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 Marrais NKS 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. The argument, simply put: 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 well call a SKY Integers thus construed are called strut-constants, of ensembles of Zero-Divisors.

7. Now for the stupid part: Zero-divisors (ZDs) 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, ZDs emerge in 16-D; the simplest Sky in which Box-Kites fly in (infinite-dimensional) fractals emerges in 32-D. Hurwitzs 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 ZDs at its ends, making zero. Only ZDs 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 (2 3 = 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 ZDs!)

11. Vents, Sails, and Box-Kites Its 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 cant 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 Zigzags 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 ZDs joined by it make zero by multiplying / with \: for S=1, in the Zigzag Sail ABC, the first product of its 6-cyle {/ \ / \ / \} is (i 3 + i 10 )*(i 6 – i 15 ) = (i 3 – i 10 )*(i 6 + i 15 ) = 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 (i 3 + i 10 )*(i 4 + i 13 ) = (i 3 – i 10 )*(i 4 – i 13 ) = 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-Kites structure can graph the substructure of a Sails Q- copies – which is not an empty execise! Why? Take the Zigzags (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 Thoms 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, were off to Chaos! Weve 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 dont mutually zero-divide, leave cell (R,C) blank. Otherwise, enter the L-index of their emanation (the 3 rd Assessor in their common Sail). (Oh, yeah: ignore the minus signs.)

17. 2 5 -ion Pléiades S = 01

18. 2 5 -ion Pléiades S = 02

19. 2 5 -ion Pléiades S = 03

20. 2 5 -ion Pléiades S = 04

21. 2 5 -ion Pléiades S = 05

22. 2 5 -ion Pléiades S = 06

23. 2 5 -ion Pléiades S = 07

24. 2 5 -ion Atlas S = 08

25. 2 5 -ion Sand Mandalas S = 09

26. 2 5 -ion Sand Mandalas S = 10

27. 2 5 -ion Sand Mandalas S = 11

28. 2 5 -ion Sand Mandalas S = 12

29. 2 5 -ion Sand Mandalas S = 13

30. 2 5 -ion Sand Mandalas S = 14

31. 2 5 -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

51. S = 28

52. S = 29

53. S = 30

54. S = 31

55. Cantor Dust, Curdled The very same properties that cause Cantor discontinua to be viewed as pathological are indispensable in a model of intermittency. - Benoit Mandelbrot

56. Georg, Cousin Moritz says hello! Given Cantors Rosicrucian theology and the proximity of his cousin Moritz Cantor – at that time a leading expert in the geometry of Egyptian art (Cantor 1880) – it may be that Georg Cantor saw the ancient Egyptian representation of the lotus creation myth and derived inspiration from this African fractal for the Cantor set. – Ron Eglash, African Fractals

57. Ethiopian Processional Crosses (L-to-R => seed + 3 iterations) Instant Fractals Course (1 slides worth)

58. Limit-case: Cesàro Double-Sweep One of the simplest (and least efficient!) plane-filling fractals, its white-space complement is clearly approached by the S=15 meta-fractal Sky!

59. S=15 Sky emerges in 32-D Pathions 6 Sedenion Assessor-dyads of S = 7 Box-Kite SPLIT UP: L (index 8) units all become L-units (index < 16 = new G) in Pathion 3-BK ensemble with S = 15 (= 8+7 ), along with prior G (=8) & S (=7), which capture U-units (index > G) from ambient turbulence, resulting in 14 Pathion Assessors

60. 2 nd Nested Sky-Box emerges in 64-D: 30 blue-sky border cells, one per each new Assessor in the 2 6 -ions (Chingons) Prior iterations row and column LABELS become blue-sky CELLS! (with label-to-cell mirror-reversal in strut-opposite boundary walls). This iterations 30 ( = 2 [N-1] - 2, N = 6) row and column LABELS will in turn become blue-sky CELLS in the next, 62-cell-edged, iteration:

62. Scale-free Boogie-Woogie (CL)AIM: Sky meta-fractal gridworks are dynamic. Like Mondrians capturing of New York Citys creative bustle in canvas-fixed oils, complex net- works can be modeled by interlocking emanation tables small-worlds and scale-free givens!

