You will find very little in the way of actual mathematics in these pages, save a few simple equations.  I can offer NO proofs. This is not mathematics as such. What I am presenting here is the raw pattern-perception that comes before math, wrestled into plain english.  Once you get the pattern, you can express it in your own math.
    The formal mathematics of the neurotypical are so admirable; you take the slimmest of facts and use them to pry the cosmos open by sheer logic.  Wonderful!   I'm no stranger to logic -- but my grasp of mathematical topics is essentially intuitive, a "savant" ability. I am a "high functioning autistic", tho not "Asperger" as such.  My lifelong practice of Zen has somewhat mitigated my early autism, I suspect, and left me much more communication-capable, and I hope you will appreciate the result.  
    I'm not a mathematician; I'm a visual artist & computer programmer, & I'm interested in all the branches and aspects of science, especially physics.   But math particularly intrigues me.  
    I've been called a "savant", but instead of the usual savant's knack for numbers, I have an apparently unique sense of shape.  Tests have shown that for pattern perception and manipulation, my abilities are literally off the scale.  
    One might therefor imagine that I may recognize a few patterns that have escaped general notice.  Possibly even patterns of significance to mathematics?  Believe it or not, yes, very much so.  I'm here to tell you that there are fundamental truths of which your mathematics are oblivious, or nearly so, yet which I strongly suspect maybe quite important.  
    These things I want to tell you about are things you ALMOST see, concepts you NEARLY have, and which I therefor may hope to nudge you into perceiving.  These are things which to me are as blatant as the sun.
    But all of my discoveries must be regarded as hypothetical, however intuitively appealing and functional they may be.   You can't do math my way and I can't do it your way.  But that doesn't mean we can't learn from each other.  I can give you major hints as to where to look. 
    These new notions of mine are fundamental, but very simple, and I believe plain english will do, if you will simply think it through and meet me halfway. I guarantee it will be worth the effort.
    You need these perceptions and concepts; you don't even recognize the 3 dimensions of valence.
    You recognize only the most literal type of polytopes, those which are both positive and real,  but there are 3 other completely independent ranges of polytope which are present in shapes in a synergetically simulated form.
    And thus you miss a fundamental fact of mathematical nature:
    Polytopes invariably occur in pairs of consecutive dimensions. 
-- with the single caveat that one must recognize the full range of polytopes, and so far you do not.
    Topologists have long recognized this pairing as a "tendency" of shape, but in the conventional view the pairing is sometimes broken, producing the "characteristic number of a surface".  
    However, whenever that pairing seems to be broken, the entire shape in fact takes over the role of the missing polytope by means of a sort of natural simulation.  If a line is missing, the entire shape will actually look like a line.  This is what I would call an "imaginary line".  
    A negative polytope on the other hand is a rather literal hole in a shape, and its level is the level required to plug the hole, so a hole in a disk is a negative plane for instance
    So, Polytopes -- the points, lines, polygons, polyhedra, etcetera, of which shapes are composed -- can actually be Negative or Imaginary, just as numbers can.  
    Negative and/or Imaginary polytopes appear in a shape as less tangible "overall" features of the shape, which describe various aspects its total connectivity, or dis-connectivity in the negative case.  They are often quite visible, in a fuzzy sort of way, appearing in their most blatant forms in those shapes which appear to be of a different dimensionality than they actually are.  
    We see such forms daily -- a length of wire appears 1 dimensional -- a grain of dust appears 0 dimensional -- a sheet of paper appears 2 dimensional -- etcetera.  All despite the fact that these are fully 3D objects.  
    In the first 2 cases we have a shape simulating a polytope which is not actually there; the latter performs the opposite illusion in which a pair of faces collapse into a single plane, hiding a polytope which IS actually there, but as we shall see, this is really part of the same game of natural mirage.  
    Once one learns to distinguish these simulated polytopes properly, one finds that they are everywhere.  In fact, every Shape whatsoever has at least one simulated polytope, for the Null Polytope itself, which topologists acknowledge to be part of every shape, is the very tip of the iceberg of Simulated Polytopes.  
    For instance, a simple unfaceted hollow torus only has 1 Literal polytope: a Real Positive planar surface, which in turn defines a Real Negative spatial interior.  And those 2 Real polytopes, by their unusual connectivities and dis-connectivities, define 5 distinct Imaginary Polytopes (2 lines, 1 point, and 2 planes) in addition to the Null Polytope, which is a part of every shape, just as the Null Set is part of every set.
    You will find more on this topic in the section labeled "Simulated Polytopes", along with information on the related concept of Levels of Form, and alternative to "dimension" as such, which is essential to a proper understanding of the full range of polytopes.  

    Another fundamental pattern that I would really like to promote is that of Geometric Rounding -- an alternative way of describing Angle, and of describing certain other aspects of shape which Angle cannot quite reach.  The basic idea is to look at the space right next to the shape, rather than the shape itself, rather as if one were shrink-wrapping it.  In this view all angles are transformed into curves, and mapped on to circles or spheres or hyperspheres as the case may be. 
    And by using circles and spheres and hyperspheres as the natural units of angle - or rather Rounding, one discovers an astonishing translatability between dimensions.  Instead of having to use different units for 2D, 3D, 4D and so on, we can simply say that the circle equals the sphere equals the hypersphere (etc.) and, amazingly, it works: we can translate freely from 2D to 3D to 4D and on, and back, without having to worry about Pi and such, and only translate into conventional units when the calculation is complete.
    This provides an incredibly versatile and intuitive way of thinking about shapes, as you will see.
    Finally, the section on Valence Notation presents the very useful concept of Topological Valence.   When one describes a cube as {4,3} in a schlafly symbol, the 4 and the 3 are the "order" of the face and the corner polytopes, respectively.  Such an "order" is a special type of valence, a circular valence.  But 'Valence' describes symmetry in general, so it can be circular, spherical, hyperspherical, and so on, and also includes the root case known as "parallel".  And when we see the entire range of possible valences for all dimensions, we find that that range itself has 3 dimensions, for each "valence" is comprised of 3 levels.
    And a proper understanding of Valence leads to a special notation system which allows one to express certain fundamental properties of topology in equation form, and derive many other interesting equations from them.

     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. I wish you luck.     I myself am reasonably certain that, however murkily expressed they may be, the methods themselves are valid, as far as they go, because I've used them a long time and tested them thoroughly enough to be quite convinced that they are at least functional and consistent. 
      And, last but not least, I am sincerely thankful for those many mathematicians, professional and otherwise, who have been open-minded enough to encourage me.







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).)