized sub-sets of sentences that may be necessary such as one each for doctors, physicists, astronomers, etc. to include the sentences only used by these particular groups in communication. Names could still be spelled out in ASCII when necessary.
The thing that makes this great increase in efficiency possible is vast gaps in our written communication of potential words that are not used as words. We could call them non-words. For example, "Ncbda" could be a word but it isn't. The existence of such non-words means that our alphabetic system has much built-in spatial inefficiency. Quite a bit of this inefficiency is because the words we use revolve around positioning of vowels and consonants.
Our wording system, if printed as a graph would look something like a map of the South Pacific or the West Indies. The words we use would be islands but there would be vast gaps, represented by sea, of potential but non-words. The way to cut out this inefficiency is to use this idea of sentence packaging, it is the ultimate codec.
In ASCII coding, a simple sentence like "I went to the store today", requires 25 bytes, one for each character, including spaces. Since a byte consists of 8 bits, that means a total of 200 bits of data. In this new system of sentence packaging, it will require only the 20 bits of one sentence package. This is an increase in efficiency by a factor of 10. Even if we use 24 bits, 3 bytes, per sentence, it brings an increase in efficiency by more than a factor of 8. Plus the fact that text encoded in this way can be easily displayed in any language.
Next Generation Technology
Have you ever thought about how primitive we really are? Let's take a look at the kinds of technology we could have.
The human brain emits and uses electrical waves. So why do we need to use cell phones (mobiles) and other electronic devices in the same way that cave men used pieces of flint as manual tools? There should be no need for buttons or switches on electronic devices nor should there be computer monitors, displays or, television screens.
All a person needs is to wear a plastic band on their head. That band will decipher what the person wants by their brain waves and will then access the information or image and put it into their brain by the same kind of waves. The person will thus see a web site like a dream or mirage with no screen needed. We will need a little practice at using our thoughts to turn things on and off but no more than is necessary for using the functions on an iphone.
We are able to put together the history of our solar system just by carefully observing it. We know that each atom is actually a solar system also with electrons revolving around the central nucleus. We should be able to do the same thing with each atom by carefully studying the placement of it's orbital electrons.
Hidden in the paths of the electrons in orbit is a history of everywhere the atom has ever been and ever done. This is simply because everything that has ever happened to an atom has had some effect on it's electron orbitals. We just need to learn to decipher that.
For example, light that hits atoms should leave some imprint in slight adjustment of the electron orbits. This means that when we become proficient enough, anything can act as a camera. We could place a scanner against the side of a pyramid to carefully measure the orbitals in each atom in the stone. Then we could glean an actual video of ancient Egyptians building the pyramid.
Why can we not make anything from anything else? We could scan the arrangements of atoms in a blueberry pie onto a computer disk. We could then place the equivalent atoms in a special chamber and have the atoms arrange themselves into an identical blueberry pie. If we got really proficient, we could scan the sub-atomic particles of the blueberry pie and so would not even need to have the same kind of atoms to make the pie.
Such copy technology would also enable people to travel between stations at the speed of light. You would step into a transportation chamber and a scan would be made of your body. The information would travel by internet or radio waves to a distant station where your body would be recreated from a different set of atoms. Your "essence", your spirit, memory and, intellect, would then be placed in the your new body.
If you are wondering if you would really be "you" if you had an identical body made of different atoms, the answer is yes. The atoms in your body are continually in transition and you may contain few, if any, of the atoms that you had a year ago. Your "old" body would be recycled for it's atoms to be used by a traveller in the opposite direction. It is true that something might go wrong when you go to be transported but the same is true when you board a plane to be transported.
Energy should not even be an issue. It should be as available as air. There is energy in everything unless it is at a temperature of absolute zero and perfectly at rest. We should be able to draw all the energy we need out of anything, anywhere. It is just that we are still on the same thought track that we were on as cavemen.
The most important physical fact in the universe is that there are two electric charges, which we term negative and positive. Like charges mutually repel, while opposite charges attract. This is the foundation of the universe, and all other facts are mere details in comparison with this one.
In my cosmological theory, everything is composed of infinitesimal electric charges. When we have a pattern of alternating charges, negative and positive forming a multi-dimensional checkerboard, we have space. Any other arrangement of the electrical charges gives us matter, instead of space.
Movement of the electrical charges in matter affect those in the adjoining space, setting up an electromagnetic wave. It is not that the wave itself is electromagnetic, it is just that the wave reveals the underlying electromagnetism of space by disturbing the prefect arrangement of alternating charges.
It appears that there is no net electrical charge to the universe, the total number of negative and positive charges ends up as exactly equal. There also seems to be no variation whatsoever between one like charge and another, no internal structure or factors whatsoever. The charges must always balance out, they will induce new charges into being if they don't. This makes it seem as if the charges that compose both the space and matter of the universe originated as a part of some pre-universe structure that has since been "trying" to reassemble itself.
Of course, the designation of negative and positive that we have given these electrical charges is entirely arbitrary. they could just as easily have been designated as the night and day charges. They tend to pair up and do not usually operate independently for long.
In trying to learn all that we can about the universe, we come up against these charges as the most basic of building blocks. Everything that we are and everything that we know are composed of these electrical charges. We cannot see anything that is not composed of these charges, and this puts strict limitations on us understanding what the charges really are. Gaining such an understanding, beyond merely seeing how the charges behave, would truly be the ultimate in science.
The ultimate in technology would be to gain the ability to manipulate and exchange these charges. If we could do that, we could make anything out of anything else, or out of empty space for that matter. We could also make any matter disappear into empty space. We could divide a pile of garbage in half, turn one side into antimatter by reversing the charges, then reap a fantastic amount of energy as the two halves mutually annihilated when brought into contact.
But we are limited by what we are and what we have. Any equipment that we can construct must necessarily be made of these electrical charges, and that puts a limit on us building anything that can manipulate the charges. We cannot look further into the two electrical charges because they are already as fundamental as we can get. An electron, for example, appears to be a mere point of negative charge with no internal structure whatsoever and so cannot be dissected further.
As it turns out, it is not entirely impossible to manipulate and exchange electrical charges. Black holes in the universe are concentrations of matter, bound together by gravity, that are so dense that not even light can escape the gravity. Hence, we can detect black hole not by direct observation but only by observation of it's gravitational effects on the surrounding area.
But it is known that black holes actually do give off radiation and gradually decay. This seems to be mystifying but, as I explained in "Black Holes And Antimatter" on the cosmology blog www.markmeekcosmology.blogspot.com , this is easily explained by my cosmological theory.
If super extreme pressure is put on electrical charges, forcing like charges together and pushing unlike charges apart, the charges will actually begin to migrate so that negative becomes positive and vice versa. But when this happens, we have matter turning into antimatter. We know that when matter and antimatter are brought into contact that they mutually destruct and give off a burst of energy.
Antimatter is the same thing as ordinary matter, except that the charges are reversed. Instead of electrons, antimatter has positively charged particles, called positrons, in orbitals around a negatively-charged nucleus that is the opposite of matter. There is now far more matter than antimatter in the universe. If you saw a galaxy in space composed of antimatter, you probably could not tell the difference just by observation.
In my theory, the negative and positive charges that composed the matter and antimatter literally rearrange themselves into the alternating checkerboard pattern of empty space. The burst of energy released is a redirection of the energy in the Big Bang that put the matter and antimatter together in the first place. This ideally explains why black holes both give off radiation and gradually decay, because matter is being turned into antimatter by charge migration brought about by the extreme pressure inside and then the two are mutually annihilating each other.
The trouble is that the pressure inside these black holes in space is far beyond anything that we know on earth. Even the tremendous pressure in the center of the sun, which is enough to crunch smaller atoms together into larger ones, is nowhere near what it would take to make charges migrate so that we could exchange and work with them.
BRAIN WAVES
The human brain emits and uses electrical waves. So why do we need to use cell phones (mobiles) and other electronic devices in the same way that cave men used pieces of flint as manual tools? There should be no need for buttons or switches on electronic devices nor should there be computer monitors, displays or, television screens.
All a person needs is to wear a plastic band on their head. That band will decipher what the person wants by their brain waves and will then access the information or image and put it into their brain by the same kind of waves. The person will thus see a web site like a dream or mirage with no screen needed. We will need a little practice at using our thoughts to turn things on and off but no more than is necessary for using the functions on an iphone.
ATOMS
We are able to put together the history of our solar system just by carefully observing it. We know that each atom is actually a solar system also with electrons revolving around the central nucleus. We should be able to do the same thing with each atom by carefully studying the placement of it's orbital electrons.
Hidden in the paths of the electrons in orbit is a history of everywhere the atom has ever been and ever done. This is simply because everything that has ever happened to an atom has had some effect on it's electron orbitals. We just need to learn to decipher that.
For example, light that hits atoms should leave some imprint in slight adjustment of the electron orbits. This means that when we become proficient enough, anything can act as a camera. We could place a scanner against the side of a pyramid to carefully measure the orbitals in each atom in the stone. Then we could glean an actual video of ancient Egyptians building the pyramid.
COPY TECHNOLOGY
Why can we not make anything from anything else? We could scan the arrangements of atoms in a blueberry pie onto a computer disk. We could then place the equivalent atoms in a special chamber and have the atoms arrange themselves into an identical blueberry pie. If we got really proficient, we could scan the sub-atomic particles of the blueberry pie and so would not even need to have the same kind of atoms to make the pie.
Such copy technology would also enable people to travel between stations at the speed of light. You would step into a transportation chamber and a scan would be made of your body. The information would travel by internet or radio waves to a distant station where your body would be recreated from a different set of atoms. Your "essence", your spirit, memory and, intellect, would then be placed in the your new body.
If you are wondering if you would really be "you" if you had an identical body made of different atoms, the answer is yes. The atoms in your body are continually in transition and you may contain few, if any, of the atoms that you had a year ago. Your "old" body would be recycled for it's atoms to be used by a traveller in the opposite direction. It is true that something might go wrong when you go to be transported but the same is true when you board a plane to be transported.
ENERGY
Energy should not even be an issue. It should be as available as air. There is energy in everything unless it is at a temperature of absolute zero and perfectly at rest. We should be able to draw all the energy we need out of anything, anywhere. It is just that we are still on the same thought track that we were on as cavemen.
ELECTRICAL CHARGES
The most important physical fact in the universe is that there are two electric charges, which we term negative and positive. Like charges mutually repel, while opposite charges attract. This is the foundation of the universe, and all other facts are mere details in comparison with this one.
In my cosmological theory, everything is composed of infinitesimal electric charges. When we have a pattern of alternating charges, negative and positive forming a multi-dimensional checkerboard, we have space. Any other arrangement of the electrical charges gives us matter, instead of space.
Movement of the electrical charges in matter affect those in the adjoining space, setting up an electromagnetic wave. It is not that the wave itself is electromagnetic, it is just that the wave reveals the underlying electromagnetism of space by disturbing the prefect arrangement of alternating charges.
It appears that there is no net electrical charge to the universe, the total number of negative and positive charges ends up as exactly equal. There also seems to be no variation whatsoever between one like charge and another, no internal structure or factors whatsoever. The charges must always balance out, they will induce new charges into being if they don't. This makes it seem as if the charges that compose both the space and matter of the universe originated as a part of some pre-universe structure that has since been "trying" to reassemble itself.
Of course, the designation of negative and positive that we have given these electrical charges is entirely arbitrary. they could just as easily have been designated as the night and day charges. They tend to pair up and do not usually operate independently for long.
In trying to learn all that we can about the universe, we come up against these charges as the most basic of building blocks. Everything that we are and everything that we know are composed of these electrical charges. We cannot see anything that is not composed of these charges, and this puts strict limitations on us understanding what the charges really are. Gaining such an understanding, beyond merely seeing how the charges behave, would truly be the ultimate in science.
The ultimate in technology would be to gain the ability to manipulate and exchange these charges. If we could do that, we could make anything out of anything else, or out of empty space for that matter. We could also make any matter disappear into empty space. We could divide a pile of garbage in half, turn one side into antimatter by reversing the charges, then reap a fantastic amount of energy as the two halves mutually annihilated when brought into contact.
But we are limited by what we are and what we have. Any equipment that we can construct must necessarily be made of these electrical charges, and that puts a limit on us building anything that can manipulate the charges. We cannot look further into the two electrical charges because they are already as fundamental as we can get. An electron, for example, appears to be a mere point of negative charge with no internal structure whatsoever and so cannot be dissected further.
As it turns out, it is not entirely impossible to manipulate and exchange electrical charges. Black holes in the universe are concentrations of matter, bound together by gravity, that are so dense that not even light can escape the gravity. Hence, we can detect black hole not by direct observation but only by observation of it's gravitational effects on the surrounding area.
But it is known that black holes actually do give off radiation and gradually decay. This seems to be mystifying but, as I explained in "Black Holes And Antimatter" on the cosmology blog www.markmeekcosmology.blogspot.com , this is easily explained by my cosmological theory.
If super extreme pressure is put on electrical charges, forcing like charges together and pushing unlike charges apart, the charges will actually begin to migrate so that negative becomes positive and vice versa. But when this happens, we have matter turning into antimatter. We know that when matter and antimatter are brought into contact that they mutually destruct and give off a burst of energy.
Antimatter is the same thing as ordinary matter, except that the charges are reversed. Instead of electrons, antimatter has positively charged particles, called positrons, in orbitals around a negatively-charged nucleus that is the opposite of matter. There is now far more matter than antimatter in the universe. If you saw a galaxy in space composed of antimatter, you probably could not tell the difference just by observation.
In my theory, the negative and positive charges that composed the matter and antimatter literally rearrange themselves into the alternating checkerboard pattern of empty space. The burst of energy released is a redirection of the energy in the Big Bang that put the matter and antimatter together in the first place. This ideally explains why black holes both give off radiation and gradually decay, because matter is being turned into antimatter by charge migration brought about by the extreme pressure inside and then the two are mutually annihilating each other.
The trouble is that the pressure inside these black holes in space is far beyond anything that we know on earth. Even the tremendous pressure in the center of the sun, which is enough to crunch smaller atoms together into larger ones, is nowhere near what it would take to make charges migrate so that we could exchange and work with them.
Our Antiquated Phone System
I find that there is no better example of how we can be technologically forward but system backward than out telephone network. Telephone numbers in North America have ten digits and look like this: 123-456-7890.
The first three digits, the 123, is the area code. This tells the telephone network the general area where the call is going. For example, in the far western part of New York State, the area code is 716. The next three digits, the 456, is known as the exchange. This is a kind of sub-area code. After the area code has been established, this tells the telephone network where in the area code the call is going. The final four digits, the 7890, is the body of the phone number.
Each exchange can have up to ten thousand numbers, from 0000 to 9999. Area codes typically contain about a hundred exchanges. For example the most common exchanges in the City of Niagara Falls, NY seem to be 283, 284 and, 297.
My point here is that the use of exchanges is inefficiency in the extreme. These exchanges are a relic of the days of mechanical switching systems to connect two callers. The early days of telephone featured operators manually plugging in a cord to join two callers. Then the mechanical relay switching systems became available in which a call was routed first to the destination area code and then to an exchange within the area code and finally to the number within the exchange. The area code is in itself an exchange.
