Keep it simple …

1. Placeholder Substructures:The Road from NKS to Small-World, Scale-Free Networks is Paved with Zero-Divisors (and a “New Kind of Number Theory”) Robert de MarraisNKS 2006 WolframScience Conference – June 17

3. Keep it simple …

4. … and keep it stupid!

5. (Eddie Van Halen on the secret of his success)

6. Complex (scale-free, small-worlds) networks are best comprehended as a side-effect of NKN (a new kind of Number Theory) which is … Based not on primes (Quantity), but bit-strings (Position). The role of primes is taken by powers of 2 (irreducible bits in “prime positions,” instead of “prime numbers”) All integers > 8 and not powers of 2 have bit-strings which can each uniquely represent a “meta-fractal,” which we’ll call a “SKY” Integers thus construed are called “strut-constants,” of ensembles of “Zero-Divisors.” The argument, simply put:

7. Now for the “stupid” part: • Zero-divisors (ZD’s) are to singularities (nested, hierarchical, invisible, yet unfoldable by morphogenesis) … • … what cycles of transformations are to groups (heat into steam into electricity into keeping this slide-show running, say…). • As we trace edges of a zero-divisor ensemble, we keep reverting not to an “identity,” but to “invisibility” (“Nobody here but us chickens”): for triangle of ZD nodes ABC, A*B = B*C = C*A = 0. • “I see your point” means a whole argument indicated by a pronoun: a Zero “place-holder” with indefinitely large (and likely nested) substructure. ( “Point well taken!” ) • An ensemble, that is, of Zero-Divisors, whose “atom” flies under the stupid name of “Box-Kite” (which flies in meta-fractal “Skies”)

8. The secret of our success? • Starting with N=4, ZD’s emerge in 16-D; the simplest Sky in which Box-Kites fly in (infinite-dimensional) fractals emerges in 32-D. • Hurwitz’s 1899 proof showed that generalizations of the Reals, to Imaginaries, Quaternions, then Octonions, by the Cayley-Dickson Process of dimension-doubling (CDP), inevitably led to Zero-Divisors (in the 16-D Sedenions) • Fields no longer could be defined, and metrics broke. (Oh my!) • So (as with the “monsters” of analysis, turned into fractal “pets” by Mandelbrot), everybody ran away screaming, and never even gave a name to the 32-D CDP numbers • But these 32-D “Pathions” (as in “pathological”) are where meta-fractal skies begin to open up! (Moral: if you want to fly a box-kite, run toward turbulence! Point your guitar into the amplifier, Eddie!)

9. Vents, Sails, and Box-Kites This is an (octahedral) Box-Kite: its 8 triangles comprise 4 Sails (shaded), made of mylar maybe, and 4 Vents through which the wind blows. Tracing an edge along a Sail multiplies the 2 ZD’s at its ends, making zero. Only ZD’s at opposite ends of a Strut (one of the 3 wooden or plastic dowels giving the Box-Kite structure) do NOT zero-divide each other.

10. Vents, Sails, and Box-Kites The strut constant (S) is the “missing Octonion”: in the 16-D Sedenions, where Box-Kites first show up, the vertices each take 2 integers, L less than the CDP “generator” (G) of the Sedenions from the Octonions (23 = 8), and U greater than it (and <> G + L). There being but 6 vertices, one Octonion must go AWOL, in one of 7 ways. Hence, there are 7 Box-Kites in the Sedenions. But 7 * 6 = 42 Assessors (the planes whose diagonals are ZD’s!)

11. Vents, Sails, and Box-Kites It’s not obvious that being missing makes it important, but one of the great surprises is the fundamental role the AWOL Octonion, or strut constant, plays. Along all 3 struts, the XOR of the opposite terms’ low-index numbers = S(which is why, graphically, you can’t trace a path for “making zero” between them!). Also, given the low-index term L at a vertex, its high-index partner = G + (L xor S): S and G, in other words, determine everything else!

