Placeholder Substructures
The Road from NKS to SmallWorld, ScaleFree Networks is Paved with ZeroDivisors (and a “New Kind of Number Theory”)
Robert de Marrais
Rightshift the bitstring representation of any integer S > 8 and not a “positional prime” (power of 2) three times. The result implies a fractal of unique dimensional signature, each of whose infinite points is linked, in turn, to three integers R, C, P all different from S, each the XOR of the others. Making such “metafractals” from suitable Ss is simple. Start with Ss’ highest ON bit and move right, using one rule, once per ON. For a square emanation table (ET), whose infinitely longedged limitcase is the fractal in question, each cell not yet “cooked” by the rule is left blank (evennumbered use) or filled (first and all odd uses), if it conforms to this “récipé”: (R or C or P = (S or 0)) mod 2K.
Here, R and C are Row and Column labels of the “spreadsheet” whose cells, when not blank, contain R and C’s XOR product, P. S, as nonzero “strut constant,” is the signature of zerodivisor (ZD) ensembles, in CayleyDickson processgenerated extensions of the imaginaries to 2N dimensions: quaternions in 4; octonions in 8; sedenions in 16, where ZDs first emerge. K is the power of 2 for a given ONbit placeholder. Cells on long diagonals (where R = C; or R, C are “strut opposites”: i.e., R XOR C = S) are always blank, so applying the rule to the first ON means “fill.” If the last line of the récipé is a fill as well, then all the whitespace remaining, whose cells haven’t been interpreted by the récipé, are left blank. For a hide finale, the reverse obtains: all whitespace is filled by its P values. (See text and graphics on page 19, plus the sourcecode appendix, of http://arxiv.org/ftp/math/papers/0603/0603281.pdf, for a “doityourself kit” in ET carpentry.) If P is filled, the ZDs that R and C index are mutual ZDs, making 0 via (+ P – P) “pair production”: we say R and C emanate P (index of a third ZD, mutual with R and C).
Infinite spreadsheets are built by ET redoublings, with edge length (2N1 – 2) for each 2Nion box in a nested sequence, with N = 4 as the simplest starter kit. Here, a septet of 6 x 6 ET multiplication tables each with a different S < 8, and 0 < (R,C<>S) < 8, display interactions among ZDs. Hiding long diagonals leaves 24 filled cells, one for each oriented edgeflow on an octahedral 6vertex figure or “box kite” (BK). 7 BKs partition all 6 x 7 = 42 primitive sedenion ZDs. For N = 5 (pathions), a 14 x 14 box houses 168 filled cells (an interlaced 7BK ensemble) when S < 8, but only 72 (3 BKs) when 8 < S < 16: “carrybit overflow” triggers a “redoubling explosion.” The simplest metafractal “sky” (socalled since box kites fly in it) emerges in this context. Thanks to ZDs having “pathologized” the study of higher imaginaries (much like the “monsters” who became Mandelbrot’s pets scared off researchers in real analysis), this is also the context, ironically, of the first hypercomplex numbers never to be given a proper name. (Until lately!)
Since fractal dimension is fixed by the ONs in S’s bitstring, and chaotic attractors are ensembles of fractals, it is possible to contemplate transformations between different modes of chaos as completely determined by cellularautomatontype rules. Corollarily, the interplay of CA and ZD theories provides a numbertheoretic, even “Fourier series”like, basis for smallworld, scalefree networks. (And thence, network recursiveness?)
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