In these days of computerized equipment with vast bandwidth and data storage capacity, why do we still use a system that was designed for the mechanical equipment available fifty years ago? There is probably no better example of how we can have space age technology operating under a virtually stone age system of organization.
Today, there is a dire need for ever-more phone numbers. Not only do most people have both land lines and cell phones (mobiles), but other devices such as alarm systems, credit card machines and, ATMs all require dedicated phone lines. Many areas are finding it necessary to create new area codes to increase the supply of available phone numbers. This causes considerable disruption and expense since everything that has a phone number written or recorded has to be changed.
With today's data processing capabilities, we need only one exchange and not two. If the exchange was eliminated and the seven digits following the area code were condidered as the body of the phone number, we would have ten million numbers available per area code instead of ten thousand per exchange. If the typical area code contains a hundred exchanges, this would give us ten times as many available phone numbers.
Countries outside North America often use city codes that can be eliminated in the same way. In fact, with ever more sophisticated equipment available, why not look forward to a day when even area codes are eliminated and the entire number is the body. This is the way the IP addresses on the internet operate, why can't telephones operate in the same way?
The first three digits, the 123, is the area code. This tells the telephone network the general area where the call is going. For example, in the far western part of New York State, the area code is 716. The next three digits, the 456, is known as the exchange. This is a kind of sub-area code. After the area code has been established, this tells the telephone network where in the area code the call is going. The final four digits, the 7890, is the body of the phone number.
Each exchange can have up to ten thousand numbers, from 0000 to 9999. Area codes typically contain about a hundred exchanges. For example the most common exchanges in the City of Niagara Falls, NY seem to be 283, 284 and, 297.
My point here is that the use of exchanges is inefficiency in the extreme. These exchanges are a relic of the days of mechanical switching systems to connect two callers. The early days of telephone featured operators manually plugging in a cord to join two callers. Then the mechanical relay switching systems became available in which a call was routed first to the destination area code and then to an exchange within the area code and finally to the number within the exchange. The area code is in itself an exchange.
In these days of computerized equipment with vast bandwidth and data storage capacity, why do we still use a system that was designed for the mechanical equipment available fifty years ago? There is probably no better example of how we can have space age technology operating under a virtually stone age system of organization.
Today, there is a dire need for ever-more phone numbers. Not only do most people have both land lines and cell phones (mobiles), but other devices such as alarm systems, credit card machines and, ATMs all require dedicated phone lines. Many areas are finding it necessary to create new area codes to increase the supply of available phone numbers. This causes considerable disruption and expense since everything that has a phone number written or recorded has to be changed.
With today's data processing capabilities, we need only one exchange and not two. If the exchange was eliminated and the seven digits following the area code were condidered as the body of the phone number, we would have ten million numbers available per area code instead of ten thousand per exchange. If the typical area code contains a hundred exchanges, this would give us ten times as many available phone numbers.
Countries outside North America often use city codes that can be eliminated in the same way. In fact, with ever more sophisticated equipment available, why not look forward to a day when even area codes are eliminated and the entire number is the body. This is the way the IP addresses on the internet operate, why can't telephones operate in the same way?
Cell Phone Projection
As far as I am concerned, the next big steps in the communications revolution are clear and obvious and relatively simple and involve cell phones rather than computers. One such step is what I will call "Cell Phone Projection".
Cell phones (mobiles) are sweeping the world even faster than computers. Why cannot cell phones simply replace most computers? The answer is the size of the display. Cell phones do not have the capacity to visually display a lot of information at once like a computer desktop, not even the two-piece flip phones that have a larger display. If cell phones could only display as much information as a computer desktop, it would certainly be a revolution in communications. Everyone could carry a full-capacity internet-connected computer with them everywhere, as well as a phone.
Why not use projection? A cell phone could have two display screens, one for direct reading as phones have now, and the other for projection. The projection screen would use liquid crystal technology to embody a working computer desktop on a lens.
To use the phone as a full-fledged computer, the user would hold the phone about 18 inches from a wall or other flat surface with the projection screen pointing toward the wall. It would project a computer desktop on the wall like a slide projector and a small laser pointer would act as a mouse. A virtual keyboard at the bottom of the screen would make typing easy. Naturally, the projection screen would require a stronger source of light than the direct-read display screen as well as reliable batteries.
That is not all. We can make "The Navigation Revolution" a part of the communications revolution. What if all cell phones were equipped with GPS receivers and maps.google.com was furnished with GPS coordinates. Anyone anywhere in the world could quickly and easily look at maps and satellite imagery of their location.
Cell phones (mobiles) are sweeping the world even faster than computers. Why cannot cell phones simply replace most computers? The answer is the size of the display. Cell phones do not have the capacity to visually display a lot of information at once like a computer desktop, not even the two-piece flip phones that have a larger display. If cell phones could only display as much information as a computer desktop, it would certainly be a revolution in communications. Everyone could carry a full-capacity internet-connected computer with them everywhere, as well as a phone.
Why not use projection? A cell phone could have two display screens, one for direct reading as phones have now, and the other for projection. The projection screen would use liquid crystal technology to embody a working computer desktop on a lens.
To use the phone as a full-fledged computer, the user would hold the phone about 18 inches from a wall or other flat surface with the projection screen pointing toward the wall. It would project a computer desktop on the wall like a slide projector and a small laser pointer would act as a mouse. A virtual keyboard at the bottom of the screen would make typing easy. Naturally, the projection screen would require a stronger source of light than the direct-read display screen as well as reliable batteries.
That is not all. We can make "The Navigation Revolution" a part of the communications revolution. What if all cell phones were equipped with GPS receivers and maps.google.com was furnished with GPS coordinates. Anyone anywhere in the world could quickly and easily look at maps and satellite imagery of their location.
International Translation Code
I have an idea that would really simplify travel to foreign countries. I am surprised that, as far as I know, no one else has thought of it. Today, most travellers have cell phones. Why not use this as a translation tool?
Many languages have well over 100,000 words. But only about 10,000 words are in common everyday use. Let's give each of these words a four-digit code from 0000 to 9999.
Numbers are the same throughout the world. Suppose you were in a building in a foreign country and you saw an incomprehensible word on a sign with a four-digit code above it such as 6347. You would simply click your cell phone (mobile) into translation mode and punch in the code. Your phone would display that the word translated as "elevators".
Suppose you were in the building and looking for the elevators. You would just click the phone into reverse translation mode and enter in the word "elevators" in your own language. Your phone would display that the International Translation Code for elevators was 6347. You would then proceed to look for that number on a sign in the building.
If you were trying to ask for information or directions and the person you were asking had no idea what you were saying, you would simply display the code for whatever it was you were asking for. The foreign person would look up the code and give you what you were requesting or point you in the right direction.
It is not important that the phone have coverage in the foreign country because the data would be stored in a memory chip in the phone. Each user would naturally have his phone programmed in his or her own language. Code numbers could be easily imprinted onto labels or placed on signs all over the world using the stick-on letters used on mail boxes.
Of course, such a simple system could not provide a genuine translation of a foreign language since grammar and syntax vary from one language to another. But for everyday travel, it would be very helpful.
Many languages have well over 100,000 words. But only about 10,000 words are in common everyday use. Let's give each of these words a four-digit code from 0000 to 9999.
Numbers are the same throughout the world. Suppose you were in a building in a foreign country and you saw an incomprehensible word on a sign with a four-digit code above it such as 6347. You would simply click your cell phone (mobile) into translation mode and punch in the code. Your phone would display that the word translated as "elevators".
Suppose you were in the building and looking for the elevators. You would just click the phone into reverse translation mode and enter in the word "elevators" in your own language. Your phone would display that the International Translation Code for elevators was 6347. You would then proceed to look for that number on a sign in the building.
If you were trying to ask for information or directions and the person you were asking had no idea what you were saying, you would simply display the code for whatever it was you were asking for. The foreign person would look up the code and give you what you were requesting or point you in the right direction.
It is not important that the phone have coverage in the foreign country because the data would be stored in a memory chip in the phone. Each user would naturally have his phone programmed in his or her own language. Code numbers could be easily imprinted onto labels or placed on signs all over the world using the stick-on letters used on mail boxes.
Of course, such a simple system could not provide a genuine translation of a foreign language since grammar and syntax vary from one language to another. But for everyday travel, it would be very helpful.
The Symbol Of Globalization
Do you remember that simple childhood game called tic-tac-toe? I believe that the square in which that game was played, a square divided into thirds on each side so that there are nine smaller squares within the large square, is the perfect symbol for globalization. Here is why.
I am convinced that cell phones (mobiles), which are taking over the world even faster than computers, are a potentially very powerful tool for translating one language into another.
In my book "The Patterns of New Ideas", I introduced what I called "The Base Language" in which everything that people say to each other is broken down into numbers according to the parts of speech and this provides a simple way of translating written communications, using computers, from one language into another.
Now let's move on to a more thorough system. The tic-tac-toe square resembles the keypad of a cell phone (mobile). Suppose we placed the tic-tac-toe square on a sign and filled each of the smaller squares with a number from 1 to 9. That would give us 362,880 possible combinations of squares. That is more than enough combinations to give each word in a language as well as basic sentences their own combinations. These combinations would be agreed upon throughout the world.
When a traveller who did not understand the local language saw such a "Global Square" sign as we will call it, the traveller would simply click his phone onto translation mode, enter in the combination of numbered squares in order and, press zero for enter. The phone would display the meaning of the word or sentence in the traveller's own language. The translation of each combination of squares would be stored on a chip in the phone. The combinations, each of which would be assigned to a word or basic sentence, could also be expressed as numbers according to which ones of the nine squares were darkened, such as 134682957.
This square, divided into nine smaller squares, would also be the perfect symbol for globalization with the various people of the world not only being brought into closer contact but also gaining more understanding of each other. In my posting below "Parallel Revolutions", I described how the communications revolution is far outpacing the transportation revolution because it is still stuck on dependence on gasoline and other fossil fuels.
However, the communications revolution has a roadblock of it's own. No matter how easy we can make global communications, the fact remains that the people of the world speak about a thousand different languages.
The next step in the transportation revolution should definitely be to get away from reliance on gasoline and other fossil fuels. The next step in the communications revolution should be the utilizing the power of cell phones and computers to get beyond the language barrier.
I am convinced that cell phones (mobiles), which are taking over the world even faster than computers, are a potentially very powerful tool for translating one language into another.
In my book "The Patterns of New Ideas", I introduced what I called "The Base Language" in which everything that people say to each other is broken down into numbers according to the parts of speech and this provides a simple way of translating written communications, using computers, from one language into another.
Now let's move on to a more thorough system. The tic-tac-toe square resembles the keypad of a cell phone (mobile). Suppose we placed the tic-tac-toe square on a sign and filled each of the smaller squares with a number from 1 to 9. That would give us 362,880 possible combinations of squares. That is more than enough combinations to give each word in a language as well as basic sentences their own combinations. These combinations would be agreed upon throughout the world.
When a traveller who did not understand the local language saw such a "Global Square" sign as we will call it, the traveller would simply click his phone onto translation mode, enter in the combination of numbered squares in order and, press zero for enter. The phone would display the meaning of the word or sentence in the traveller's own language. The translation of each combination of squares would be stored on a chip in the phone. The combinations, each of which would be assigned to a word or basic sentence, could also be expressed as numbers according to which ones of the nine squares were darkened, such as 134682957.
This square, divided into nine smaller squares, would also be the perfect symbol for globalization with the various people of the world not only being brought into closer contact but also gaining more understanding of each other. In my posting below "Parallel Revolutions", I described how the communications revolution is far outpacing the transportation revolution because it is still stuck on dependence on gasoline and other fossil fuels.
However, the communications revolution has a roadblock of it's own. No matter how easy we can make global communications, the fact remains that the people of the world speak about a thousand different languages.
The next step in the transportation revolution should definitely be to get away from reliance on gasoline and other fossil fuels. The next step in the communications revolution should be the utilizing the power of cell phones and computers to get beyond the language barrier.
Underwater Communications
There is one thing that I simply do not understand about communications with submarines and other underwater devices. Such communication has always been difficult. The radio waves that are used for communication on land do not travel well through water. For details look up "Submarine Communications" at http://www.wikipedia.com/
Efforts have been made to develop blue lasers for underwater communication because blue light is the color (colour) that can travel furthest through water without being absorbed. But this method cannot really be considered as satisfactory because the exact position of the sub must be known. Sound waves travel well through water and this has made sonar the undersea version of radar. The disadavantage of sonar, of course, is that in wartime the sub risks giving away it's position by using it.
I simply do not understand why we do not use the same principle that operates a microwave oven for underwater communications. This just seems so obvious to me. A molecule of water consists of an oxygen atom and two hydrogen atoms. However, this arrangement is not symmetrical and means that there will be a different charge each side of the molecule. In other words, the water molecule is polar.
Yet, a quantity of water is electrically neutral and manifests no electric charge. This can only mean that the water molecules align themselves negative to positive and this is why water molecules cling together in drops.
I believe that water should be our ally, instead of our enemy, in underwater communications. The trouble is that we have approached such communications with a terrestrial mindset, trying to adapt land-based methods of communication to the sea. Now, let's try a different approach.
Polar water molecules line up negative to positive into an equilibrium which the presence of an electric charge will disturb. This means that an alternating electric charge will create electrical waves in the water which can be used for communication. A microwave oven is a development rooted in the so-called "travelling wave tube" used in early radar sets.
In the microwave oven, alternating negative and positive charges are produced by electrons moving along a circular track. This causes water molecules in the food to flip over many times a second, producing the heat that cooks the food from within. The microwave cooking process is based on the electrical polarity of water molecules.
It should be simple to use this microwave principle for communications or range-finding under the water. All we have to do is broadcast the electric waves in the water outward instead of inward onto food.
If a portion of the hull of a submarine or an antenna were given alternating electric charges, waves in the position of the polar water molecules would emanate outward. No heat would be produced because each molecule would shift only once in each wave. The energy would thus go not into heat but into waves moving outward from the source.
Keep in mind that these are not electromagnetic waves that we would produce. They would be waves in the alignment of water molecules based on their electric polarity. This is something that we cannot relate to on land because air is not polar as water is.
Neither is it an electric current through the water. The transmitting antenna would not have electrons travelling from top to bottom and back as would a radio antenna. Rather, it would be an object which would bounce from having a negative charge to positive and back again. This would change the directional alignment of nearby water molecules, which would change the alignment of the molecules further out, and so on, thus creating a wave that can readily be used for communications.
Efforts have been made to develop blue lasers for underwater communication because blue light is the color (colour) that can travel furthest through water without being absorbed. But this method cannot really be considered as satisfactory because the exact position of the sub must be known. Sound waves travel well through water and this has made sonar the undersea version of radar. The disadavantage of sonar, of course, is that in wartime the sub risks giving away it's position by using it.
I simply do not understand why we do not use the same principle that operates a microwave oven for underwater communications. This just seems so obvious to me. A molecule of water consists of an oxygen atom and two hydrogen atoms. However, this arrangement is not symmetrical and means that there will be a different charge each side of the molecule. In other words, the water molecule is polar.