63. SKY PILOT! A SKY is made of many layers, like the sheets of a Riemann surface in complex analysis. Navigate between sky-box-bounded spreadsheets like David Niven in Around the World in 80 Days:

64. SKY PILOT! Go up higher? Push G a bit to the left! (David drops a sand-bag over the edge, to same effect.) Drop down lower? Shift G to the right! (David vents hot air by pulling a cord: same thing.)

65. SKY PILOT! Remember: if you know S and G, the L-index is all you see in the ET, and all you need to see: the U-index is always just G + (L xor S) … with G an infinite constant in the meta-fractal limit-case! (Up, up and away !)

66. SKY as first fractal?

67. Cooking with Récipés Strut-Constant-Emanated Number Theory (SCENT) is the basis of R, C, Ps: simple formulas specifying the relations between Row and Column labels, and their XOR Products housed in the spreadsheet-like cells of Emanation Tables (ETs). For all S > 8 and not a power of 2, there exists a unique meta-fractal or Sky, whose ET has a simple algorithm. For any cell, consider the bit-representation of S; the cell is filled or empty (shows or hides P) depending upon a series of bits to the left tests, starting with the highest, and stopping at the lowest (if the 3 rightmost bits > 0) or next-to-lowest (if S = multiple of 8 and not a power of 2). Well build to the general case from the simplest ones.

68. The Simplest Flip Book Récipé Pathion Flip-Books obey this formula: R|C|P = S|0 mod 8 This (and all more complex formulas) always implicitly assume a zero th rule: All long-diagonal cells are empty (The \ has all R=C, so all P = R xor C = 0, and i 0 is the Real unit!) (The / has R and C as strut-opposites, which also cant mutually zero-divide, since P = R xor C = S … and S is suppressed in ETs). The Flip-Book Formula then says: after blanking out diagonals, fill all cells with Row or Column or cell-value Product equal to (S – 8), or 8 itself, then leave the rest empty. For 8 < S < 16, the 1 st and last rows and columns start off defining a square, then move toward each other (with Ps forming diagonals joining the central cross when S = 15), as S is incremented.

69. Add Parentheses, Get Balloon Rides For 8 < S < 16 in the 64-D Chingons, recursive structure emerges, based on the Pathion flip-book: R and C labels of the latter become bordering cell values along left and top edges (and mirrored on the other sides) of an exact copy of the Pathion ET. With each dimension doubling, the ETs edge tends toward doubling too (= 2 [N-1] – 2, for 2 N -ions); the count of filled rows and columns tends likewise ( #R = #C = 2 * {2 [N-4] – 1} ), with prior iterations labels mapped to current cell values along all skybox edges. And always, the formula is just what we saw above, but with parentheses added: (R|C|P = S|0 ) mod 8 That is: all multiples of 8 < G, and S modulo each, now appear: for S=15, this means {8,7} plus {16, 23; 24, 31} for 2 6 -ions (G=32); for 2 7 -ions (G=64), these plus {32, 39; 40, 47; 48, 55; 56, 63}; etc.

70. Next, Add Rules (one per hi-bit) For Chingons with S > 16, we get new behavior, best appreciated by contrasting the cases S = 24 and S = 25. As a bit-string, 24 = 11000, with 2 hi-bits; but, with no low-bits, the 8- bit is treated as one, yielding one fill rule (and an empty central box): ( R|C|P = 8 | 0 ) mod 16. But 25 = 11001, with 2 effective hi-bits, hence a fill rule (off the 16- bit) and a hide rule (off the 8-bit). As with cooking a stew, once you toss something in, you cant take it out: a cell, once a rule has been applied to it, can not have its hide/fill status changed by later rules. 24 ends with a fill rule, so all cells not ruled upon are left empty; but 25 ends with a hide rule, so all untouched cells are filled.

71. Canonical Récipés If S, as string, has hi-bits b 1,b 2,…,b k in L-to-R positions from 2 H to 2 L (L > 3): base a fill rule on all ON bits b i where i = odd; base a hide rule on all ON bits b j where j = even. If the last rule is hide, then fill all cells untouched by a rule; if the last rule is fill, then hide all cells untouched by a rule. For any hi-bit 2 A, the rule has form ( R|C|P = (S|0) ) mod 2 A, with all nominated cells filled or hidden according to case. To see recipes at work, the simplest abutment of 2-rule and 3-rule S values ( S = 56 and 57, respectively, in the 128-D 2 7 -ions, or Routions) are illustrated in stepwise detail in what follows.