Keep it simple …

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

Keep it simple …

… and keep it stupid!

(Eddie Van Halen on the secret of his success)

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 we’ll call a “SKY”

• Integers thus construed are called “strut-constants,” of ensembles of “Zero-Divisors.”

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”)

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!)

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.

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!)

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!

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.

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

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!

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”)

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.)

2 4 6 7 5 32 6 -4 5 -74 6 -2 3 -76 -4 -2 3 57 5 3 -2 -45 -7 3 -2 63 -7 5 -4 6

25-ion “Pléiades”

S = 01

S = 02

S = 03

S = 04

S = 05

S = 06

S = 07

25-ion “Atlas”

S = 08

25-ion “Sand Mandalas”

S = 09

S = 10

S = 11

S = 12

S = 13

S = 14

S = 15

S = 09

S = 10

S = 11

S = 12

S = 13

S = 14

S = 15

S = 16

S = 17

S = 18

S = 19

S = 20

S = 21

S = 22

S = 23

S = 24

S = 25

S = 26

S = 27

S = 28

S = 29

S = 30

S = 31

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

Georg, Cousin Moritz says hello!

“Given Cantor’s 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

Ethiopian Processional Crosses (L-to-R => seed + 3 iterations)

Instant Fractals Course (1 slide’s worth)

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!

-1 2 3 -4 -5 6 -7 8 -9 10 11 -12 -13 14-1 -7 6 -9 8 142 -7 -5 10 8 -133 -7 -4 11 8 -12

-4 -7 3 -12 8 11-5 -7 2 -13 8 106 -7 -1 14 8 -9

-7 6 -5 -4 3 2 -1 14 -13 -12 11 10 -9 88 -9 10 11 -12 -13 14 -1 2 3 -4 -5 6 -7

-9 8 14 -1 -7 610 8 -13 2 -7 -511 8 -12 3 -7 -4

-12 8 11 -4 -7 3-13 8 10 -5 -7 214 8 -9 6 -7 -1

14 -13 -12 11 10 -9 8 -7 6 -5 -4 3 2 -1

S=15 Sky emerges

in 32-D Pathions

6 Sedenion Assessor-dyads of S = 7 Box-Kite SPLIT UP:L (index < 8) and U (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

-1 2 3 -4 -5 6 -7 -16 17 -18 -19 20 21 -22 23 -24 25 -26 -27 28 29 -30 31 8 -9 10 11 -12 -13 14-1 -7 6 17 -16 -23 22 -25 24 31 -30 -9 8 142 -7 -5 -18 -16 -23 -21 26 24 31 29 10 8 -133 -7 -4 -19 -16 -23 -20 27 24 31 28 11 8 -12

-4 -7 3 20 -23 -16 19 -28 31 24 -27 -12 8 11-5 -7 2 21 -23 -16 18 -29 31 24 -26 -13 8 106 -7 -1 -22 -23 -16 -17 30 31 24 25 14 8 -9

-7 6 -5 -4 3 2 -1 23 22 -21 -20 19 18 -17 -16 31 30 -29 -28 27 26 -25 -24 14 -13 -12 11 10 -9 8-16 17 -18 -19 20 21 -22 23 -1 2 3 -4 -5 6 -7 8 -9 10 11 -12 -13 14 -24 25 -26 -27 28 29 -30 3117 -16 -23 22 -1 -7 6 -9 8 14 -25 24 31 -30

-18 -16 -23 -21 2 -7 -5 10 8 -13 26 24 31 29-19 -16 -23 -20 3 -7 -4 11 8 -12 27 24 31 2820 -23 -16 19 -4 -7 3 -12 8 11 -28 31 24 -2721 -23 -16 18 -5 -7 2 -13 8 10 -29 31 24 -26

-22 -23 -16 -17 6 -7 -1 14 8 -9 30 31 24 2523 22 -21 -20 19 18 -17 -16 -7 6 -5 -4 3 2 -1 14 -13 -12 11 10 -9 8 31 30 -29 -28 27 26 -25 -24