Yet, a quantity of water is electrically neutral and manifests no electric charge. This can only mean that the water molecules align themselves negative to positive and this is why water molecules cling together in drops.
I believe that water should be our ally, instead of our enemy, in underwater communications. The trouble is that we have approached such communications with a terrestrial mindset, trying to adapt land-based methods of communication to the sea. Now, let's try a different approach.
Polar water molecules line up negative to positive into an equilibrium which the presence of an electric charge will disturb. This means that an alternating electric charge will create electrical waves in the water which can be used for communication. A microwave oven is a development rooted in the so-called "travelling wave tube" used in early radar sets.
In the microwave oven, alternating negative and positive charges are produced by electrons moving along a circular track. This causes water molecules in the food to flip over many times a second, producing the heat that cooks the food from within. The microwave cooking process is based on the electrical polarity of water molecules.
It should be simple to use this microwave principle for communications or range-finding under the water. All we have to do is broadcast the electric waves in the water outward instead of inward onto food.
If a portion of the hull of a submarine or an antenna were given alternating electric charges, waves in the position of the polar water molecules would emanate outward. No heat would be produced because each molecule would shift only once in each wave. The energy would thus go not into heat but into waves moving outward from the source.
Keep in mind that these are not electromagnetic waves that we would produce. They would be waves in the alignment of water molecules based on their electric polarity. This is something that we cannot relate to on land because air is not polar as water is.
Neither is it an electric current through the water. The transmitting antenna would not have electrons travelling from top to bottom and back as would a radio antenna. Rather, it would be an object which would bounce from having a negative charge to positive and back again. This would change the directional alignment of nearby water molecules, which would change the alignment of the molecules further out, and so on, thus creating a wave that can readily be used for communications.
The Alphabet As Symbols
We have many terms that we use to describe the shapes or patterns in the world around us. Of course, there are the basic geometric shapes like circle, square and, triangle. Some things may be described as existing in layers or loops or, concentric circles. Others as egg-shaped or elliptical.
Commonly used terms to describe layout plans or patterns are star or tree. A highway interchange is referred to as a cloverleaf because of the similarity in shape. We also make use of the letters of the alphabet and the number 8 to describe the shapes of things. You may have heard terms such as; V-formation, S-curve. A-frame, I-beam or, Y-intersection. The letters D, C, L and, T are also used often to describe shapes or forms that would otherwise be more difficult to describe.
Our alphabet originated from earlier hieroglyphics. The first people to use an alphabet was probably the Phoenicians, an ancient nation around what is now Lebanon that was best-known for it's seafaring.
I got to thinking; now that we are in modern times, wouldn't it be a good idea to get the most use out of the alphabet by having all of the letters resembling some pattern or form that is in everyday use? Being able to simply refer to a letter to describe a common shape would make our language more efficient. The letters of the alphabet other than the ones I have listed above seem to have little or no form-symbol value.
It would make it easier to communicate efficiently if there was a letter like an upside down T, since this is the form of antennae, towers and, signposts supported on a horizontal base. A canyon-shape would be useful, like a V but three sides of a square. The division symbol used in arithmetic would also be useful, a line with a dot on each side of the line.
Another common form seen in roads with ramps is the joining of two perpendicular lines but with a partial loop instead of a right angle. Yet another practical modern symbol would be a circle like an O but with a dot in the middle, this form appears often with coaxial cables and also represents the radius a given distance from a central point. A crescent within a rectangle is a form frequently seen on street maps of cities and would make a useful letter.
A straight line meeting a curving line would be similarly useful. A straight line with two or three perpendicular lines meeting it would represent a common form in both street patterns and electrical wiring. Then, when we wanted to describe any such shapes or forms we could simply refer to the letter.
Commonly used terms to describe layout plans or patterns are star or tree. A highway interchange is referred to as a cloverleaf because of the similarity in shape. We also make use of the letters of the alphabet and the number 8 to describe the shapes of things. You may have heard terms such as; V-formation, S-curve. A-frame, I-beam or, Y-intersection. The letters D, C, L and, T are also used often to describe shapes or forms that would otherwise be more difficult to describe.
Our alphabet originated from earlier hieroglyphics. The first people to use an alphabet was probably the Phoenicians, an ancient nation around what is now Lebanon that was best-known for it's seafaring.
I got to thinking; now that we are in modern times, wouldn't it be a good idea to get the most use out of the alphabet by having all of the letters resembling some pattern or form that is in everyday use? Being able to simply refer to a letter to describe a common shape would make our language more efficient. The letters of the alphabet other than the ones I have listed above seem to have little or no form-symbol value.
It would make it easier to communicate efficiently if there was a letter like an upside down T, since this is the form of antennae, towers and, signposts supported on a horizontal base. A canyon-shape would be useful, like a V but three sides of a square. The division symbol used in arithmetic would also be useful, a line with a dot on each side of the line.
Another common form seen in roads with ramps is the joining of two perpendicular lines but with a partial loop instead of a right angle. Yet another practical modern symbol would be a circle like an O but with a dot in the middle, this form appears often with coaxial cables and also represents the radius a given distance from a central point. A crescent within a rectangle is a form frequently seen on street maps of cities and would make a useful letter.
A straight line meeting a curving line would be similarly useful. A straight line with two or three perpendicular lines meeting it would represent a common form in both street patterns and electrical wiring. Then, when we wanted to describe any such shapes or forms we could simply refer to the letter.
Information From Heat
I believe that we could be getting a lot more from heat, more properly infrared radiation, than just warmth. Infrared is the section of the electromagnetic spectrum just to the longer wavelength/lower frequency side of visible light. If we took red light and made it's wavelength longer until our eyes no longer saw it, we would have infrared.
We can feel infrared as heat but cannot see it and this affects our attitudes toward it. Infrared radiation is one of the three ways that heat can travel; conduction, convection and radiation. Infrared seems to us to be so monolithic but actually has many different frequencies just as visible light does. We have a vast amount to gain with some new thinking about this section of the spectrum.
Infrared can be easily sensed to measure the temperature of an object. Astronomers have long used data from infrared radiation to study the universe. There are night-vision goggles that reveal enemy soldiers hiding in vegetation by the difference between their body temperature and that of the background.
I would like to point out that even if the temperature of two objects is the same, they will radiate different frequencies of infrared, or IR, according to their atomic or molecular composition and structure. Each atom and molecule has a natural vibrational rythm like a pendulum. Objects typically absorb IR and re-radiate it at different frequencies. The amount and frequency of the re-radiated IR can tell us a lot about the inside of an object.
This concept of gaining information from heat that would otherwise be inaccessible is made possible by the fact that different objects absorb and re-radiate IR at different frequencies according to their internal composition. Smaller molecules will vibrate faster with heat energy and will thus radiate IR at a higher frequency, although with less energy, than larger molecules.
There are tremendous advantages to this new way of thinking about IR in order to get information about the composition of an object from the heat it radiates. Visible light cannot tell us what is inside an object, IR can. All objects above a temperature of absolute zero radiate IR.
The technique of using X-rays to see through an object is similar in concept but use of IR does not require the object to be brought into a laboratory. IR can be a very effective form of passive X-rays. Everything about the structure of an object could be discerned by it's IR signature if we could prefect this technique.
We consider heat, or infrared radiation, as a very simple entity that we tend to measure only in quantity. We can feel heat but cannot see it and usually only recognize it when the temperature is above what we consider as room temperature. But I recognize heat as far more complex than that and the idea of getting information from heat has a vast amount of unrealized potential.
The everyday radiation of heat around the environment is far more complex than is generally supposed. Keep in mind that heat moves in three ways. Conduction, convection and, radiation. The conduction and convection of heat is relatively simple. It is the radiation of heat by infrared electromagnetic radiation that requires much more understanding. The simplicity of conduction and convection movement of heat hides the complexity of radiant heat.
As we know, objects tend to absorb heat and then radiate it at a different frequency. Every atom and molecule radiates heat at a it's own frequency of infrared. This is the section of the electromagnetic spectrum longer than radio waves but shorter in wavelength than visible light.
Radiant heat is absorbed or passes through a material it encounters according to how the frequency of the infrared radiation matches the resonant frequency of the atoms. More infrared radiation is reflected or passes through a material whose atoms do not vibrate at or near the frequency of the IR coming in. This operates much like the tuned circuit in a radio receiver but in the case of radiant heat, the atoms themselves are the receivers.
Suppose we heat a lump of iron. Now if we take two lumps of identical size and shape, one of iron and one of another metal and put both at exactly the same distance from the hot lump of iron. The cold lump of iron will absorb more radiant heat from the hot iron than will the cold lump of the other metal. This is because the atoms of the hot iron will radiate a frequency of infrared that the cold lump of iron is more naturally tuned into than the other lump of metal.
Water is slow to absorb radiant heat because it is not at the resonant frequency of the water molecules. Water does however, hold onto heat once it has it because it's molecules are so mobile that it transmits heat to neighboring molecules by conduction instead of radiating it away. This gives water it's tremendous heat capacity. Land and rock absorb heat from the sun faster than water because the natural vibrational period of it's molecules are closer to the predominant frequencies of IR coming in.
What all of this means to us is that each object must have a unique heat spectrum signature. We could develop sensitive receivers that are tuned into the IR frequency radiated out by gold atoms. For that matter, every person and animal has different DNA atoms that radiate out slightly different frequencies of IR. Every different atom and molecule radiates heat at a different frequency regardless of the frequencies it absorbed to gain the heat. I believe that a heat spectrum scan of the body can tell us far more than X-rays can.
This is an idea with mind-boggling potential. We recognize that radio waves consist of many different frequencies and light of many shades and colors (colours), why don't we see heat in the same way?
More Creative Mathematics
I believe that in the teaching of mathematics, there is far too much use of pre-fabricated problems and not enough on creativity in setting up a problem. Figuring something out using numbers or geometry involves setting up the problem and then solving it. Students, in my opinion, learn much more of how to solve a problem that has already been set up for them than how to observe the real world around them and apply the math to it by setting up the problem.
In figuring out the things that I have presented on these series of blogs, I find that real-life mathematical problems are often relatively simple but require considerable creativity to set up effectively. I wish the teaching of math was more like art, such as painting or drawing. A proficient mathematician is not one who can solve any pre-prepared problem that is presented to him or her but one who can observe the real world and effectively apply mathematics to it by setting up the problem to be solved.
Calculators make the setting up of the problem even more important relative to the solving of it. Calculators can solve problems quickly but only after it has been correctly set up. No matter how proficient one may be with a calculator, it is useless unless it's user can observe the real world and select the mathematics to apply to it. There may be several ways that a real-world calculation can be accomplished.
I am sure that at least half of the mathematics one learns in school will never be used in real life, no matter what occupation the student ends up in. Remember that imaginary number, i, supposedly the square root of -1, which cannot actually exist, from algebra class? I am still mystified as to what that is supposed to be actually used for. The same for factoring polynomials, although it is a useful mental exercise.
In the calculations that I have done for these series of blogs, the spatial branches of mathematics such as geometry and trigonometry have been by far the most useful. I have rarely used algebra and never calculus. I do find fractions to be very valuable since this is often how the real world operates.
In making real-world decisions, it is often a "sense" of numbers or geometry that is required, rather than an actual calculation. I was never brilliant in math class but later found that I had a knack for creating my own mathematics and applying it to the real world and universe.
One day, I was figuring something out and wanted a quick way to add up all the numbers up to a certain number, 1+2+3... There was the factorial function on a calculator (!) to multiply all the numbers up to a given number but I had never heard of a similar function for addition. However, after a few minutes of trial, I noticed that if you divide a number in half, add one half (.5) to it and multiply it by the original number, it will provide the answer. This means that the numbers from 1 to 10 should add up to 55 and they do.
Another time, just as a mental exercise, I wondered what the odds would be of winning a game in which the odds of winning were one in three and we played the game three times. I knew that the answer would have to be more than one in three but less than 100%. I had never heard of this in class but I realized that the only practical answer would be 1/3 + (2/3 x 1/3) + (1/3 x 1/3). In other words, 6/9 or 2/3.
Mathematics is the underlying patterns of how reality works. The objective of math class should be more like art class, to see how the system operates and then create your own. There should be more emphasis on setting up problems in the real world as opposed to solving pre-prepared problems.
In figuring out the things that I have presented on these series of blogs, I find that real-life mathematical problems are often relatively simple but require considerable creativity to set up effectively. I wish the teaching of math was more like art, such as painting or drawing. A proficient mathematician is not one who can solve any pre-prepared problem that is presented to him or her but one who can observe the real world and effectively apply mathematics to it by setting up the problem to be solved.
Calculators make the setting up of the problem even more important relative to the solving of it. Calculators can solve problems quickly but only after it has been correctly set up. No matter how proficient one may be with a calculator, it is useless unless it's user can observe the real world and select the mathematics to apply to it. There may be several ways that a real-world calculation can be accomplished.
I am sure that at least half of the mathematics one learns in school will never be used in real life, no matter what occupation the student ends up in. Remember that imaginary number, i, supposedly the square root of -1, which cannot actually exist, from algebra class? I am still mystified as to what that is supposed to be actually used for. The same for factoring polynomials, although it is a useful mental exercise.
In the calculations that I have done for these series of blogs, the spatial branches of mathematics such as geometry and trigonometry have been by far the most useful. I have rarely used algebra and never calculus. I do find fractions to be very valuable since this is often how the real world operates.
In making real-world decisions, it is often a "sense" of numbers or geometry that is required, rather than an actual calculation. I was never brilliant in math class but later found that I had a knack for creating my own mathematics and applying it to the real world and universe.
One day, I was figuring something out and wanted a quick way to add up all the numbers up to a certain number, 1+2+3... There was the factorial function on a calculator (!) to multiply all the numbers up to a given number but I had never heard of a similar function for addition. However, after a few minutes of trial, I noticed that if you divide a number in half, add one half (.5) to it and multiply it by the original number, it will provide the answer. This means that the numbers from 1 to 10 should add up to 55 and they do.
Another time, just as a mental exercise, I wondered what the odds would be of winning a game in which the odds of winning were one in three and we played the game three times. I knew that the answer would have to be more than one in three but less than 100%. I had never heard of this in class but I realized that the only practical answer would be 1/3 + (2/3 x 1/3) + (1/3 x 1/3). In other words, 6/9 or 2/3.
Mathematics is the underlying patterns of how reality works. The objective of math class should be more like art class, to see how the system operates and then create your own. There should be more emphasis on setting up problems in the real world as opposed to solving pre-prepared problems.
The Power Of Multiplication
Most people remember the classic example used in school to illustrate the surprising power of successive multiplication. Suppose a laborer (labourer) did some work for someone for a month. The employer asks the laborer how much he would like to be paid. The laborer answers " a penny on the first day doubled every day so that I would be paid two cents on the second day and four cents on the third day and so on and I must be scheduled for at least thirty days.
"Wow, what a bargain this is" thought the employer as he agreed to pay the laborer as he requested. However, the employer did not stop to consider the power of successive multiplication and had to pay the labourer over five million dollars (or pounds, euros, rupees, etc,) for thirty days work.