12. A different view, with numbers too! Arbitrarily label the vertices of one Sail A, B, C (the “Zigzag”). Label the vertices of its strut-opposite Vent F, E, D respectively. The L-indices of each Sail form an Octonion triple, or Q-copy, since such triples are isomorphic to the Quaternions. But the L-index at one vertex also makes a Q-copy with the H-indices of its “Sailing partners.” Using lower- and upper-case letters, we can write, e.g., (a,b,c); (a,B,C); (A,b,C); (A,B,c ) for the Zigzag’s Q-copies. And similarly, for the other 3 “Trefoil” Sails.

13. A different view, with numbers too! Note the edges of the Zigzag and the Vent opposite it are red, while the other 6 edges are blue. If the edge is red, then the ZD’s joined by it “make zero” by multiplying ‘/’ with ‘\’: for S=1, in the Zigzag Sail ABC, the first product of its 6-cyle {/ \ / \ / \} is (i3 + i10)*(i6 – i15) = (i3 – i10)*(i6 + i15) = A*B = {+ C – C} = 0 For a blue edge, ‘/’*’/’ or ‘\’*’\’ make 0 instead: again for S=1, in Trefoil Sail ADE, the first product of its 6-cycle { / / / \ \ \ } is (i3 + i10)*(i4 + i13) = (i3 – i10)*(i4 – i13) = A*D = {+ E – E} = 0

14. A different view, with numbers too! One surprisingly deep aspect among many in this simple structure: the route to fractals is already in evidence! The 4 Q-copies in a Sail split into 1 “pure” Octonion triple and 3 “mixed” triples of 1 Octonion + 2 Sedenions; the 4 Sails also split: into one with 3 “red” edges, and 3 with 1 “red,” 2 “blue.” Implication: the Box-Kite’s structure can graph the substructure of a Sail’s Q-copies – which is not an empty execise! Why? Take the Zigzag’s (A,a); (B,b); (C,c) Assessors and imagine them agitated or “boiled” until they split apart. Send L and U terms to strut-opposite positions, then let them “catch” higher 32-D terms, with a higher-order G=32 instead of 16. We are now in the Pathions – the on-ramp to the Metafractal Highway!

15. Strut Opposites and Semiotic Squares René Thom’s disciple, Jean Petitot, has been translating the structures of literary and mythic theory – Algirdas Greimas’ “Semiotic Square,” Lévi-Strauss’ “Canonical Law of Myth” – into Catastrophe Theory models; here, we translate these into Box-Kite strut-opposite logic: ZD “representation theory” as semiotics. From here, we’re off to Chaos! We’ve just one stop left: another “representation” of Box-Kite dynamics – the ZD “multiplication table” called an ET (for “Emanation Table”)

16. The Simplest (Sedenion) Emanation Tables For S=1 Box-Kite, put L-indices of the 6 vertices as labels of Rows and Columns of a ZD “multiplication table,” entering them in left-right (top-down) order, with smallest first, and its strut-opposite in the mirror-opposite slot: 2 xor 3 = 4 xor 5 = 6 xor 7 = 1 = S. If R and C don’t mutually zero-divide, leave cell (R,C) blank. Otherwise, enter the L-index of their emanation (the 3rd Assessor in their common Sail). (Oh, yeah: ignore the minus signs.)

17. 25-ion “Pléiades” S = 01

18. 25-ion “Pléiades” S = 02

19. 25-ion “Pléiades” S = 03

20. 25-ion “Pléiades” S = 04

21. 25-ion “Pléiades” S = 05

22. 25-ion “Pléiades” S = 06

23. 25-ion “Pléiades” S = 07

24. 25-ion “Atlas” S = 08

25. 25-ion “Sand Mandalas” S = 09

26. 25-ion “Sand Mandalas” S = 10

27. 25-ion “Sand Mandalas” S = 11

28. 25-ion “Sand Mandalas” S = 12

29. 25-ion “Sand Mandalas” S = 13

30. 25-ion “Sand Mandalas” S = 14

31. 25-ion “Sand Mandalas” S = 15

32. S = 09

33. S = 10

34. S = 11

35. S = 12

36. S = 13

37. S = 14

38. S = 15

39. S = 16

40. S = 17

41. S = 18

42. S = 19

43. S = 20

44. S = 21

45. S = 22

46. S = 23

47. S = 24

48. S = 25

49. S = 26

50. S = 27