-24 -25 26 27 -28 -29 30 31 8 -9 10 11 -12 -13 14 -1 2 3 -4 -5 6 -7 -16 -17 18 19 -20 -21 22 2325 24 31 30 -9 8 14 -1 -7 6 -17 -16 -23 -22

-26 24 31 -29 10 8 -13 2 -7 -5 18 -16 -23 21-27 24 31 -28 11 8 -12 3 -7 -4 19 -16 -23 2028 31 24 27 -12 8 11 -4 -7 3 -20 -23 -16 -1929 31 24 26 -13 8 10 -5 -7 2 -21 -23 -16 -18

-30 31 24 -25 14 8 -9 6 -7 -1 22 -23 -16 1731 -30 29 28 -27 -26 25 -24 14 -13 -12 11 10 -9 8 -7 6 -5 -4 3 2 -1 23 -22 21 20 -19 -18 17 -16

8 -9 10 11 -12 -13 14 -24 -25 26 27 -28 -29 30 31 -16 -17 18 19 -20 -21 22 23 -1 2 3 -4 -5 6 -7-9 8 14 25 24 31 30 -17 -16 -23 -22 -1 -7 610 8 -13 -26 24 31 -29 18 -16 -23 21 2 -7 -511 8 -12 -27 24 31 -28 19 -16 -23 20 3 -7 -4

-12 8 11 28 31 24 27 -20 -23 -16 -19 -4 -7 3-13 8 10 29 31 24 26 -21 -23 -16 -18 -5 -7 214 8 -9 -30 31 24 -25 22 -23 -16 17 6 -7 -1

14 -13 -12 11 10 -9 8 31 -30 29 28 -27 -26 25 -24 23 -22 21 20 -19 -18 17 -16 -7 6 -5 -4 3 2 -1

2nd Nested “Sky-Box” emerges in 64-D:30 “blue-sky” border cells, one per each new

Assessor in the 26-ions (“Chingons”)

Prior iteration’s row and column LABELS become blue-sky CELLS! (with label-to-cell mirror-reversal in “strut-opposite” boundary walls). This iteration’s 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:

1 2 3 4 5 6 7 16 17 18 19 20 21 22 23 32 33 34 35 36 37 38 39 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 40 41 42 43 44 45 46 47 24 25 26 27 28 29 30 31 8 9 10 11 12 13 141 -7 6 17 -16 -23 22 33 -32 -39 38 49 -48 -55 54 -57 56 63 -62 -41 40 47 -46 -25 24 31 -30 -9 82 -7 -5 -18 -16 -23 -21 -34 -32 -39 -37 -50 -48 -55 -53 58 56 63 61 42 40 47 45 26 24 31 29 10 83 -7 -4 -19 -16 -23 -20 -35 -32 -39 -36 -51 -48 -55 -52 59 56 63 60 43 40 47 44 27 24 31 28 11 84 -7 3 20 -23 -16 19 36 -39 -32 35 52 -55 -48 51 -60 63 56 -59 -44 47 40 -43 -28 31 24 -27 -12 85 -7 2 21 -23 -16 18 37 -39 -32 34 53 -55 -48 50 -61 63 56 -58 -45 47 40 -42 -29 31 24 -26 -13 86 -7 -1 -22 -23 -16 -17 -38 -39 -32 -33 -54 -55 -48 -49 62 63 56 57 46 47 40 41 30 31 24 25 14 87 6 -5 -4 3 2 -1 23 22 -21 -20 19 18 -17 -16 39 38 -37 -36 35 34 -33 -32 55 54 -53 -52 51 50 -49 -48 63 62 -61 -60 59 58 -57 -56 47 46 -45 -44 43 42 -41 -40 31 30 -29 -28 27 26 -25 -24 14 -13 -12 11 10 -9