One day while messing around with a scientific calculator I discovered what I consider as a more space age example of the power of successive multiplication. Consider a roll of pennies that is stored in cash registers. There are fifty pennies in a roll. We could vary the position of the pennies in the roll and also flip each penny so that the heads side is facing one way and tails the other or vice versa.
I calculated that in this roll of pennies, there are more possible combinations than there are atoms in the entire universe.
The number of atoms in the universe is belived to be about 10 (79) or a 1 followed by 79 zeroes. If we take the factorial of 50 (50!), which is 50 x 49 x 48... back to 1, it will give us the number of possible combinations in the sequence of pennies in the roll. I come up with 3.04140932 x 10 (64).
Remember that we can also flip each penny in the sequence for heads or tails to face in a given direction. So, we would take 2 multiplied by itself fifty times and multiply it by the first number. We get 1.125899907 x 10 (15). This gives us a total of 3.42432247 x 10 (79). In other words, within our roll of pennies, there are three possible combinations for each and every atom in the entire universe.
"Wow, what a bargain this is" thought the employer as he agreed to pay the laborer as he requested. However, the employer did not stop to consider the power of successive multiplication and had to pay the labourer over five million dollars (or pounds, euros, rupees, etc,) for thirty days work.
One day while messing around with a scientific calculator I discovered what I consider as a more space age example of the power of successive multiplication. Consider a roll of pennies that is stored in cash registers. There are fifty pennies in a roll. We could vary the position of the pennies in the roll and also flip each penny so that the heads side is facing one way and tails the other or vice versa.
I calculated that in this roll of pennies, there are more possible combinations than there are atoms in the entire universe.
The number of atoms in the universe is belived to be about 10 (79) or a 1 followed by 79 zeroes. If we take the factorial of 50 (50!), which is 50 x 49 x 48... back to 1, it will give us the number of possible combinations in the sequence of pennies in the roll. I come up with 3.04140932 x 10 (64).
Remember that we can also flip each penny in the sequence for heads or tails to face in a given direction. So, we would take 2 multiplied by itself fifty times and multiply it by the first number. We get 1.125899907 x 10 (15). This gives us a total of 3.42432247 x 10 (79). In other words, within our roll of pennies, there are three possible combinations for each and every atom in the entire universe.
The Geographic Formula
In my book "The Patterns of New Ideas", I introduced the idea of a "geographical formula" that would make it easy to calculate the distance between any two points on earth given the latitude and longitude of each point. This function would be a very useful addition to a scientific calculator. Suppose you wish to measure the distance between two places on a map. All you have to do is measure the distance with a ruler and then multiply according to the scale of the map. For example, a map may have a scale of one inch=45 miles. Or scale-1/10,000.
But what do you do if you have two maps and wish to measure the distance between a place on each map? There are a number of ways to do it. I believe that I have found the simplest and easiest way.
First, define what we will call C. C is the circumference if the earth. If you want your answer to be in kilometers, then you would start by defining C in kilometers (or kilometres, depending on which country you live in.) We will define C in miles.
The circumference of the earth is about 25,000 miles. You can easily apply this formula to any celestial body, such as the moon, by simply defining the circumference, which is pi (3.1415926) times the diameter of the body. You can use whichever units of length you wish.
Latitude is the degrees north or south of the equator. The total circumference of the earth or any other sphere is 360 degrees. The equator is 0 degrees. The north pole is 90 degrees north. The south pole is 90 degrees south.
Longitude is the degrees east or west of the Prime Meridian. This line passes from the north to the south pole through Greenwich, a suburb of London, and is defined as 0 degrees. On the other side of the world, the International Date Line, 180 degrees, corresponds to the Prime Meridian.
This is also from where we measure time, you may have heard the phrase "Greenwich Mean Time" or "GMT". Greenwich was the site of the astronomical observatory where the latitude-longitude system was set up. Contrary to what someone might think, Big Ben is not on, nor has anything to do with the Prime Meridian.
Let's measure the distance from where I live, Niagara Falls, NY to the place I was born, Lydbrook, Gloucestershire, England. The location of Niagara Falls on the earth's surface is 43 degrees north, 79 degrees west. That is where it lies in relation to the equator and the Prime Meridian. Lydbrook is roughly 52 degrees north, 2 degrees west.
First, find the distance between the latitudes of the two places. This is easier than finding the distance between two longitude lines because the distance between latitude lines is the same anywhere in the world while longitude lines are closer near the poles and further near the equator. The latitudes of the two places are 52-43 = 9 degrees apart. This means a distance of 25,000 miles/360 x 9 = 625 miles.
Second, find the distance between the longitudes of the two places along the lower of the two latitudes (43 degrees latitude). The two places are 79-2 = 77 degrees longitude apart. However, we must understand that the earth varies in circumference at different latitudes. At the equator, the earth is the full 25,000 miles in circumference. But right at a pole, it becomes actually zero in circumference.
Notice that a trigonometric function, the cosine, goes from 1 at zero degrees (the equator) to 0 at 90 degrees (the pole). So, that is the tool that we must use here to get the right answer. Let's divide the longitudinal distance, 77 degrees, by the circumference, 360 degrees. 77/360 = .214. Now we must multiply C by that number, 25,000 x .214 = 5,350 miles.
Since the circumference of the earth is less at 43 degrees than it is at the equator, we now multiply our answer by the cosine of 43 degrees, .731. 5350 x .731 = 3,911 miles. This is the distance between the longitudes of the two places at the lower of the two latitudes.
Third, we find the distance between the two longitudes at the higher of the two latitudes. Take our 5350 miles (the distance that it would be if it was at the equator) and multiply it by the cosine of 52 degrees. This gives us 5,350 x .616 = 3,296 miles.
Fourth, find the average of the two lateral distances between the two lines of longitude. The average of 3,911 miles and 3,296 miles is 3,604 miles. You must find the lateral difference between the two lines of longitude at both latitudes and then find the average of the two. You cannot simply average the distance in degrees between the two lines of latitude and then calculate the distance or you will get a wrong answer.
Fifth, now construct a rectangle with this average, 3,604 miles and the 625 miles between the lines of latitude of the two places. Using the Pythagorean Theorem, C squared = A squared + B squared. We get our final answer. 3,606 x 3,606 = 13,003,236. 625 x 625 = 390,625. Add the two results and we get 13,393,861.
Sixth, Now, all we have to do is to find the square root of that number and we get our answer. Niagara Falls and Lydbrook are roughly 3,660 miles apart.
Seventh, we can now easily calculate the exact compass direction from one of the places to the other using simple trigonometry. Divide the short side of the rectangle, 625 miles, by the long side, 3,604 miles. We get .173 as an answer. Since the sine of an angle starts at zero at 0 degrees and goes to 1 at 90 degrees, if we could find out what angle .173 is a sine of, we would have our answer. The answer is 10 degrees.
From Niagara Falls, if we went in a direction of ten degrees northeast, directly east-west being 0 degrees and directly north-south being 90 degrees, we would get to Lydbrook after travelling 3,660 miles. Think what a plus this function would be if it was available on scientific calculators. However, the seven-step formula is simple enough for geography students or anyone who uses maps to memorize and use.
But what do you do if you have two maps and wish to measure the distance between a place on each map? There are a number of ways to do it. I believe that I have found the simplest and easiest way.
First, define what we will call C. C is the circumference if the earth. If you want your answer to be in kilometers, then you would start by defining C in kilometers (or kilometres, depending on which country you live in.) We will define C in miles.
The circumference of the earth is about 25,000 miles. You can easily apply this formula to any celestial body, such as the moon, by simply defining the circumference, which is pi (3.1415926) times the diameter of the body. You can use whichever units of length you wish.
Latitude is the degrees north or south of the equator. The total circumference of the earth or any other sphere is 360 degrees. The equator is 0 degrees. The north pole is 90 degrees north. The south pole is 90 degrees south.
Longitude is the degrees east or west of the Prime Meridian. This line passes from the north to the south pole through Greenwich, a suburb of London, and is defined as 0 degrees. On the other side of the world, the International Date Line, 180 degrees, corresponds to the Prime Meridian.
This is also from where we measure time, you may have heard the phrase "Greenwich Mean Time" or "GMT". Greenwich was the site of the astronomical observatory where the latitude-longitude system was set up. Contrary to what someone might think, Big Ben is not on, nor has anything to do with the Prime Meridian.
Let's measure the distance from where I live, Niagara Falls, NY to the place I was born, Lydbrook, Gloucestershire, England. The location of Niagara Falls on the earth's surface is 43 degrees north, 79 degrees west. That is where it lies in relation to the equator and the Prime Meridian. Lydbrook is roughly 52 degrees north, 2 degrees west.
First, find the distance between the latitudes of the two places. This is easier than finding the distance between two longitude lines because the distance between latitude lines is the same anywhere in the world while longitude lines are closer near the poles and further near the equator. The latitudes of the two places are 52-43 = 9 degrees apart. This means a distance of 25,000 miles/360 x 9 = 625 miles.
Second, find the distance between the longitudes of the two places along the lower of the two latitudes (43 degrees latitude). The two places are 79-2 = 77 degrees longitude apart. However, we must understand that the earth varies in circumference at different latitudes. At the equator, the earth is the full 25,000 miles in circumference. But right at a pole, it becomes actually zero in circumference.
Notice that a trigonometric function, the cosine, goes from 1 at zero degrees (the equator) to 0 at 90 degrees (the pole). So, that is the tool that we must use here to get the right answer. Let's divide the longitudinal distance, 77 degrees, by the circumference, 360 degrees. 77/360 = .214. Now we must multiply C by that number, 25,000 x .214 = 5,350 miles.
Since the circumference of the earth is less at 43 degrees than it is at the equator, we now multiply our answer by the cosine of 43 degrees, .731. 5350 x .731 = 3,911 miles. This is the distance between the longitudes of the two places at the lower of the two latitudes.
Third, we find the distance between the two longitudes at the higher of the two latitudes. Take our 5350 miles (the distance that it would be if it was at the equator) and multiply it by the cosine of 52 degrees. This gives us 5,350 x .616 = 3,296 miles.
Fourth, find the average of the two lateral distances between the two lines of longitude. The average of 3,911 miles and 3,296 miles is 3,604 miles. You must find the lateral difference between the two lines of longitude at both latitudes and then find the average of the two. You cannot simply average the distance in degrees between the two lines of latitude and then calculate the distance or you will get a wrong answer.
Fifth, now construct a rectangle with this average, 3,604 miles and the 625 miles between the lines of latitude of the two places. Using the Pythagorean Theorem, C squared = A squared + B squared. We get our final answer. 3,606 x 3,606 = 13,003,236. 625 x 625 = 390,625. Add the two results and we get 13,393,861.
Sixth, Now, all we have to do is to find the square root of that number and we get our answer. Niagara Falls and Lydbrook are roughly 3,660 miles apart.
Seventh, we can now easily calculate the exact compass direction from one of the places to the other using simple trigonometry. Divide the short side of the rectangle, 625 miles, by the long side, 3,604 miles. We get .173 as an answer. Since the sine of an angle starts at zero at 0 degrees and goes to 1 at 90 degrees, if we could find out what angle .173 is a sine of, we would have our answer. The answer is 10 degrees.
From Niagara Falls, if we went in a direction of ten degrees northeast, directly east-west being 0 degrees and directly north-south being 90 degrees, we would get to Lydbrook after travelling 3,660 miles. Think what a plus this function would be if it was available on scientific calculators. However, the seven-step formula is simple enough for geography students or anyone who uses maps to memorize and use.
A Very Useful Tool
There is a very simple measurement tool that I thought of that can quickly and easily accomplish tasks that are very cumbersome and time-consuming with existing methods. This tool can be very easily homemade and I believe that anyone involved in any kind of building, constructing or, surveying would find it invaluable. I have decided, for various reasons, not to pursue a patent for it any longer. So, I have decided to put it here in the public domain so that anyone can make their own and no one else can get a patent on it.
One day, I drove past some large oil storage tanks in Tonawanda, NY near the South Grand Island Bridges. Just as a mental exercise, I tried to dream up a way to quickly measure the circumference of such tanks or another large, circular object. I started thinking of measuring the curvature over a given linear distance with the idea that the less the curvature per linear distance, the larger the circumference.
But then another idea clicked into my head. What if someone got an ordinary magnetic compass and enlarged either the compass itself or it's mounting so that it was circular and of a known circumference, such as a yard or a meter? Suppose we then placed the edge of the compass against the side of a large oil tank and noted the directional reading given by the compass needle. Then we would note the point on the side of the compass that was in contact with the side of the tank. If we proceeded to rotate the compass over a complete circle and noted the change in the directional reading of the needle, we would have all the information needed to quickly and easily calculate the circumference of the oil tank.
If we placed the compass against the side of the oil tank and noted that the directional reading of the needle was 192 degrees and then rotated the compass a complete circle so that the point on it's edge that had originally contacted the side of the tank was back in the same place, all we would have to do would be to take the fraction of a complete circle that the needle changed during the rotation and multiply it by the circumference of the compass mounting and we would have the answer, the circumference of the tank.
For example, If the compass mounting was one yard in circumference and, upon completion of the rotation the needle had moved from 192 degrees to 196 degrees, the circumference of the tank would be 360/4 times one yard. In other words, 90 yards.
As I stopped to have dinner in a restaurant, my mind really began racing. I realized that I was onto something. I asked the waitress for something I could use for a sheet of paper and by the time I was done, I had filled a side of the paper with all manner of tasks that such a simple device could quickly and easily accomplish. I decided that the device would be called "The Compass Ruler".
I made one of my own by getting a Wal-Mart hiking compass, breaking off the casing and, gluing it onto a piece of plywood I had cut with a jig saw to a circumference of one meter. I knew enough about building and construction to know that such a tool was not in common use. However, I checked extensively to see if such a tool was in use anywhere and found no sign that it was. There was once such a thing as a surveyor's compass, that had fallen into disuse, but it was a compass mounted on a stand and was in no way used like my Compass Ruler would be. On my device, measurements would be taken by actually contacting the side of a structure.
The principle of the operation of the Compass Ruler is simple. Just as a plumb, a weight tied to a string, uses the earth's gravity as a fixed reference point for measurment of vertical angles, the Compass Ruler uses the earth's magnetic field as a fixed reference point for measurement of horizontal angles. The Compass Ruler obviously must be marked around the circumference edge in degrees, just as a protractor would be.
There is an even simpler version of the Compass Ruler. Simply take a square of wood, 1 x 4 for example, and glue a compass in the middle of it. For best results, be sure that it is indeed a square and that each cardinal direction faces toward the middle of one side of the wood. Suppose you have built a corner between two walls or fences and you want to be sure that it does indeed form a right angle. Simply hold one side of your Compass Ruler against one wall and note the directional reading of the needle. Then hold the same side against the other wall. You should get a change in the needle of ninety degrees. Simple.
This method is just as useful if the two walls do not actually contact each other, or for that matter do not even come near each other. This makes the old standby, the builder's square seem awkward and obsolete by comparison. Verifying a right angle by the 3-4-5 Pythagorean Theorem method is also awkward and time-consuming.
