WARNING TO MATHEMATICIANS
You will find very little in the way of FORMAL mathematics in these pages, other than a few, very simple equations. They contain only descriptions of patterns which I have observed and studied and concluded are of direct pertinence to math. No proof of anything will even be attempted.
So, turn off the math filters.
If you waste your time trying to relate what I'm saying to standard mathematical concepts, you will never grasp it. You must set aside everything you know, and really look and listen and think as if for the first time. Once you get the basic idea, THEN will be the time to relate it to known ideas. If you don't get the gist on its own terms first, you will never really understand.
I am a mildly autistic savant (similar to an Asperger, but not), and a bit of of a mutation with multiple birth defects. Tests have shown that I have a "visual IQ" (so to speak; an ability to recognize and manipulate patterns) which is literally off the scale.
Think about that for a minute. Certainly one would expect such a person to notice things that others had overlooked. Well, I'm here to tell you that there are fundamental facts of which the 'normal' human is rather oblivious. I intend here to rescue a few very handy concepts from the edge of oblivion.
There are ideas I wouldn't begin to know how to communicate to persons of normal perception. But these are concepts which you are already so tantalizingly close to understanding that I really can hardly believe you haven't put them together yet. But that's life; who can think of everything?
The concept I intend to present first, that of the 4 types of Simulated Polytope, is so very blatant that it is, in a way, already acknowledged by the English language.
Everyone has seen many instances of "imaginary" polytopes in such common objects; we speak of "telephone Lines" and the "glimmering Points of stars" for example. These plainly visible low-dimensional geometric forms are generated by complex structures of at least 3 dimensions.
It is incredible; millennia of geometry and centuries of topology, and you have never noticed that polytopes have SIGN. Polytopes are rather like numbers that way; they can be positive or negative, real or imaginary -- the analogy is imperfect, but irresistible.
Anyway, the negative and/or imaginary polytopes, along with the Null Polytope, comprise what I like to call the Simulated Polytopes. These are synergetic aspects of a shape generated by its overall structure. They are entirely contextual, and never quite localized.
Because you have not explicitly recognized the non-literal polytopes, you count whole clusters of them as "holes" & "cuffs" and other topological complications of a shape, without recognizing the exact types of polytopes of which they consist, or their proper levels.
When one recognizes ALL the polytopes in a shape, both literal and simulated, one finds that they invariably occur in pairs of adjacent dimension -- point and line, line and plane, plane and space, etcetera, with one point always paired with the Null Polytope. Any time this pairing appears to be broken, it is only because some simulated polytope is not being counted. This is a truly fundamental feature of shape, comparable in importance to the pairing of particle and anti-particle in physics.
For instance, a wire or pipe or hose may be called a "line" because they simulate a line and function as conduits of linear flow, when in fact they have cylindrical surfaces in which there is no literal Edge or Line. The apparent linearity is "buttered all over the surface", so to speak.
Similarly, a distant planet resembles a point, although we know that their surfaces are roughly spherical. Again the "pointiness" is so distributed all over the surface that it is only visible from a great distance. Generally speaking, when the actual details of an object are invisible, and yet we can still see the object, what remains visible are its "imaginary" polytopes -- functional images of polytopes without any formal reality, but which nicely summarize the overall connectivity of the shape.
If you cut a hole in a piece of cardboard or other plane, the cutout is indeed a simple negative plane; however, the "hole" in a simple unfaceted torus consists of a cluster of 5 distinct polytopes which we may call "negative" or "imaginary" or both, as will be shown later -- and thats not counting the Null Polytope, mother of all simulated polytopes, which is part of every shape.
Dimension, as such, is a fine and useful concept, but not appropriate for absolutely every topic -- rather like the old joke about the man who only has a hammer so he treats everything like a nail.
One simply cannot properly understand the full range of polytopes, both literal and simulated, in terms of dimension, as we will see.
Level of Form looks very similar to Dimension, but is slightly more general and inclusive -- a significantly different concept. The 'Level' of a polytope is equal to the minimum number of points from which one can form that polytope.
For instance, it takes at least 3 points to form a plane, 2 points to form a line, and the point defines itself as the topological unit of Level 1. On the other hand, the Null Polytope, conventionally of dimension "-1" (a meaningless number in terms of dimension, i'm told, a mere "tag") is far more logically understood as being of Level 0, requiring NO points. The Null Polytope, being the topological empty set, is in fact rather literally Zero. The zero of dimension is the unit of form, but the zero of form is truly Nothing.
And since the Null Polytope is the only thing that all 4 ranges of polytopes have in common -- i.e., they in fact have Nothing in common at all -- that makes it the crux & origin & the proper center of the whole system, a true Zero indeed. As with the Copernican versus the Aristotelian models of the Solar System, finding the correct center (e.g. the sun) eliminates a great deal of unnecessary complication and promotes a far more intuitive view of the system's operation.
I do realize how utterly ridiculous it sounds, to suggest that in the last several thousand years, nobody else has ever looked at things the way I do -- or, at least, been able to formulate these particular concepts, simple as they are -- but I think that the results speak for themselves.
So you're welcome to attempt to prove or disprove my several hypotheses described herein. I wish you luck. I myself am reasonably certain that, however murkily expressed they may be, my concepts and methods themselves are valid, as far as they go (for nothing in science or math is ever really the final word), because I've used them a long time and tested them quite thoroughly enough to be convinced that they do work, and indeed function quite consistently.
And, last but not least, I must say that I am sincerely thankful for those several mathematicians, professional and otherwise, who have been open-minded enough to encourage me -- particularly Dr. Michael D. Taylor of the University of Central Florida.
You can e-mail me, if you have comments or questions, at shapesavant (at) gmail.com.
(I've written it this way to discourage the spam-bots; just substitute the usual @ sign for (at).)