16 17 -18 -19 20 21 -22 23 -1 2 3 -4 -5 6 -7 48 49 -50 -51 52 53 -54 55 -32 -33 34 35 -36 -37 38 -39 40 -41 42 43 -44 -45 46 47 -56 57 -58 -59 60 61 -62 -63 8 -9 10 11 -12 -13 14 -24 25 -26 -27 28 29 -3017 -16 -23 22 -1 -7 6 -49 -48 -55 54 -33 -32 -39 38 -41 40 47 46 -57 56 63 62 -9 8 14 -25 24 3118 -16 -23 -21 2 -7 -5 50 -48 -55 -53 34 -32 -39 -37 42 40 47 -45 58 56 63 -61 10 8 -13 26 24 3119 -16 -23 -20 3 -7 -4 51 -48 -55 -52 35 -32 -39 -36 43 40 47 -44 59 56 63 -60 11 8 -12 27 24 3120 -23 -16 19 -4 -7 3 -52 -55 -48 51 -36 -39 -32 35 -44 47 40 43 -60 63 56 59 -12 8 11 -28 31 2421 -23 -16 18 -5 -7 2 -53 -55 -48 50 -37 -39 -32 34 -45 47 40 42 -61 63 56 58 -13 8 10 -29 31 2422 -23 -16 -17 6 -7 -1 54 -55 -48 -49 38 -39 -32 -33 46 47 40 -41 62 63 56 -57 14 8 -9 30 31 2423 22 -21 -20 19 18 -17 -16 -7 6 -5 -4 3 2 -1 -55 54 -53 -52 51 50 -49 -48 -39 38 -37 -36 35 34 -33 -32 47 46 -45 -44 43 42 -41 40 63 62 -61 -60 59 58 -57 56 14 -13 -12 11 10 -9 8 31 30 -29 -28 27 26 -2532 33 -34 -35 36 37 -38 39 48 -49 50 51 -52 -53 54 -55 -1 2 3 -4 -5 6 -7 -16 17 -18 -19 20 21 -22 23 -24 25 -26 -27 28 29 -30 31 8 -9 10 11 -12 -13 14 56 -57 58 59 -60 -61 62 -63 -40 41 -42 -43 44 45 -4633 -32 -39 38 49 -48 -55 54 -1 -7 6 17 -16 -23 22 -25 24 31 -30 -9 8 14 -57 56 63 -62 -41 40 4734 -32 -39 -37 -50 -48 -55 -53 2 -7 -5 -18 -16 -23 -21 26 24 31 29 10 8 -13 58 56 63 61 42 40 4735 -32 -39 -36 -51 -48 -55 -52 3 -7 -4 -19 -16 -23 -20 27 24 31 28 11 8 -12 59 56 63 60 43 40 4736 -39 -32 35 52 -55 -48 51 -4 -7 3 20 -23 -16 19 -28 31 24 -27 -12 8 11 -60 63 56 -59 -44 47 4037 -39 -32 34 53 -55 -48 50 -5 -7 2 21 -23 -16 18 -29 31 24 -26 -13 8 10 -61 63 56 -58 -45 47 4038 -39 -32 -33 -54 -55 -48 -49 6 -7 -1 -22 -23 -16 -17 30 31 24 25 14 8 -9 62 63 56 57 46 47 4039 38 -37 -36 35 34 -33 -32 55 54 -53 -52 51 50 -49 -48 -7 6 -5 -4 3 2 -1 23 22 -21 -20 19 18 -17 -16 31 30 -29 -28 27 26 -25 -24 14 -13 -12 11 10 -9 8 63 62 -61 -60 59 58 -57 -56 47 46 -45 -44 43 42 -4148 49 -50 -51 52 53 -54 55 -32 -33 34 35 -36 -37 38 -39 -16 17 -18 -19 20 21 -22 23 -1 2 3 -4 -5 6 -7 8 -9 10 11 -12 -13 14 -24 25 -26 -27 28 29 -30 31 40 -41 42 43 -44 -45 46 47 -56 57 -58 -59 60 61 -6249 -48 -55 54 -33 -32 -39 38 17 -16 -23 22 -1 -7 6 -9 8 14 -25 24 31 -30 -41 40 47 46 -57 56 6350 -48 -55 -53 34 -32 -39 -37 -18 -16 -23 -21 2 -7 -5 10 8 -13 26 24 31 29 42 40 47 -45 58 56 6351 -48 -55 -52 35 -32 -39 -36 -19 -16 -23 -20 3 -7 -4 11 8 -12 27 24 31 28 43 40 47 -44 59 56 6352 -55 -48 51 -36 -39 -32 35 20 -23 -16 19 -4 -7 3 -12 8 11 -28 31 24 -27 -44 47 40 43 -60 63 5653 -55 -48 50 -37 -39 -32 34 21 -23 -16 18 -5 -7 2 -13 8 10 -29 31 24 -26 -45 47 40 42 -61 63 5654 -55 -48 -49 38 -39 -32 -33 -22 -23 -16 -17 6 -7 -1 14 8 -9 30 31 24 25 46 47 40 -41 62 63 5655 54 -53 -52 51 50 -49 -48 -39 38 -37 -36 35 34 -33 -32 23 22 -21 -20 19 18 -17 -16 -7 6 -5 -4 3 2 -1 14 -13 -12 11 10 -9 8 31 30 -29 -28 27 26 -25 -24 47 46 -45 -44 43 42 -41 40 63 62 -61 -60 59 58 -5756 -57 58 59 -60 -61 62 63 40 -41 42 43 -44 -45 46 47 -24 -25 26 27 -28 -29 30 31 8 -9 10 11 -12 -13 14 -1 2 3 -4 -5 6 -7 -16 -17 18 19 -20 -21 22 23 -32 -33 34 35 -36 -37 38 -39 -48 -49 50 51 -52 -53 5457 56 63 62 -41 40 47 46 25 24 31 30 -9 8 14 -1 -7 6 -17 -16 -23 -22 -33 -32 -39 38 -49 -48 -5558 56 63 -61 42 40 47 -45 -26 24 31 -29 10 8 -13 2 -7 -5 18 -16 -23 21 34 -32 -39 -37 50 -48 -5559 56 63 -60 43 40 47 -44 -27 24 31 -28 11 8 -12 3 -7 -4 19 -16 -23 20 35 -32 -39 -36 51 -48 -5560 63 56 59 -44 47 40 43 28 31 24 27 -12 8 11 -4 -7 3 -20 -23 -16 -19 -36 -39 -32 35 -52 -55 -4861 63 56 58 -45 47 40 42 29 31 24 26 -13 8 10 -5 -7 2 -21 -23 -16 -18 -37 -39 -32 34 -53 -55 -4862 63 56 -57 46 47 40 -41 -30 31 24 -25 14 8 -9 6 -7 -1 22 -23 -16 17 38 -39 -32 -33 54 -55 -4863 -62 61 60 -59 -58 57 -56 47 46 -45 -44 43 42 -41 40 31 -30 29 28 -27 -26 25 -24 14 -13 -12 11 10 -9 8 -7 6 -5 -4 3 2 -1 23 -22 21 20 -19 -18 17 -16 -39 38 -37 -36 35 34 -33 -32 55 -54 53 52 -51 -50 4940 -41 42 43 -44 -45 46 47 -56 -57 58 59 -60 -61 62 63 8 -9 10 11 -12 -13 14 -24 -25 26 27 -28 -29 30 31 -16 -17 18 19 -20 -21 22 23 -1 2 3 -4 -5 6 -7 -48 -49 50 51 -52 -53 54 55 -32 -33 34 35 -36 -37 3841 40 47 46 57 56 63 62 -9 8 14 25 24 31 30 -17 -16 -23 -22 -1 -7 6 -49 -48 -55 -54 -33 -32 -3942 40 47 -45 -58 56 63 -61 10 8 -13 -26 24 31 -29 18 -16 -23 21 2 -7 -5 50 -48 -55 53 34 -32 -3943 40 47 -44 -59 56 63 -60 11 8 -12 -27 24 31 -28 19 -16 -23 20 3 -7 -4 51 -48 -55 52 35 -32 -3944 47 40 43 60 63 56 59 -12 8 11 28 31 24 27 -20 -23 -16 -19 -4 -7 3 -52 -55 -48 -51 -36 -39 -3245 47 40 42 61 63 56 58 -13 8 10 29 31 24 26 -21 -23 -16 -18 -5 -7 2 -53 -55 -48 -50 -37 -39 -3246 47 40 -41 -62 63 56 -57 14 8 -9 -30 31 24 -25 22 -23 -16 17 6 -7 -1 54 -55 -48 49 38 -39 -3247 -46 45 44 -43 -42 41 -40 -63 62 -61 -60 59 58 -57 56 14 -13 -12 11 10 -9 8 31 -30 29 28 -27 -26 25 -24 23 -22 21 20 -19 -18 17 -16 -7 6 -5 -4 3 2 -1 -55 54 -53 -52 51 50 -49 48 39 -38 37 36 -35 -34 3324 -25 26 27 -28 -29 30 31 8 -9 10 11 -12 -13 14 56 -57 58 59 -60 -61 62 63 40 -41 42 43 -44 -45 46 47 -32 -33 34 35 -36 -37 38 -39 -48 -49 50 51 -52 -53 54 -55 -1 2 3 -4 -5 6 -7 -16 -17 18 19 -20 -21 2225 24 31 30 -9 8 14 -57 56 63 62 -41 40 47 46 -33 -32 -39 38 -49 -48 -55 54 -1 -7 6 -17 -16 -2326 24 31 -29 10 8 -13 58 56 63 -61 42 40 47 -45 34 -32 -39 -37 50 -48 -55 -53 2 -7 -5 18 -16 -2327 24 31 -28 11 8 -12 59 56 63 -60 43 40 47 -44 35 -32 -39 -36 51 -48 -55 -52 3 -7 -4 19 -16 -2328 31 24 27 -12 8 11 -60 63 56 59 -44 47 40 43 -36 -39 -32 35 -52 -55 -48 51 -4 -7 3 -20 -23 -1629 31 24 26 -13 8 10 -61 63 56 58 -45 47 40 42 -37 -39 -32 34 -53 -55 -48 50 -5 -7 2 -21 -23 -1630 31 24 -25 14 8 -9 62 63 56 -57 46 47 40 -41 38 -39 -32 -33 54 -55 -48 -49 6 -7 -1 22 -23 -1631 -30 29 28 -27 -26 25 -24 14 -13 -12 11 10 -9 8 -63 -62 61 60 -59 -58 57 -56 47 46 -45 -44 43 42 -41 40 -39 38 -37 -36 35 34 -33 -32 55 -54 53 52 -51 -50 49 48 -7 6 -5 -4 3 2 -1 23 -22 21 20 -19 -18 17