Suppose it is necessary to measure the angle beyween any two walls that do not actually intersect. With a builder's square it is impossible. With tape measures it is tedious, time-consuming and, prone to error. With a surveying crew, it is expensive. With my Compass Ruler, it is almost effortless.
What if you have built a long wall or fence and want to verify it's straightness? All you have to do is walk down the wall, taking periodic measurements with the Compass Ruler by placing it's edge against the wall. If the wall is indeed straight, you will get the same directional reading of the needle on every measurement. If it is not straight, by measuring the wall with the Compass Ruler at given intervals, you can tell by how much it curves.
This is also useful for a vast number of other such similar measurements. How would you verify that two parallel walls are truly parallel? Just take a reading on one wall with the Compass Ruler. Then, go to the other wall and put the same edge against that wall. If the walls are parallel, you will get a difference in the directional readings of 180 degrees.
Suppose you wished to set up a series of signs along a road and wished them to all have the same directional orientation. How would you do it? What if you were setting up a sign along the road and wanted it to be set at 45 degrees to the road to give maximum exposure. Or suppose you were building a wall or fence and wished it to run parallel (or perpendicular) to the road.
All of these tasks would be difficult, impossible or expensive with existing methods. With my Compass Ruler, all would be simple and easy. To measure the directional orientation of the road with the Compass Ruler, simply place the device on the road surface alongside a traffic line on the road.
Measurement of curvature is just as easy with the Compass Ruler. Just take readings against the curved structure at regular intervals. Curvature can be expressed as change in the directional orientation of the needle per given linear distance. Another advantage of either version of the Compass Ruler, either the circular or the simpler square version of the device, is that contact measurements, such as those described above, are not hampered if two structures to be measured and compared are not visible from each other or if there is an obstacle, like a row of bushes, between two structures.
Surveying is easy with the Compass Ruler. Suppose you want to get an accurate measurement of the distance to a certain remote point. First, you would either set up or pick out a remote visible reference point to use in the measurements. Then you would mark the local point from which you would take the measurement to the remote point. Then you would establish a measurement point a convenient distance away so that a line from the local point (Point A) to the measurement point (Point B) would form a right angle with a line from point A to the remote point (Point C).
Using a straight-edge, such as a perfectly straight 1 x 4 board, you would sight on the remote point C from the local point A looking straight down the straight-edge. You would use the Compass Ruler to note the directional orientation of the straight-edge as it points from Point A to Point C. You would then go to the measurement Point B that you have selected and take another sighting on the remote Point C from there.
All you would than have to do is take the difference in the angular reading of the two measurements. Using a scientific calculator, you would get the cotangent of the angular difference. You would then multiply the cotangent by the distance from Point A, the local point, to the measurement Point B. That would give you the distance from Point A to the remote Point C.
Obviously, for best results in surveying using the Compass Ruler, measurements must be taken carefully. The distance from Point A to Point B must be accurately measured. And, the same spot on the remote point must be sighted upon. The longer the carefully measured distance from Point A to Point B is in relation to the distance from Point A to the remote Point C is, the better the result will be. It should always be at least 10% of the distance.
It is not necessary to have a right angle between the two lines from points A to C and from A to B, but if not, the simplicity of a cotangent calculation will be lost and a graphical calculation will become necessary. If possible, the baseline for the measurement from Point A to Point B can make use of a pre-existant line, such as a road.
The straight-edge can be built onto the Compass Ruler if it is to be used for surveying. For even better results, the straight-edge can be fitted with a small telescope, a laser pointer, or, both. A vertically diagonal mirror can make it possible to see the compass on the Compass Ruler at the same time that the sighting is being done. For a finishing touch, the entire device can be set on a mounting.
To set up a marker, such as a traffic cone, at a given distance in a given direction from a starting point, use the reverse of this method. Pre-set a sighting from a Point B to that distance and have a rodman walk with the marker until he is in the sight. Then use hand signals or radio/phone communication to have the marker set up at the correct point.
Suppose you are out on the water in a boat and wish to measure how far you are from shore because you notice a shipwreck or some other object of interest under the water and wish to record the position. You would pick out two easily recognizable objects on shore such as trees or large rocks. The two objects should be in a line perpendicular to the line between you and one of the objects. Measure the angle between the two objects from where you are in the boat and record it.
Later, you would carefully measure the distance between the two objects using a tape measure or a map. Then you would take the cotangent of the angle measured from the boat and multiply it by that distance. Alternatively, you could simply take the directional readings of any two (or more) prominently visible, fixed position objects. The position on the water could then be charted using a map or satellite photo of the area.
Astronomers have long used this technique to measure the distance to stars, it is known as parallax. The carefully measured distance from Point A to Point B is referred to as the baseline. The same principle can be used with the Compass Ruler to map an entire area. Simply pick out visible objects such as trees, houses, etc. Measure the distances from a central point to the objects and then measure the angular distances between those objects from the central point. The map then can be easily made using a ruler and protractor. Of course, on complex maps, more than one central point can be used. If the terrain to be mapped is hilly, the logical place for the central points would obviously be on the high ground.
Aside from the contact measurement and surveying versions, there is yet version of the Compass Ruler, the drawing version. Simply fasten or glue a small compass to a straight-edge such as a ruler and it makes the protractor used in geometric drawings just as obsolete as the builder's square is in construction. To draw two lines at a certain angle to each other, simply draw one, noting the angle indicated on the compass dial. Then move the straight-edge so that the difference showing on the dial is now the desired angle and then draw the second line. To measure the angle between two existing lines, simply reverse this process.
If the drawing paper is securely taped down, aligned either east-west or north-south with the compass for best results, an entire geometric drawing can be made with unprecedented accuracy using the drawing version of the Compass Ruler. Parallel lines will have the same compass dial reading anywhere in the drawing. Perpendicular lines will differ 90 degrees in reading. Existing methods are far inferior to this. Of course, it would be simple to draw a map that was surveyed using the surveying version of the tool, just drawing the measurements on paper.
There are certainly many more everyday applications of this simple but extremely useful device. The device could also bring geometry and trigonometry classes to life. The lessons that now consist of drawing lines and circles on paper could occasionally be done as actual measurements in the gym or schoolyard.
Anyone can make their own of any version of the Compass Ruler, the circular or the simpler square version for contact measurements. Or the surveying or drawing versions with an attached or accompanying straight-edge. It can also be manufactured and sold although it will not be patentable now that I have put it in the public domain.
I will not earn any money on this but I just want to have contributed something to the world. We read of George Washington Carver and how he revealed many things that the humble peanut can be used for. I would like to do the same thing for the simple device known as the compass. Just as GPS systems are becoming ubiquitous and it seems to be of little use any more, we see that there is a whole world of tasks that it can accomplish most effectively. Any simple compass would have it's usefulness multiplied if it were encased with a straight side to perform some of the measurements listed above.
One day, I drove past some large oil storage tanks in Tonawanda, NY near the South Grand Island Bridges. Just as a mental exercise, I tried to dream up a way to quickly measure the circumference of such tanks or another large, circular object. I started thinking of measuring the curvature over a given linear distance with the idea that the less the curvature per linear distance, the larger the circumference.
But then another idea clicked into my head. What if someone got an ordinary magnetic compass and enlarged either the compass itself or it's mounting so that it was circular and of a known circumference, such as a yard or a meter? Suppose we then placed the edge of the compass against the side of a large oil tank and noted the directional reading given by the compass needle. Then we would note the point on the side of the compass that was in contact with the side of the tank. If we proceeded to rotate the compass over a complete circle and noted the change in the directional reading of the needle, we would have all the information needed to quickly and easily calculate the circumference of the oil tank.
If we placed the compass against the side of the oil tank and noted that the directional reading of the needle was 192 degrees and then rotated the compass a complete circle so that the point on it's edge that had originally contacted the side of the tank was back in the same place, all we would have to do would be to take the fraction of a complete circle that the needle changed during the rotation and multiply it by the circumference of the compass mounting and we would have the answer, the circumference of the tank.
For example, If the compass mounting was one yard in circumference and, upon completion of the rotation the needle had moved from 192 degrees to 196 degrees, the circumference of the tank would be 360/4 times one yard. In other words, 90 yards.
As I stopped to have dinner in a restaurant, my mind really began racing. I realized that I was onto something. I asked the waitress for something I could use for a sheet of paper and by the time I was done, I had filled a side of the paper with all manner of tasks that such a simple device could quickly and easily accomplish. I decided that the device would be called "The Compass Ruler".
I made one of my own by getting a Wal-Mart hiking compass, breaking off the casing and, gluing it onto a piece of plywood I had cut with a jig saw to a circumference of one meter. I knew enough about building and construction to know that such a tool was not in common use. However, I checked extensively to see if such a tool was in use anywhere and found no sign that it was. There was once such a thing as a surveyor's compass, that had fallen into disuse, but it was a compass mounted on a stand and was in no way used like my Compass Ruler would be. On my device, measurements would be taken by actually contacting the side of a structure.
The principle of the operation of the Compass Ruler is simple. Just as a plumb, a weight tied to a string, uses the earth's gravity as a fixed reference point for measurment of vertical angles, the Compass Ruler uses the earth's magnetic field as a fixed reference point for measurement of horizontal angles. The Compass Ruler obviously must be marked around the circumference edge in degrees, just as a protractor would be.
There is an even simpler version of the Compass Ruler. Simply take a square of wood, 1 x 4 for example, and glue a compass in the middle of it. For best results, be sure that it is indeed a square and that each cardinal direction faces toward the middle of one side of the wood. Suppose you have built a corner between two walls or fences and you want to be sure that it does indeed form a right angle. Simply hold one side of your Compass Ruler against one wall and note the directional reading of the needle. Then hold the same side against the other wall. You should get a change in the needle of ninety degrees. Simple.
This method is just as useful if the two walls do not actually contact each other, or for that matter do not even come near each other. This makes the old standby, the builder's square seem awkward and obsolete by comparison. Verifying a right angle by the 3-4-5 Pythagorean Theorem method is also awkward and time-consuming.
Suppose it is necessary to measure the angle beyween any two walls that do not actually intersect. With a builder's square it is impossible. With tape measures it is tedious, time-consuming and, prone to error. With a surveying crew, it is expensive. With my Compass Ruler, it is almost effortless.
What if you have built a long wall or fence and want to verify it's straightness? All you have to do is walk down the wall, taking periodic measurements with the Compass Ruler by placing it's edge against the wall. If the wall is indeed straight, you will get the same directional reading of the needle on every measurement. If it is not straight, by measuring the wall with the Compass Ruler at given intervals, you can tell by how much it curves.
This is also useful for a vast number of other such similar measurements. How would you verify that two parallel walls are truly parallel? Just take a reading on one wall with the Compass Ruler. Then, go to the other wall and put the same edge against that wall. If the walls are parallel, you will get a difference in the directional readings of 180 degrees.
Suppose you wished to set up a series of signs along a road and wished them to all have the same directional orientation. How would you do it? What if you were setting up a sign along the road and wanted it to be set at 45 degrees to the road to give maximum exposure. Or suppose you were building a wall or fence and wished it to run parallel (or perpendicular) to the road.
All of these tasks would be difficult, impossible or expensive with existing methods. With my Compass Ruler, all would be simple and easy. To measure the directional orientation of the road with the Compass Ruler, simply place the device on the road surface alongside a traffic line on the road.
Measurement of curvature is just as easy with the Compass Ruler. Just take readings against the curved structure at regular intervals. Curvature can be expressed as change in the directional orientation of the needle per given linear distance. Another advantage of either version of the Compass Ruler, either the circular or the simpler square version of the device, is that contact measurements, such as those described above, are not hampered if two structures to be measured and compared are not visible from each other or if there is an obstacle, like a row of bushes, between two structures.
Surveying is easy with the Compass Ruler. Suppose you want to get an accurate measurement of the distance to a certain remote point. First, you would either set up or pick out a remote visible reference point to use in the measurements. Then you would mark the local point from which you would take the measurement to the remote point. Then you would establish a measurement point a convenient distance away so that a line from the local point (Point A) to the measurement point (Point B) would form a right angle with a line from point A to the remote point (Point C).
Using a straight-edge, such as a perfectly straight 1 x 4 board, you would sight on the remote point C from the local point A looking straight down the straight-edge. You would use the Compass Ruler to note the directional orientation of the straight-edge as it points from Point A to Point C. You would then go to the measurement Point B that you have selected and take another sighting on the remote Point C from there.
All you would than have to do is take the difference in the angular reading of the two measurements. Using a scientific calculator, you would get the cotangent of the angular difference. You would then multiply the cotangent by the distance from Point A, the local point, to the measurement Point B. That would give you the distance from Point A to the remote Point C.
Obviously, for best results in surveying using the Compass Ruler, measurements must be taken carefully. The distance from Point A to Point B must be accurately measured. And, the same spot on the remote point must be sighted upon. The longer the carefully measured distance from Point A to Point B is in relation to the distance from Point A to the remote Point C is, the better the result will be. It should always be at least 10% of the distance.
It is not necessary to have a right angle between the two lines from points A to C and from A to B, but if not, the simplicity of a cotangent calculation will be lost and a graphical calculation will become necessary. If possible, the baseline for the measurement from Point A to Point B can make use of a pre-existant line, such as a road.
The straight-edge can be built onto the Compass Ruler if it is to be used for surveying. For even better results, the straight-edge can be fitted with a small telescope, a laser pointer, or, both. A vertically diagonal mirror can make it possible to see the compass on the Compass Ruler at the same time that the sighting is being done. For a finishing touch, the entire device can be set on a mounting.
To set up a marker, such as a traffic cone, at a given distance in a given direction from a starting point, use the reverse of this method. Pre-set a sighting from a Point B to that distance and have a rodman walk with the marker until he is in the sight. Then use hand signals or radio/phone communication to have the marker set up at the correct point.
Suppose you are out on the water in a boat and wish to measure how far you are from shore because you notice a shipwreck or some other object of interest under the water and wish to record the position. You would pick out two easily recognizable objects on shore such as trees or large rocks. The two objects should be in a line perpendicular to the line between you and one of the objects. Measure the angle between the two objects from where you are in the boat and record it.
Later, you would carefully measure the distance between the two objects using a tape measure or a map. Then you would take the cotangent of the angle measured from the boat and multiply it by that distance. Alternatively, you could simply take the directional readings of any two (or more) prominently visible, fixed position objects. The position on the water could then be charted using a map or satellite photo of the area.
Astronomers have long used this technique to measure the distance to stars, it is known as parallax. The carefully measured distance from Point A to Point B is referred to as the baseline. The same principle can be used with the Compass Ruler to map an entire area. Simply pick out visible objects such as trees, houses, etc. Measure the distances from a central point to the objects and then measure the angular distances between those objects from the central point. The map then can be easily made using a ruler and protractor. Of course, on complex maps, more than one central point can be used. If the terrain to be mapped is hilly, the logical place for the central points would obviously be on the high ground.
Aside from the contact measurement and surveying versions, there is yet version of the Compass Ruler, the drawing version. Simply fasten or glue a small compass to a straight-edge such as a ruler and it makes the protractor used in geometric drawings just as obsolete as the builder's square is in construction. To draw two lines at a certain angle to each other, simply draw one, noting the angle indicated on the compass dial. Then move the straight-edge so that the difference showing on the dial is now the desired angle and then draw the second line. To measure the angle between two existing lines, simply reverse this process.
