Wednesday, January 9, 2013

Mirror-Shadow Polytopes

Mirror-Shadow Polytopes

L. Edgar Otto    08 January, 2013

We suspect that a cube is not a fundamental structure if we spin a cube model from a vertex to one diagonal to it.  The shape is not convex but seems bounded by half cones and between them we find a concave band. This vaguely reminds me of hyperbolic situations of which we may find the spinning object from the geometry as many try to picture the dynamics from an algebraic view that these concern absolute values of these abstract coordinates of zero and null singularities.

I found this a stretch or leap on the frontier of my intuition, of what it is to make of the shock wave front of physics.  I feel strongly the cycle of five things, a recursion of duality of sorts, will be an essential part of this picture, the idea on many scales and compounded refinement for actual measurement over many scales, and relatively fixed scales as in our dimensionless constants- that is, the question and how it was looked at from concrete measure by Dirac and the answer as suggested by Eddington.  If this sort of vector algebra does reflect QFT, then in these matters we observe and suspect pattern analogies of nuclei.

I color coded the division of a standard dodecahedron over its subcells in the count as presented by the soon abandoned choice of colors in the Paint program.  But two sequences I use here... one the color spectrum of the primaries plus cyan and magenta and the other the full range of the paint colors with the exclusion of black and white.  Of course I could have divided the spectrum more elegantly so as to find the true inverse compliments of colors, the essential pattern of 24 and so on from a relative quasifinite view.  I note also the dodecahedron was one of the finite universe models still in the running not long ago.

Figures A, B, and C uses the part of the paint spectrum for those counts of 11, 14, 24 or even 36 in sub-brane sets as sub-branes (My Otto-Conway types of general matrices).
Excluding black and white we use 26 of the paint colors to describe this mirror-shadow dodecahedron object in its depth.  The orders and the placement of colors in all the figures are in the main arbitrary at this time in the examples.

I find it surprising that the dodecahedron can be seen as a representation of a hypercube. Is this something I have missed in my musings and could be a lack in my studies?  In this representation I would have thought the icosahedron as closer to the general structures but new concepts for me at least concerning duality suggests a focusing or summing up of things into the patterns of interactive and adjacent dimensions.  It is in these abstract brane-like differences that we may begin to define mensuration where it appears concrete as physicality so to shore up what may be in the teleological box of theories and not lost into Churchman's so called fourth box.  What is the nature then of a fifth and greater box?  Can there be analogs after all to these Platonic like polytopes beyond five fold spaces?  I have often, if naivety thought so or tried to see why it not so.

Perhaps the 24 dimensional patterns, that unique polytope with no analogs in higher or lower dimensions in my quasifinite universe, it self dual, holds more foundational surprises still, otherwise why the surprise in the higher arithmetical patterns- and why indeed in this way of counting things were certain patterns of internal equilateral triangles overlooked by angles of three? Why years after I found a corner solution to the queen's chess problem it came out in the Journal of Recreational Mathematics with a side comment wonder why no one had noticed these groups before?

Creativity in itself as part of an inquiring system can be a state of attention and concentration in regards to the system of logic say in playing chess with the machine. In this sober and casual stance, avoiding blunders or shifts of positions of general strategy I was able to bring the machine set to maximum difficulty to a draw after a few tries of feeling what was programmed (the scientific part of the game it is said) in the end game.  In a world of nonnecessary causality there is still a place for sober reasoning.  Such a state of concentration is an ongoing form of creativity itself, and a sensible relation to emotions.

Still, in the case where chips learn, first by reversals to mirror from other chips then the unique reversal, a certain intelligence in design seems given outside our usual definitions as human.  If the chips can design themselves is it not remarkable it can function without clocks and may depend on isolated transistors?  So why did the string theorists abandon the idea of tachyons?  What sense of reductionism as sanity shuns the idea of the teleological.  In my teleoscope of patterns most easily seen in two space these were binary after all- we can measure things and explain them in all their complexity with the usual tool of but statistics alone- that is if we wish to grasp the nature of foundations explicitly of what we call particle physics.

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