Super-elegant Space

Super-elegant Space L. Edgar Otto April 20, 2001

The generalization of our ideas of space and dimension as represented in claims for a standard theory or a unified theory can be made clear by a deeper view of elements in a space array where the colors, labels, and the arithmetic with relevance of operations sensible between them have logical topological structure models in which our normal space is embedded within. This space is best seen by alphanumeric's and the interpretation of the arrangement of elements matches closely natural principles of physics.

[ I call this space, grounded in a certain order of binary cells in a "quasic" plane "elegant" in honor of John Horton Conway who remarked to me upon my giving him an arrangement of labels of five matching things on the faces of a dodecahedron which was my solution as an inversion thru the center rather than just a few possibilities of surface solutions he deduced. His comment to me was that it was very elegant. I do not know if he ever explored what he called the non-standard arrangement of the "Conway Matrix" of which I now propose a general space for all such solutions, a space with coherence and to some level of understanding our looked for unification. I regard Conway as the greatest mathematician in our time. ]

The Conway matrix is a 6 x 6 array embedded in an 8 x 8 array. In the binary ordering as this procedure can be viewed as in superelegant space as four dimensions embedded in five in a geometric analogy to the matrix of 6 matching faces of the color cubes of Malhon in the 6 x 6 array as three dimensions embedded in four where the 28 surrounding cells are permutations and crossovers in four space.

As it involves 5 and ten fold symmetry we can demonstrate the coherence of the matrix relations which are orientation and matching in all directions considered of three and four axes of space, we can use the decimal base numbers as well.

Thus for the array we have these decimal integers. I begin at 1 in the upper left hand corner but it does not matter except that with the compliment at the lower left end of the cell it sums to 65. We can begin at zero, or for that matter there are cases where some pattern intrinsic to integers are seen if we begin by a prime or if we extend the notation at that first cell into some negative integer directions.

[see illustration above]

In the illustration above we observe there are four quadrants of integers. In one of the quadrants, the upper right, we have nine numbers and these can be added as compliments together to find a number that fits the quadrant. this gives four other numbers adding across the sub-quadrants to 102 as complimentary compliments.

These numbers I have labeled with my own matrix notation by alphanumeric letters that describe pairs of complimentary colors, thus as with Conway we exclude from the 36 the main diagonal that gives the cells of pairs of the same color. This leaves the 30 cubes and the problem is to match them in three space as well as Conway has done in two space. This is a hard and super-elegant problem.

But as the quasic plane already arranges the sub-elements, the points and edges, faces and volumes, of orthogons, we should not be surprised that such a more generalized and higher dimensionally interpreted grid will simply this problem.

Where then I label these cells in the 3 x 3 quadrant and give them a three axis name like LHC DIJ GKE and so on, we find that the sum of these labels for the six possible cases (and eventually the other nine) sum to 76.

Reading across a quasic matrix those things in a line, either in the up or down direction, represent the relation of vectors of a symmetrical space system or its relations. In this sense the significant number is 24 for three and four space. It is also the number of orientations of Malhoons color matching square of four colors. I is also the number of 2D faces in a hypercube.

Reading across the grid [in my illustration of yesterday as a series of red dots in an array] I observe that there are five standard alphanumeric label matrices to consider- in fact there are 15 or 30 of them in general reading the totality of the plane in a plane and each element a unique thing, beginning at triality such that the diagonals, both of them, of the plane have spatial and computational significance. Three of the labels are in a sense excluded as only used once and not in pairs leaving the arrangements of 12 things.

The reading across the superelegant (or hyperconway) array gives the unique conditions for a positive orientation of color objects each a unique state. Of course if we take one of the six colors of these five hyper arrays as the center we establish one of the five (or 15) standard Conway arrays. In this extension from simplicity we may begin to realize that things like the number of cubes or tetrahedra or rhombuses across polyhedra of icosahedral symmetry goes a little deeper in our conception beginning with asking what exactly a cube is and is it the most fundamental of structures. [This, with all such questions on the foundation of physics, is not an obvious one to ask and harder to find proof and acceptance by those who so fundamentally cannot understand, or may understand a different tradition.]

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[ A Bit of Philosophy: Of Vertigo and Algorithmic Man this part of the post also informal between intrusive brackets and partly inspired by Ullas post today.] [ . . .

1* One amazing thing about the Omnic view as to the improbable preservation and uniqueness of the soul, about the spirit of philosophy and science that unearths and grounds it in a totality, is that a direct proof of God or gods is not needed for the mechanisms (where one might conclude or believe pre-set being as intelligent design) in reality that precipitates out, records, such structures. But it is still in our evolved core superstitions and intuitions dwells our rituals for the dead and for souls. There may well be, insisting on some faith or warring philosophy that not the best of character we may take with us in the here or hearafter.

2* A finite intelligible accounting does not necesssarily distinguish the depth or span but can arrange structure with some distinctions such as partition theory (that is of teleoscoping quasic structures information theory -wise.)

3* The complexity of the new physics and possibly the generalization to an omnic background while vaster in relation to organism encoding even more than we now suspect is a diversity manageable by the new physics as far as I can see for now.

4* The relation of notions of physical theories have parallels as if they too a quasic matrix or superelegant space of their intelligible connections.

5* A hCw matrix encodes 5D axes as 3D in 8 dimensions of vector variance. Is it the Pentality of Coxeters delta honeycombs and general orientations not broken but are 5-cyclic over some superelegant linear array of all the possible orientations? Would their then not be a further generalization of such flat honeycombs?

6* 25x30 = 750, 750-720 = 30, (6!+ 300)/6 = 5^3 ; 4 x 136 = 2^9, 2^5 = 4^2.5 ;

7* The integer weights of some crossovers of integers are invariant (quasi-partitions)

This second, informal creative philosophy section came before the more formal parts posted above- also another attempt for a song as a break called Through Blinds, Glass-ly.

So, the EILNO stands out against the 10 of A BF CGM DHJK by which around the five we can generate the 15 color cubes, this where the flat 45 or 90 orientations come up on two space in three space as if we distinguish the z axis but do not know which to color in a standard system before hand. That is for example: HCADB in an array with EILNO then JKMFG giving HEJ CIK ALM DNF BOG but these shifted such that instead of say ALM we set AJO or ANK for the 45 for such a 3 x 5 set of axial arrangements.

This found by considering "counter clockwise complementarity of reversed color sequence spectra to encode and place the letter labels giving a reduction in three space over one letter like E red violet for the face pairs AJO AND BIM ;
BHN and CHL where the significant reduced 3 x 3 matrix is FEM CHL OJA, that is the other way FCO EHJ and MLA to choose one of the 5 or 15 from superelegant space matrices.

Interestingly one can discover the triplification of natural dimensions either from a higher or lower view of the natural dimension in regards to surfaces and volumes.

These ideas preceded the more formal parts of the post above.

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