If the drawing paper is securely taped down, aligned either east-west or north-south with the compass for best results, an entire geometric drawing can be made with unprecedented accuracy using the drawing version of the Compass Ruler. Parallel lines will have the same compass dial reading anywhere in the drawing. Perpendicular lines will differ 90 degrees in reading. Existing methods are far inferior to this. Of course, it would be simple to draw a map that was surveyed using the surveying version of the tool, just drawing the measurements on paper.
There are certainly many more everyday applications of this simple but extremely useful device. The device could also bring geometry and trigonometry classes to life. The lessons that now consist of drawing lines and circles on paper could occasionally be done as actual measurements in the gym or schoolyard.
Anyone can make their own of any version of the Compass Ruler, the circular or the simpler square version for contact measurements. Or the surveying or drawing versions with an attached or accompanying straight-edge. It can also be manufactured and sold although it will not be patentable now that I have put it in the public domain.
I will not earn any money on this but I just want to have contributed something to the world. We read of George Washington Carver and how he revealed many things that the humble peanut can be used for. I would like to do the same thing for the simple device known as the compass. Just as GPS systems are becoming ubiquitous and it seems to be of little use any more, we see that there is a whole world of tasks that it can accomplish most effectively. Any simple compass would have it's usefulness multiplied if it were encased with a straight side to perform some of the measurements listed above.
New Trigonometric Functions
As you may know, the branch of mathematics known as trigonometry deals with triangles. Specifically, right triangles. That is, a triangle containing one 90 degree angle. It is very useful for measurement of the world and universe around us.
Consider a straight line that we will call x. Now let's draw a line perpendicular to x and call it y. From the point where x and y meet, the origin, we can draw another line we will refer to as r, for radius. The line, r can be drawn at any angle out from the origin from 0 degrees to 90 degrees. If it is drawn at 0 degrees, r will be one and the same as the line, x. If r is drawn at 90 degrees, r will be one and the same as the line, y. If r is drawn at 45 degrees, it will divide the original angle xy into two equal angles.
We can also think of it as a square or rectangle. One side of the rectangle is the x side. The perpendicular side is the y side. The line, r is the diagonal line that could be drawn between opposite angles of the rectangle. You may notice that unless r is drawn at either 0 or 90 degrees, it must always be longer than either x or y.
The length of r, as opposed to x and y, depend on the relative lengths of x and y. If these two axes are equal, r will form a 45 degree angle to x or y in the origin. If x and y in the rectangle are not equal, r will form an angle other than 45 degrees to x or y.
In trigonometry, we define x as the horizontal axis and y as the vertical axis. The angle of r is measured from the x axis and intersects the two axes at the origin. We have what we refer to as "trigonometric functions". These are the sine, cosine and, tangent. Each angle has a value for each of these three functions.
The sine is defined as the ratio y/r for any given angle. The sine starts at zero at 0 degrees and goes to one at 90 degrees. The cosine is defined as the ratio x/r and does the opposite. It starts at one at 0 degrees and decreases to zero at 90 degrees. The tangent is defined as the ratio y/x for a given angle. It starts at zero at 0 degrees, reaches one at 45 degrees and goes to infinity at 90 degrees.
There are the lesser-used so-called "inverse functions". The cosecant is the inverse of the sine. So, it is defined as the ratio r/y. The secant is the inverse of the cosine and is defined as r/x. The cotangent is x/y. You may notice that three of the six possible functions have the prefix co- in front of them. These co- functions are the ones whose values decrease as the angle from 0 degrees to 90 degrees gets larger. The trigonometric functions are based on a 90 degree angle. This, of course, makes sense because in our universe, the dimensions of space form 90 degree angles.
There are two additions that I wish to make to trigonometry. I have noticed that, as useful as trigonometry is, it could be even more useful. The trigonometry that we use now is merely the 90 degree set of functions. The nature of the space we inhabit brings two more sets of meaningful trigonometric functions into being that we are not using as of yet.
The 90 degree set of functions will always remain the most useful simply because that is the angle at which the dimensions we inhabit intersect. But I have developed two more useful trigonometric functions. In my book "The Patterns of New Ideas", one of the ideas I introduced was "The Trigonometric Product". If we multiply the sine by the cosine of an angle from 0 to 90 degrees, we find that the product starts at zero at 0 degrees, peaks at 0.5 at 45 degrees and, goes back to zero at 90 degrees. It would be more convenient to multiply the product by two to have it peak at one at 45 degrees.
What we thus obtain is a trigonometric function of a different set than the traditional functions. This new function is obtained by multiplying two of the original 90 degree functions but it is based on an angle of 45 degrees rather than 90 degrees. It may be true that the spatial dimensions of our universe are based upon an angle that we have defined as 90 degrees. But the nature of this space also causes a number of everyday situations to fit into a description based on 45 degree functions.
The standard 90 degree functions assume that x and y are equal to start with. The new 45 degree function that I wish to introduce is for situations in which the two axes, x and y, are not equal. To give a few examples of the usefulness of the 45 degree trigonometric function, consider the following: You are at one corner of a rectangle, say an athletic field. You wish to go to the opposite corner of the rectangle. How much travel distance will you save, expressed as a ratio, by cutting diagonally across the field instead of going around the perimeter?
The answer depends on the ratio of the lengths of the two perpendicular sides of the rectangle. If the opposing sides are equal, the direction to the opposite corner will be 45 degrees and the efficiency of the savings will be at a maximum. We will express this efficiency as 1. If the rectangle has one side vastly longer than the perpendicular side so that it is a long thin strip, the efficiency will be much less than 1.
If one side could be infinitely long and the side perpendicular to it was infinitely short, the efficiency of a diagonal crossing would be 0. The efficiency of the diagonal cut, peaking at 1, can be expressed as a ratio, the short side divided by the long side. It is at a maximum when the two perpendicular sides are equal so that the angle of the diagonal is 45 degrees.
At that point, if we multiply the sine of the 45 degree angle by the cosine of the angle, we come up with 0.5 as a result. For convenience, we will obtain the trigonometric product by performing this multiplication and then multiplying it by a factor of two, so that it starts at 0 and peaks at 1 in the same way as we are accustomed to with the 90 degree functions.
Let's look at another application of the 45 degree function, the trigonometric product. Suppose you have a stack of fence panels and you wish to enclose the maximum possible area with these panels. We could refer to the problem as area enclosed per given length of perimeter. The enclosed area would be at a maximum when the two perpendicular sides were equal and would be expressed as short side/long side.
The example of the trigonometric product for 45 degree functions that I used in my book "The Patterns of New Ideas" is that of a cannon being fired into the air. It provides yet another example. It's range on level ground or sea would be at a maximum when the cannon was aimed equidistant between the horizontal and the vertical, in other words at 45 degrees. It's impact point on the ground could be brought closer than the maximum range by aiming it either higher or lower than 45 degrees.
If you have ever studied calculus, you may have noticed by now that this 45 degree function can replace quite a bit of calculus and is much simpler and easier. Calculus uses a graphed curve to find maxima and minima of the curve. We seek to find which point on a curve in calculus is the one in which the curve stops it's climb or descent and begins to move in the other direction.
You have seen how the peaking of the trigonometric product in the 45 degree function does the same thing. Just picture the zero to one and back to zero again of the function as a calculus graph. And to find such things as distance traveled, it is much easier to calculate the area of a right triangle of certain given angles than it is to find the area under a graph in calculus.
The next addition that I would like to add to trigonometry is another function. Now that we have the original 90 degree functions and the new 45 degree functions, let's add the 180 degree functions. Just as the distance across a rectangle involves the 45 degree function, the distance across a circle involves the 180 degree function.
Put another way, the familiar 90 degree functions involve pre-existing equal dimensions. The new 45 degree functions involve pre-existing potentially unequal dimensions. The 180 degree functions involve the relationship between lines and pre-existing circles. In 90 degree functions, the radius, r, draws a circle. In the 180 degree function, the circle already exists as our "field" and we draw a chord across the circle.
We could call the origional 90 degree functions, the "primary functions" and the new 45 and 180 degree functions, the "secondary functions". Imagine a circle, such as a circular racetrack. Suppose you were at one point on the circle and wished to go to another point. How much efficiency would you gain by taking a shortcut directly across the circle to the destination point?
The answer would depend on how far ahead was your destination point and thus how much of the circle you were to cut out. The closer the destination point was to the present point, the less would be the efficiency of cutting straight across. The efficiency would be at a maximum if you were cutting directly across the circle to the point diametrically opposite you.
So, we could say that the expression of efficiency begins at 0 if our destination point is the point immediately ahead of our present position and goes to 1 if our destination point is on the diametrically opposite side of the circle. In other words, 180 degrees ahead. Thus, we have a new function that starts at zero for 0 degrees and goes to 1 for 180 degrees.
Such a straight line from a point on a circle, across the circle to another point on the circle, is known as a chord. In a chord, the angle of the outside circle occupied by the chord is equal to the sum of the instantaneous angles formed in the two places where the chord intersects the circle. Obviously, the instantaneous angles formed by chords cannot be over 90 degrees in a circle because 90 multiplied by two equals 180, which is the number of degrees cut off the circle by the longest possible chord, which is a diameter.
By "instantaneous", I mean the immediate angle at only the contact point of the circle. Suppose you had an infinitely large circle. The intersection of a diameter line or a chord would form a clearly measurable angle with the circle. The difference between a chord and a full diameter of a circle is that the instantaneous angles formed in a diameter line are perpendicular, 90 degrees, and in a chord are less than 90 degrees.
This is the basis of the 180 degree functions. The instantaneous angle of the radius, r, and the circle thus formed is always 90 degrees in the 90 degree functions. In our new 180 degree functions, it varies from 0 to 90 degrees. The instantaneous angle is equal to the angle between the diameter and the chord, which can range from 0 to 90 degrees. Or half the angle of the outside circle that is within the chord, which can range from 0 to 180 degrees.
The length of the radius, r, always stays the same in the 90 degree functions. However, the length of the chord can vary from 0 to 1 in the 180 degree functions. a length of 1 would, of course, be equivalent to a diameter of the circle. As the efficiency becomes greater, the length of the chord becomes longer until at an efficiency of 1, the chord has it's greatest diameter and has become a diameter of the circle. In terms of distance, the efficiency of cutting diametrically across a circle would be 2/pi or 1.57. Now for the actual formula for the 180 degree trigonometric function. It is the sine of the angle (from 0 to 180) divided by two.
For example, if you wanted to find the 180 degree function of 130 degrees, you would take the sine of 65 degrees. As we draw a chord of varying lengths across part of a circle from our starting point, we notice that the increase in the length of the chord is much greater from 0 to 90 degrees than it is from 90 to 180 degrees. If a diameter (180 degrees) is considered as having a value of 1, a chord will have a length of .707 when it covers 90 degrees of the circle. It will increase in length only .293 more as it goes from 90 to 180 degrees.
This is simply because as the length of the chord increases in the second quadrant (90-180 degrees), it is decreasing in the first quadrant from .707 when the chord is 90 degrees to 0.5 when it is 180 degrees. At 180 degrees, of course, the chord of 1 will consist of 0.5 length in each of the two quadrants.
This new function is useful whenever there are interior lines forming a chord or a diameter in a circle. For example, two planets in their orbits. Or a specific point on the earth's surface to a particular point on the moon's surface at a particular instant in time. Of course, the orbits of the planets tend to be ellipses instead of perfect circles.
But this function can be easily modified for two ellipses by defining the aphelion and perihelion (furthest and closest points of approach) in terms of relative distance and expressing the orbits as comprising 360 degrees. The difference in horizontal plane of two orbits can easily be expressed in terms of 90 degree trigonometric functions.
To summarize, the 45 degree function describes a radius from an origin consisting of a perpendicular x and y axes. The function equals the sine multiplied by the cosine of the angle of the radius from the horizontal x axis and then multiplied by 2. This function starts at zero at 0 degrees, peaks at 1 at 45 degrees and goes back to zero at 90 degrees. The 180 degree function describes a straight line between two points on the inside of a circle. It starts at zero for an infinitesimal chord and goes to 1 for a diameter, or 180 degree chord.
The efficiency as well as the length of a given chord from zero to the maximum (diametrical) value are both expressed by the same number from 0 to 1. The function is given by the sine of (the angle divided by two). We could call these two new functions, the "Compound Functions". The sine, cosine and, tangent are the main functions. The cosecant, secant and, cotangent are the inverse functions.
The new 180 degree function could possibly be called the "Planetary Function" because it is ideal for describing the directional relationship between two planets, one rotating inside the orbit of the other. Actually the best way to illustrate this function is the view of the moon from the earth. I believe this is also a better way to give an example of trigonometry than anything concerning the traditional 90 degree functions.
The area of the moon that is lighted, as seen from earth, is a function of the angular distance in the sky between the earth and the sun. This can only be described by my 180 degree function and not by any of the 90 degree functions. The proportion of the moon that appears illuminated to us starts at zero at new moon, goes to complete at full moon and then back to zero at new moon.
New moon is, of course, when the moon is in the same place in the sky as the sun and full moon is when the moon is 180 degrees opposite the sun. Take the proportion of the moon that appears lit by the sun, multiply by 180 degrees and that will give the angular distance in the sky between moon and sun.
I believe at this point that there are no more sets of useful trigonometric functions to be found beyond the origional 90 degree functions and the new 45 and 180 degree secondary functions that I have introduced. A meaningful 270 degree (3/4 of a circle) set of functions could not exist because it could not have an equivalent rise and drop and would be a repetition of existing functions. 360 degree functions would be linear and equivalent to 0 degree functions. We could actually define 0 degree trigonometric functions as those functions in only one dimension.
Thus when you measure a straight line or express a linear distance without using trigonometry, you are actually using what we might call the 0 degree function. Although the very definition of the word "trigonometry" means triangle and this is not possible in one dimension. We could, however, refer to measurement of a one-dimensional line as making use of the 0 degree function and say that trigonometry only applies to two-dimensional space.
Consider a straight line that we will call x. Now let's draw a line perpendicular to x and call it y. From the point where x and y meet, the origin, we can draw another line we will refer to as r, for radius. The line, r can be drawn at any angle out from the origin from 0 degrees to 90 degrees. If it is drawn at 0 degrees, r will be one and the same as the line, x. If r is drawn at 90 degrees, r will be one and the same as the line, y. If r is drawn at 45 degrees, it will divide the original angle xy into two equal angles.
We can also think of it as a square or rectangle. One side of the rectangle is the x side. The perpendicular side is the y side. The line, r is the diagonal line that could be drawn between opposite angles of the rectangle. You may notice that unless r is drawn at either 0 or 90 degrees, it must always be longer than either x or y.
The length of r, as opposed to x and y, depend on the relative lengths of x and y. If these two axes are equal, r will form a 45 degree angle to x or y in the origin. If x and y in the rectangle are not equal, r will form an angle other than 45 degrees to x or y.
