Friday, July 30, 2010
Cube Representation of the Pentagonal Dipyramid
* * * [I think I made an error here in the illustration coloring the cubes but will check on them later because of tired eyes... but you have the gist of this method of symmetry classification. In a sense the use of color here is very abstract as if the cubes are an elaborate notation rather than something we feel we can physically hold and feel close to in our world- much like that era of crystallographers who always felt a kinship with even the ball and stick models on their desk.]
Cube Representation of the Pentagonal Dipyramid
Yesterday I looked for a better cube representation than I had which required imagining things in four space. (As if the pentagonal symmetry in three space were not hard enough to visualize, but I finally understood the significance of the drawings in Rowlands book.)
I computed the 121 bi and mono colors for labels of the axial pairs in the models of the five and ten stacked dimensions- which Rowlands points out may apply to DNA alternative stackings. Indeed, in 1967 after reading Klein's The Icosahedron and Equations of the Fifth Degree and having a small model I made of DNA hanging from the ceiling I immediately envisioned the flow along the structure as if an icosahedron and tried to relate the 20 sidedness- yet we also have to consider topological properties of the 24ness of which mapping into the cube with the five or ten tets and cubes possible seems to do.
One thing for sure is that the quintic spaces are not to be considered broken as symmetry nor as trivial. In a sense the linear group uniqueness that Rowlands intuits from the fact of three space plus one can be a more general property across many dimensions. I am not sure at this point if we in fact can go beyond the fifth degree but I imagine so.
The illustration above has some rather arbitrary colors for its spectrum. I will probably take a hint from Western music and assign the 11 (the broken symmetry is the folding symmetry of cubes btw but not in the sense the golden section has the monopoly on these ideas of symmetry breaking- rather quasi or semi-symmetry breaking.) one 12 note colors or perhaps 18 (24 counting semi compactified self duals) in the 16th dimension (roughly 17 + 1 as the types of centered and extended matrices are considered, and of course quasicity.)
Certainly there are uses for this modification and extension of group theories. It is not as complicated in our familiar world to sort out as one might think but the variations if useful are enormous. It has been a mental workout that only now begins to see new things as a measure of the value of a theory that the old ones seem simple in retrospect.
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Imagine further the dihedral groups (as Rowlands in his careful speculations said may be an important part of the picture in his book) A dihedron has two faces and no volume is does come from the possibilities of what numbers can apply to simply connected solids. But we can have a null dihedron with neither faces nor volume and maybe no points. But the permutation of the points are much like two interlaced and opposite tetrahedra. Now as the tetrahedron can break down into two circuits of handed right angles so can the analogy 5-cell or pentatope so be split into two circuits. Imagine then eight colors in a cube and one in the center and that center is a quasi singularity of three points. The cubes I propose today simplifies that four dimensional into three space picture.
We can imagine further a cube of five faces and the sixth a vacuum or nil dihedron and these two can come together much like two cubes in hyperspace for a ten sided structure. But the principle is that concrete monads can enclose vacuum windows as well as a concrete monad having no window but metaphysically all not it is the vacuum.
I also imagine in this proposed lithon that the points of the cube meet with different values, 5 and 5 at the apexes of the dipyramid, 3 and 3 and two sets of 4 and 4 for a total of 32 points reduced to 8.
Let us imagine also the general importance of 32 squared (as in the dreams of that number and structure moderated by Jung of the quantum physicist) and of Dirac. This is 1024 which for me is important for four space chess of two players. In this 8 natural or time-like dimensional space in the counting of the multicolor label 8x17^2 minus 17 divided by 15 equals 153 and that minus 17 is 136. 11 of course has a special place where things add or multiply in some sense of counting for the powers of it are 1 11 121 1331 and so on.
The ambiguity where in the extended matrices of abstractly dimensional numbers that does distinguish between evenness and oddness of dimensions centered or not, primary or virtual or not, in the sense of complex space not clear of the sign- there is a process by which we may not assume the differences in signs equal and a method to distinguish the structure of some of them where the differences approach unity of a theory and not the local assumption of say the broken symmetry of and beyond octonians.
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Dear Lubos of TheReferenceFrame blogspot:
I have been looking into some of these SUSY ideas on my pesla.blogspot lately. Clearly supersting theory implies SUSY.
I am much in agreement with you post. I find that the usual mathematics assumes some privileged position for some of the functions but this is not clearly sorted out as to how much physics itself is independent of something like string theories. Most certainly in computing the concrete privileged positions permutation of point like things with some assumption of order is required.