The Road from NKS to Small-World, Scale-Free Networks is Paved with Zero-Divisors (and a “New Kind of Number Theory”)

Robert de Marrais

Right-shift the bit-string 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 “meta-fractals” 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 long-edged limit-case is the fractal in question, each cell not yet “cooked” by the rule is left blank (even-numbered 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 zero-divisor (ZD) ensembles, in Cayley-Dickson process-generated 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 ON-bit 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 source-code appendix, of http://arxiv.org/ftp/math/papers/0603/0603281.pdf, for a “do-it-yourself 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 (2N-1 – 2) for each 2N-ion 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 edge-flow on an octahedral 6-vertex 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 7-BK ensemble) when S < 8, but only 72 (3 BKs) when 8 < S < 16: “carry-bit overflow” triggers a “redoubling explosion.” The simplest meta-fractal “sky” (so-called 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 bit-string, and chaotic attractors are ensembles of fractals, it is possible to contemplate transformations between different modes of chaos as completely determined by cellular-automaton-type rules. Corollarily, the interplay of CA and ZD theories provides a number-theoretic, even “Fourier series”-like, basis for small-world, scale-free networks. (And thence, network recursiveness?)