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.