In trigonometry, we define x as the horizontal axis and y as the vertical axis. The angle of r is measured from the x axis and intersects the two axes at the origin. We have what we refer to as "trigonometric functions". These are the sine, cosine and, tangent. Each angle has a value for each of these three functions.
The sine is defined as the ratio y/r for any given angle. The sine starts at zero at 0 degrees and goes to one at 90 degrees. The cosine is defined as the ratio x/r and does the opposite. It starts at one at 0 degrees and decreases to zero at 90 degrees. The tangent is defined as the ratio y/x for a given angle. It starts at zero at 0 degrees, reaches one at 45 degrees and goes to infinity at 90 degrees.
There are the lesser-used so-called "inverse functions". The cosecant is the inverse of the sine. So, it is defined as the ratio r/y. The secant is the inverse of the cosine and is defined as r/x. The cotangent is x/y. You may notice that three of the six possible functions have the prefix co- in front of them. These co- functions are the ones whose values decrease as the angle from 0 degrees to 90 degrees gets larger. The trigonometric functions are based on a 90 degree angle. This, of course, makes sense because in our universe, the dimensions of space form 90 degree angles.
There are two additions that I wish to make to trigonometry. I have noticed that, as useful as trigonometry is, it could be even more useful. The trigonometry that we use now is merely the 90 degree set of functions. The nature of the space we inhabit brings two more sets of meaningful trigonometric functions into being that we are not using as of yet.
45 DEGREE TRIGONOMETRIC FUNCTION
The 90 degree set of functions will always remain the most useful simply because that is the angle at which the dimensions we inhabit intersect. But I have developed two more useful trigonometric functions. In my book "The Patterns of New Ideas", one of the ideas I introduced was "The Trigonometric Product". If we multiply the sine by the cosine of an angle from 0 to 90 degrees, we find that the product starts at zero at 0 degrees, peaks at 0.5 at 45 degrees and, goes back to zero at 90 degrees. It would be more convenient to multiply the product by two to have it peak at one at 45 degrees.
What we thus obtain is a trigonometric function of a different set than the traditional functions. This new function is obtained by multiplying two of the original 90 degree functions but it is based on an angle of 45 degrees rather than 90 degrees. It may be true that the spatial dimensions of our universe are based upon an angle that we have defined as 90 degrees. But the nature of this space also causes a number of everyday situations to fit into a description based on 45 degree functions.
The standard 90 degree functions assume that x and y are equal to start with. The new 45 degree function that I wish to introduce is for situations in which the two axes, x and y, are not equal. To give a few examples of the usefulness of the 45 degree trigonometric function, consider the following: You are at one corner of a rectangle, say an athletic field. You wish to go to the opposite corner of the rectangle. How much travel distance will you save, expressed as a ratio, by cutting diagonally across the field instead of going around the perimeter?
The answer depends on the ratio of the lengths of the two perpendicular sides of the rectangle. If the opposing sides are equal, the direction to the opposite corner will be 45 degrees and the efficiency of the savings will be at a maximum. We will express this efficiency as 1. If the rectangle has one side vastly longer than the perpendicular side so that it is a long thin strip, the efficiency will be much less than 1.
If one side could be infinitely long and the side perpendicular to it was infinitely short, the efficiency of a diagonal crossing would be 0. The efficiency of the diagonal cut, peaking at 1, can be expressed as a ratio, the short side divided by the long side. It is at a maximum when the two perpendicular sides are equal so that the angle of the diagonal is 45 degrees.
At that point, if we multiply the sine of the 45 degree angle by the cosine of the angle, we come up with 0.5 as a result. For convenience, we will obtain the trigonometric product by performing this multiplication and then multiplying it by a factor of two, so that it starts at 0 and peaks at 1 in the same way as we are accustomed to with the 90 degree functions.
Let's look at another application of the 45 degree function, the trigonometric product. Suppose you have a stack of fence panels and you wish to enclose the maximum possible area with these panels. We could refer to the problem as area enclosed per given length of perimeter. The enclosed area would be at a maximum when the two perpendicular sides were equal and would be expressed as short side/long side.
The example of the trigonometric product for 45 degree functions that I used in my book "The Patterns of New Ideas" is that of a cannon being fired into the air. It provides yet another example. It's range on level ground or sea would be at a maximum when the cannon was aimed equidistant between the horizontal and the vertical, in other words at 45 degrees. It's impact point on the ground could be brought closer than the maximum range by aiming it either higher or lower than 45 degrees.
If you have ever studied calculus, you may have noticed by now that this 45 degree function can replace quite a bit of calculus and is much simpler and easier. Calculus uses a graphed curve to find maxima and minima of the curve. We seek to find which point on a curve in calculus is the one in which the curve stops it's climb or descent and begins to move in the other direction.
You have seen how the peaking of the trigonometric product in the 45 degree function does the same thing. Just picture the zero to one and back to zero again of the function as a calculus graph. And to find such things as distance traveled, it is much easier to calculate the area of a right triangle of certain given angles than it is to find the area under a graph in calculus.
180 DEGREE TRIGONOMETRIC FUNCTION
The next addition that I would like to add to trigonometry is another function. Now that we have the original 90 degree functions and the new 45 degree functions, let's add the 180 degree functions. Just as the distance across a rectangle involves the 45 degree function, the distance across a circle involves the 180 degree function.
Put another way, the familiar 90 degree functions involve pre-existing equal dimensions. The new 45 degree functions involve pre-existing potentially unequal dimensions. The 180 degree functions involve the relationship between lines and pre-existing circles. In 90 degree functions, the radius, r, draws a circle. In the 180 degree function, the circle already exists as our "field" and we draw a chord across the circle.
We could call the origional 90 degree functions, the "primary functions" and the new 45 and 180 degree functions, the "secondary functions". Imagine a circle, such as a circular racetrack. Suppose you were at one point on the circle and wished to go to another point. How much efficiency would you gain by taking a shortcut directly across the circle to the destination point?
The answer would depend on how far ahead was your destination point and thus how much of the circle you were to cut out. The closer the destination point was to the present point, the less would be the efficiency of cutting straight across. The efficiency would be at a maximum if you were cutting directly across the circle to the point diametrically opposite you.
So, we could say that the expression of efficiency begins at 0 if our destination point is the point immediately ahead of our present position and goes to 1 if our destination point is on the diametrically opposite side of the circle. In other words, 180 degrees ahead. Thus, we have a new function that starts at zero for 0 degrees and goes to 1 for 180 degrees.
Such a straight line from a point on a circle, across the circle to another point on the circle, is known as a chord. In a chord, the angle of the outside circle occupied by the chord is equal to the sum of the instantaneous angles formed in the two places where the chord intersects the circle. Obviously, the instantaneous angles formed by chords cannot be over 90 degrees in a circle because 90 multiplied by two equals 180, which is the number of degrees cut off the circle by the longest possible chord, which is a diameter.
By "instantaneous", I mean the immediate angle at only the contact point of the circle. Suppose you had an infinitely large circle. The intersection of a diameter line or a chord would form a clearly measurable angle with the circle. The difference between a chord and a full diameter of a circle is that the instantaneous angles formed in a diameter line are perpendicular, 90 degrees, and in a chord are less than 90 degrees.
This is the basis of the 180 degree functions. The instantaneous angle of the radius, r, and the circle thus formed is always 90 degrees in the 90 degree functions. In our new 180 degree functions, it varies from 0 to 90 degrees. The instantaneous angle is equal to the angle between the diameter and the chord, which can range from 0 to 90 degrees. Or half the angle of the outside circle that is within the chord, which can range from 0 to 180 degrees.
The length of the radius, r, always stays the same in the 90 degree functions. However, the length of the chord can vary from 0 to 1 in the 180 degree functions. a length of 1 would, of course, be equivalent to a diameter of the circle. As the efficiency becomes greater, the length of the chord becomes longer until at an efficiency of 1, the chord has it's greatest diameter and has become a diameter of the circle. In terms of distance, the efficiency of cutting diametrically across a circle would be 2/pi or 1.57. Now for the actual formula for the 180 degree trigonometric function. It is the sine of the angle (from 0 to 180) divided by two.
For example, if you wanted to find the 180 degree function of 130 degrees, you would take the sine of 65 degrees. As we draw a chord of varying lengths across part of a circle from our starting point, we notice that the increase in the length of the chord is much greater from 0 to 90 degrees than it is from 90 to 180 degrees. If a diameter (180 degrees) is considered as having a value of 1, a chord will have a length of .707 when it covers 90 degrees of the circle. It will increase in length only .293 more as it goes from 90 to 180 degrees.
This is simply because as the length of the chord increases in the second quadrant (90-180 degrees), it is decreasing in the first quadrant from .707 when the chord is 90 degrees to 0.5 when it is 180 degrees. At 180 degrees, of course, the chord of 1 will consist of 0.5 length in each of the two quadrants.
This new function is useful whenever there are interior lines forming a chord or a diameter in a circle. For example, two planets in their orbits. Or a specific point on the earth's surface to a particular point on the moon's surface at a particular instant in time. Of course, the orbits of the planets tend to be ellipses instead of perfect circles.
But this function can be easily modified for two ellipses by defining the aphelion and perihelion (furthest and closest points of approach) in terms of relative distance and expressing the orbits as comprising 360 degrees. The difference in horizontal plane of two orbits can easily be expressed in terms of 90 degree trigonometric functions.
To summarize, the 45 degree function describes a radius from an origin consisting of a perpendicular x and y axes. The function equals the sine multiplied by the cosine of the angle of the radius from the horizontal x axis and then multiplied by 2. This function starts at zero at 0 degrees, peaks at 1 at 45 degrees and goes back to zero at 90 degrees. The 180 degree function describes a straight line between two points on the inside of a circle. It starts at zero for an infinitesimal chord and goes to 1 for a diameter, or 180 degree chord.
The efficiency as well as the length of a given chord from zero to the maximum (diametrical) value are both expressed by the same number from 0 to 1. The function is given by the sine of (the angle divided by two). We could call these two new functions, the "Compound Functions". The sine, cosine and, tangent are the main functions. The cosecant, secant and, cotangent are the inverse functions.
The new 180 degree function could possibly be called the "Planetary Function" because it is ideal for describing the directional relationship between two planets, one rotating inside the orbit of the other. Actually the best way to illustrate this function is the view of the moon from the earth. I believe this is also a better way to give an example of trigonometry than anything concerning the traditional 90 degree functions.
The area of the moon that is lighted, as seen from earth, is a function of the angular distance in the sky between the earth and the sun. This can only be described by my 180 degree function and not by any of the 90 degree functions. The proportion of the moon that appears illuminated to us starts at zero at new moon, goes to complete at full moon and then back to zero at new moon.
New moon is, of course, when the moon is in the same place in the sky as the sun and full moon is when the moon is 180 degrees opposite the sun. Take the proportion of the moon that appears lit by the sun, multiply by 180 degrees and that will give the angular distance in the sky between moon and sun.
I believe at this point that there are no more sets of useful trigonometric functions to be found beyond the origional 90 degree functions and the new 45 and 180 degree secondary functions that I have introduced. A meaningful 270 degree (3/4 of a circle) set of functions could not exist because it could not have an equivalent rise and drop and would be a repetition of existing functions. 360 degree functions would be linear and equivalent to 0 degree functions. We could actually define 0 degree trigonometric functions as those functions in only one dimension.
Thus when you measure a straight line or express a linear distance without using trigonometry, you are actually using what we might call the 0 degree function. Although the very definition of the word "trigonometry" means triangle and this is not possible in one dimension. We could, however, refer to measurement of a one-dimensional line as making use of the 0 degree function and say that trigonometry only applies to two-dimensional space.
Infinite Geometry
Standard geometry is known as "Euclidean Geometry" because it was that which was taught by Euclid in Alexandria around 300 B.C. Euclidean geometry is based upon an assumption that is presumed to be factual even though it is difficult to prove mathematically. This assumption is that if there is a point outside a line, there is one and only one line that can be constructed through the point which is parallel to the given line.
This is considered to be so self-evident that it really does not need to be mathematically proven. There are so-called non-euclidean geometries such as the one created by the German mathematician Bernhard Riemann, which pre-dated Einstein and is useful for describing geometrically curved space. This geometry has served well ever since.
However, much has been learned about the universe and the nature of reality since Euclid's time. The introduction or first chapter of a book of geometry usually has the theoretical basis underlying lines and points. A geometric point is an infinitesimal structure that has no dimensions at all except for it's location and parallel lines are said to eventually meet in infinity.
Today I would like to update not the general rules of practical geometry themselves, the three angles of a triangle will still add up to 180 degrees and when two straight lines cross, the opposite angles will still be equal, but the theoretical underpinnings of it.
To be useful, mathematical systems such as numbers and geometry must be more expansive than the entities being measured or described. Thus, we should presume the mathematical universe to be an infinite array of infinitesimal points in an infinite number of dimensions. This is what the universe is, or could potentially be.
Mathematics should be based on infinity because it represents all that exists or could potentially exist. A good way to represent infinity would be 1/0 because one can be divided by zero an infinite number of times. My belief is that the infinite cannot be defined in terms of the finite but the finite can be defined in terms of the infinite.
All euclidean geometry, all straight lines, all angles less than 360 degrees are finite patches of the infinite mathematical universe. The base numbers of the mathematical universe are zero and infinity, all others are just for the finite. The main change in the underlying geometric theory that I would like to propose is that the circle comes before the straight line and not the other way around.
First in geometry is the point, what comes next is not straight lines but the locus of points a given distance from the original point, in other words a circle. It is easier to describe a circle than a straight line in terms of geometry, the circle requires only one defining point while the line requires two. If we have a lower-dimensional surface, such as a curved sheet of paper, in higher-dimensional space, a straight line is not even the shortest distance between two points with regard to the higher-dimensional space. Thus, I consider it as self-evident that the circle must be the next geometric entity after the point and must come before the straight line.
I would like to define a geometric straight line in terms of the circle that comes before it. A straight line must be an arc of an infinite circle and a flat plane is a sector of an infinite sphere. This is easy to imagine because there can be a short straight line or square of land on the earth, while the earth as a whole is a sphere. Any line that we can define is an arc of a circle centered at an infinitely distant point in a perpendicular direction to the line.
Finite beings can define exactly only one of the two aspects of an infinite circle or sphere, either the arc (or plane) or the center but not both. Even a small circle itself is a reflection of infinity and we can prove this by the fact that the value of pi, 3.1415927..., goes on to an infinite number of decimal places with no known repetition. Any such circle that we can define can be considered as the bottom of a cone whose apex is an infinitely distant point.
Geometric shapes are a function of the number of dimensions of space. This was not well understood in Euclid's time, it was Einstein that pointed out our three dimensions of space. I have noticed that the Pythagorean Theorem used on right triangles, the diagonal squared is equal to the one side squared plus the other side squared, works in any number of dimensions.
The origination of lines from circles that I have proposed is also a function of the number of dimensions. A one-dimensional line originates from an infinite two-dimensional circle and a two-dimensional plane originates from an infinite three-dimensional sphere. It is only in one-dimensional space that the straight line comes before the circle because a circle requires two dimensions.
There must be an infinity of geometric shapes that exist or potentially exist but that we cannot imagine due to our dimensional limit. We are most finite not in size but in the number of dimensions that we are able to access. As for time, it is a straight line but it just a property of ourselves and we do not measure it geometrically.