8 -9 10 11 -12 -13 14 -24 -25 26 27 -28 -29 30 31 -40 -41 42 43 -44 -45 46 47 -56 -57 58 59 -60 -61 62 63 -48 -49 50 51 -52 -53 54 55 -32 -33 34 35 -36 -37 38 39 -16 -17 18 19 -20 -21 22 23 -1 2 3 -4 -5 69 8 14 25 24 31 30 41 40 47 46 57 56 63 62 -49 -48 -55 -54 -33 -32 -39 -38 -17 -16 -23 -22 -1 -7

10 8 -13 -26 24 31 -29 -42 40 47 -45 -58 56 63 -61 50 -48 -55 53 34 -32 -39 37 18 -16 -23 21 2 -711 8 -12 -27 24 31 -28 -43 40 47 -44 -59 56 63 -60 51 -48 -55 52 35 -32 -39 36 19 -16 -23 20 3 -712 8 11 28 31 24 27 44 47 40 43 60 63 56 59 -52 -55 -48 -51 -36 -39 -32 -35 -20 -23 -16 -19 -4 -713 8 10 29 31 24 26 45 47 40 42 61 63 56 58 -53 -55 -48 -50 -37 -39 -32 -34 -21 -23 -16 -18 -5 -714 8 -9 -30 31 24 -25 -46 47 40 -41 -62 63 56 -57 54 -55 -48 49 38 -39 -32 33 22 -23 -16 17 6 -7

Scale-free Boogie-Woogie

(CL)AIM: Sky meta-fractal gridworks are dynamic.Like Mondrian’s capturing of New York City’s

creative bustle in canvas-fixed oils, “complex net-works” can be modeled by interlocking emanation

tables’ “small-worlds” and “scale-free” givens!

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:

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.)

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!)

SKY as first fractal?

Cooking with Récipés

• Strut-Constant-Emanated Number Theory (SCENT) is the basis of R, C, P’s: simple formulas specifying the relations between Row and Column labels, and their XOR Products housed in the spreadsheet-like cells of Emanation Tables (ET’s).