Diagonal distance is a function of the number of dimensions of space. A finite distance must encompass a finite number of dimensions. If time can be defined as motion, it must be meaningless in an infinite number of dimensions because it would take forever to get anywhere.
Let's go on an exercise in mind expansion today, by trying to wrap our minds around the concept of infinity.
Infinity is supposedly a number, the highest number that there is. Yet, it is a realm in which numbers have absolutely no meaning. At the other end of the number line, zero is also a number. But it is likewise a realm in which numbers have no meaning. This reveals something about numbers, to have meaning we have to be at a point on the scale of numbers where there are numbers both above and below us. If we have numbers on one side but not the other, at either zero or infinity, then numbers become meaningless.
Numbers are themselves infinite, meaning that they continue indefinitely, they must be or else infinity could not be the number that it is. That is at least the theory. But the limitation lies in ourselves. To be meaningful to us numbers must be manifested in some way, if only as figures on paper. But that creates an impenetrable barrier between us and infinity. We could fill the whole universe with numbers on paper. But since the universe that we inhabit is finite, whatever number we could thus create would also have to be finite and so would fall short of infinity. To be infinite, a number must just be. It can never be infinite if it must be generated or manifested in any way.
Any finite number not only falls short of infinity, it must fall infinitely short of infinity. No matter what we do with finite numbers, we can never get even an iota closer to infinity. We can spend our whole lives multiplying numbers until we have a number that fills the whole universe, and we will be not a bit closer to infinity than when we started. If we could somehow get closer to it, then infinity would not be infinite. You can only make progress to a destination if it is a finite distance away. There can never be any common ground between the finite and the infinite.
Infinity is not just an imaginary concept, it is very real. In geometry, we are taught that parallel lines are sets of lines that are in the same plane but which never meet in our finite realm. They do, however, eventually meet at infinity. Parallel lines must meet somewhere. To claim that they do not is for the finite to reach the infinite. Parallel lines may never meet in the geometry textbook illustrations, or in the world, or in the universe. But infinity is so far, in fact infinitely far, that nothing finite like a pair of parallel lines can ever reach it. The most perfect pair of parallel lines that our universe can manifest must eventually meet. The parallel lines do not have to meet in the finite universe to which they belong. But for the parallel lines of a finite universe to reach infinity without meeting is to reach infinity by finite means, and we know that such a thing is impossible.
Infinity actually can be expressed with finite numbers, but we must go to the opposite end of the number scale to do it. Any fraction with zero as a denominator is representative of infinity, such as 1/0. Since zero is nothing, one divided by zero must be infinity. An infinite number of zero can fit into one. If 1/0 was not tantamount to infinity, then zero would have to equal something and if it did then it wouldn't be zero. This reveals that a finite something is as far removed from nothing as the finite something is removed from infinity. This is why we can only express infinity with finite numbers if one of those numbers is zero.
The opposite of the infinite is the infinitesimal. Something that is infinitesimal is something that is just about zero. In fact, any finite quantity can be divided into an infinite number of infinitesimal divisions. Like the infinite, the infinitesimal can never be described with finite numbers. All finite numbers are just as meaningless with the infinitesimal as with the infinite. If we can apply numbers to something, it is neither infinitesimal nor infinite. Just as we are limited by the fact that we are composed of matter in a universe of space and matter from reaching infinity, we are also prevented from reaching the infinitesimal. An electron, a mere point particle with no discernible internal structure, is the closest we come to the infinitesimal, just as the entire universe is the closest we come to the infinite.
Upon reaching infinity, we would find that numbers have become utterly meaningless. If any number has any meaning at all, then we have not reached infinity. 5 would equal 23, or 36,754,013, if you prefer. Numbers are meaningless at the other end of the scale, at zero, because there is nothing to manifest numbers and numbers, or any mathematical entity, must be manifested in some way to be real. Neither would numbers make any sense at infinity, because no finite number would be manifested.
If we have zero at one end of the number scale, and infinity at the other end, there should be some halfway point between zero and infinity. The halfway point appears to be the number 1/2, one half. First of all consider the time version of infinity, which is eternity. An eternal being, which has existed for eternity past and will exist for eternity future, will always be at the halfway point of it's existence. No matter how far into the past or future, the eternal being will still be at the halfway point of it's existence. A truly eternal being, meaning both past and future, can never be anywhere but at the halfway point of it's existence.
For another example of how the number 1/2 relates to eternity, consider the statistics of repetitive odds. If you play a game in which there is a 1/2 chance of winning, and you play the game twice, your odds of winning are 3/4. This is because your chance of winning the first play is 1/2. That leaves 1/2 remaining, and your chances of winning that one the second play is 1/2. So, 1/2 + 1/2 of 1/2, or 1/4, = 3/4. Now, suppose that we play another game in which the odds of winning are only 1/4, but we play it four times. The odds of winning become 1/4 + (3/4 x 1/4) + (1/2 x 1/4) + (1/4 x 1/4) = 10/16, or 5/8. Thus, the odds of winning are less than if we played the game of 1/2 odds twice. As the number of the odds game gets higher, for example the odds of 1/100 played 100 times, the odds of winning get progressively lower. But the odds of winning never go below 1/2, no matter how high the number. If the odds of winning were one in a million, but we played a million times, our odds of winning would be a shade over 1/2. When we get to infinity, and played a game in which the odds of winning were infinitesimal, or 1/infinity, but we played the game an infinite number of times, the odds of winning would be exactly 1/2. This is another way in which one-half relates to infinity as the halfway point one the number scale between zero and infinity.
What if the finite could be made infinite? It would mean that everything would have to exist. If the universe was infinite, there could be nothing which could possibly exist which did not exist somewhere. There would have to be exact copies of our earth and solar system out there, in fact, an infinite number of exact copies of our earth and solar system. There would also have to be copies of the earth and solar system with every possible variation, such as solar systems with earth and Venus exchanging places and earths with Australia attached to the coast of Africa.
This is considered to be so self-evident that it really does not need to be mathematically proven. There are so-called non-euclidean geometries such as the one created by the German mathematician Bernhard Riemann, which pre-dated Einstein and is useful for describing geometrically curved space. This geometry has served well ever since.
However, much has been learned about the universe and the nature of reality since Euclid's time. The introduction or first chapter of a book of geometry usually has the theoretical basis underlying lines and points. A geometric point is an infinitesimal structure that has no dimensions at all except for it's location and parallel lines are said to eventually meet in infinity.
Today I would like to update not the general rules of practical geometry themselves, the three angles of a triangle will still add up to 180 degrees and when two straight lines cross, the opposite angles will still be equal, but the theoretical underpinnings of it.
To be useful, mathematical systems such as numbers and geometry must be more expansive than the entities being measured or described. Thus, we should presume the mathematical universe to be an infinite array of infinitesimal points in an infinite number of dimensions. This is what the universe is, or could potentially be.
Mathematics should be based on infinity because it represents all that exists or could potentially exist. A good way to represent infinity would be 1/0 because one can be divided by zero an infinite number of times. My belief is that the infinite cannot be defined in terms of the finite but the finite can be defined in terms of the infinite.
All euclidean geometry, all straight lines, all angles less than 360 degrees are finite patches of the infinite mathematical universe. The base numbers of the mathematical universe are zero and infinity, all others are just for the finite. The main change in the underlying geometric theory that I would like to propose is that the circle comes before the straight line and not the other way around.
First in geometry is the point, what comes next is not straight lines but the locus of points a given distance from the original point, in other words a circle. It is easier to describe a circle than a straight line in terms of geometry, the circle requires only one defining point while the line requires two. If we have a lower-dimensional surface, such as a curved sheet of paper, in higher-dimensional space, a straight line is not even the shortest distance between two points with regard to the higher-dimensional space. Thus, I consider it as self-evident that the circle must be the next geometric entity after the point and must come before the straight line.
I would like to define a geometric straight line in terms of the circle that comes before it. A straight line must be an arc of an infinite circle and a flat plane is a sector of an infinite sphere. This is easy to imagine because there can be a short straight line or square of land on the earth, while the earth as a whole is a sphere. Any line that we can define is an arc of a circle centered at an infinitely distant point in a perpendicular direction to the line.
Finite beings can define exactly only one of the two aspects of an infinite circle or sphere, either the arc (or plane) or the center but not both. Even a small circle itself is a reflection of infinity and we can prove this by the fact that the value of pi, 3.1415927..., goes on to an infinite number of decimal places with no known repetition. Any such circle that we can define can be considered as the bottom of a cone whose apex is an infinitely distant point.
Geometric shapes are a function of the number of dimensions of space. This was not well understood in Euclid's time, it was Einstein that pointed out our three dimensions of space. I have noticed that the Pythagorean Theorem used on right triangles, the diagonal squared is equal to the one side squared plus the other side squared, works in any number of dimensions.
The origination of lines from circles that I have proposed is also a function of the number of dimensions. A one-dimensional line originates from an infinite two-dimensional circle and a two-dimensional plane originates from an infinite three-dimensional sphere. It is only in one-dimensional space that the straight line comes before the circle because a circle requires two dimensions.
There must be an infinity of geometric shapes that exist or potentially exist but that we cannot imagine due to our dimensional limit. We are most finite not in size but in the number of dimensions that we are able to access. As for time, it is a straight line but it just a property of ourselves and we do not measure it geometrically.
Diagonal distance is a function of the number of dimensions of space. A finite distance must encompass a finite number of dimensions. If time can be defined as motion, it must be meaningless in an infinite number of dimensions because it would take forever to get anywhere.
Let's go on an exercise in mind expansion today, by trying to wrap our minds around the concept of infinity.
Infinity is supposedly a number, the highest number that there is. Yet, it is a realm in which numbers have absolutely no meaning. At the other end of the number line, zero is also a number. But it is likewise a realm in which numbers have no meaning. This reveals something about numbers, to have meaning we have to be at a point on the scale of numbers where there are numbers both above and below us. If we have numbers on one side but not the other, at either zero or infinity, then numbers become meaningless.
Numbers are themselves infinite, meaning that they continue indefinitely, they must be or else infinity could not be the number that it is. That is at least the theory. But the limitation lies in ourselves. To be meaningful to us numbers must be manifested in some way, if only as figures on paper. But that creates an impenetrable barrier between us and infinity. We could fill the whole universe with numbers on paper. But since the universe that we inhabit is finite, whatever number we could thus create would also have to be finite and so would fall short of infinity. To be infinite, a number must just be. It can never be infinite if it must be generated or manifested in any way.
Any finite number not only falls short of infinity, it must fall infinitely short of infinity. No matter what we do with finite numbers, we can never get even an iota closer to infinity. We can spend our whole lives multiplying numbers until we have a number that fills the whole universe, and we will be not a bit closer to infinity than when we started. If we could somehow get closer to it, then infinity would not be infinite. You can only make progress to a destination if it is a finite distance away. There can never be any common ground between the finite and the infinite.
Infinity is not just an imaginary concept, it is very real. In geometry, we are taught that parallel lines are sets of lines that are in the same plane but which never meet in our finite realm. They do, however, eventually meet at infinity. Parallel lines must meet somewhere. To claim that they do not is for the finite to reach the infinite. Parallel lines may never meet in the geometry textbook illustrations, or in the world, or in the universe. But infinity is so far, in fact infinitely far, that nothing finite like a pair of parallel lines can ever reach it. The most perfect pair of parallel lines that our universe can manifest must eventually meet. The parallel lines do not have to meet in the finite universe to which they belong. But for the parallel lines of a finite universe to reach infinity without meeting is to reach infinity by finite means, and we know that such a thing is impossible.
Infinity actually can be expressed with finite numbers, but we must go to the opposite end of the number scale to do it. Any fraction with zero as a denominator is representative of infinity, such as 1/0. Since zero is nothing, one divided by zero must be infinity. An infinite number of zero can fit into one. If 1/0 was not tantamount to infinity, then zero would have to equal something and if it did then it wouldn't be zero. This reveals that a finite something is as far removed from nothing as the finite something is removed from infinity. This is why we can only express infinity with finite numbers if one of those numbers is zero.
The opposite of the infinite is the infinitesimal. Something that is infinitesimal is something that is just about zero. In fact, any finite quantity can be divided into an infinite number of infinitesimal divisions. Like the infinite, the infinitesimal can never be described with finite numbers. All finite numbers are just as meaningless with the infinitesimal as with the infinite. If we can apply numbers to something, it is neither infinitesimal nor infinite. Just as we are limited by the fact that we are composed of matter in a universe of space and matter from reaching infinity, we are also prevented from reaching the infinitesimal. An electron, a mere point particle with no discernible internal structure, is the closest we come to the infinitesimal, just as the entire universe is the closest we come to the infinite.
Upon reaching infinity, we would find that numbers have become utterly meaningless. If any number has any meaning at all, then we have not reached infinity. 5 would equal 23, or 36,754,013, if you prefer. Numbers are meaningless at the other end of the scale, at zero, because there is nothing to manifest numbers and numbers, or any mathematical entity, must be manifested in some way to be real. Neither would numbers make any sense at infinity, because no finite number would be manifested.
If we have zero at one end of the number scale, and infinity at the other end, there should be some halfway point between zero and infinity. The halfway point appears to be the number 1/2, one half. First of all consider the time version of infinity, which is eternity. An eternal being, which has existed for eternity past and will exist for eternity future, will always be at the halfway point of it's existence. No matter how far into the past or future, the eternal being will still be at the halfway point of it's existence. A truly eternal being, meaning both past and future, can never be anywhere but at the halfway point of it's existence.
For another example of how the number 1/2 relates to eternity, consider the statistics of repetitive odds. If you play a game in which there is a 1/2 chance of winning, and you play the game twice, your odds of winning are 3/4. This is because your chance of winning the first play is 1/2. That leaves 1/2 remaining, and your chances of winning that one the second play is 1/2. So, 1/2 + 1/2 of 1/2, or 1/4, = 3/4. Now, suppose that we play another game in which the odds of winning are only 1/4, but we play it four times. The odds of winning become 1/4 + (3/4 x 1/4) + (1/2 x 1/4) + (1/4 x 1/4) = 10/16, or 5/8. Thus, the odds of winning are less than if we played the game of 1/2 odds twice. As the number of the odds game gets higher, for example the odds of 1/100 played 100 times, the odds of winning get progressively lower. But the odds of winning never go below 1/2, no matter how high the number. If the odds of winning were one in a million, but we played a million times, our odds of winning would be a shade over 1/2. When we get to infinity, and played a game in which the odds of winning were infinitesimal, or 1/infinity, but we played the game an infinite number of times, the odds of winning would be exactly 1/2. This is another way in which one-half relates to infinity as the halfway point one the number scale between zero and infinity.
What if the finite could be made infinite? It would mean that everything would have to exist. If the universe was infinite, there could be nothing which could possibly exist which did not exist somewhere. There would have to be exact copies of our earth and solar system out there, in fact, an infinite number of exact copies of our earth and solar system. There would also have to be copies of the earth and solar system with every possible variation, such as solar systems with earth and Venus exchanging places and earths with Australia attached to the coast of Africa.
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