• 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).

• We’ll build to the general case from the simplest ones.

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

“zeroth rule”: All long-diagonal cells are empty

• (The ‘\’ has all R=C, so all P = R xor C = 0, and i0 is the Real unit!)

• (The ‘/’ has R and C as strut-opposites, which also can’t 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 1st and last rows and columns start off defining a square, then move toward each other (with P’s forming diagonals joining the central cross when S = 15), as S is incremented.

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 ET’s edge tends toward doubling too (= 2[N-1] – 2, for 2N-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 26-ions (G=32); for 27-ions (G=64), these plus {32, 39; 40, 47; 48, 55; 56, 63}; etc.

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 can’t 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.

Canonical Récipés

If S, as string, has hi-bits b1,b2,…,bk in L-to-R positions from 2H to 2L (L > 3):

• base a fill rule on all “ON” bits bi where i = odd;

• base a hide rule on all “ON” bits bj 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 2A, the rule has form ( R|C|P = (S|0) ) mod 2A,

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 27-ions, or “Routions”) are illustrated in stepwise detail in what follows.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 39 38 37 36 35 34 33 32 47 46 45 44 43 42 41 40 55 54 53 52 51 50 49 48 63 62 61 60 59 58 571 3 2 5 4 7 6 -11 -10 -13 -12 -15 -14 -19 -18 -21 -20 -23 -22 -25 -24 27 26 29 28 31 30 -38 -39 -36 -37 -34 -35 32 33 46 47 44 45 42 43 54 55 52 53 50 51 -62 -63 -60 -61 -58 -592 3 1 6 7 4 5 -11 -9 -14 -15 -12 -13 -19 -17 -22 -23 -20 -21 -26 27 -24 25 30 31 28 29 -37 -36 -39 -38 -33 32 -35 34 45 44 47 46 41 43 53 52 55 54 49 51 -61 -60 -63 -62 -57 -593 2 1 7 6 5 4 -10 -9 -15 -14 -13 -12 -18 -17 -23 -22 -21 -20 -27 26 25 -24 31 30 29 28 -36 -37 -38 -39 32 -33 -34 35 44 45 46 47 41 42 52 53 54 55 49 50 -60 -61 -62 -63 -57 -584 5 6 7 1 2 3 -13 -14 -15 -9 -10 -11 -21 -22 -23 -17 -18 -19 -28 29 30 31 -24 25 26 27 -35 -34 -33 32 -39 -38 -37 36 43 42 41 47 46 45 51 50 49 55 54 53 -59 -58 -57 -63 -62 -615 4 7 6 1 3 2 -12 -15 -14 -9 -11 -10 -20 -23 -22 -17 -19 -18 -29 28 31 30 25 -24 27 26 -34 -35 32 -33 -38 -39 -36 37 42 43 41 46 47 44 50 51 49 54 55 52 -58 -59 -57 -62 -63 -606 7 4 5 2 3 1 -15 -12 -13 -10 -11 -9 -23 -20 -21 -18 -19 -17 -30 31 28 29 26 27 -24 25 -33 32 -35 -34 -37 -36 -39 38 41 43 42 45 44 47 49 51 50 53 52 55 -57 -59 -58 -61 -60 -637 6 5 4 3 2 1 -14 -13 -12 -11 -10 -9 -22 -21 -20 -19 -18 -17 -31 30 29 28 27 26 25 -24 32 -33 -34 -35 -36 -37 -38 39 41 42 43 44 45 46 49 50 51 52 53 54 -57 -58 -59 -60 -61 -628 -24 -16 40 329 -11 -10 -13 -12 -15 -14 3 2 5 4 7 6 -24 27 26 29 28 31 30 17 -19 -18 -21 -20 -23 -22 46 47 44 45 42 43 -41 -38 -39 -36 -37 -34 -35 32 -62 -63 -60 -61 -58 -59 54 55 52 53 50 51

10 -11 -9 -14 -15 -12 -13 3 1 6 7 4 5 27 -24 25 30 31 28 29 18 -19 -17 -22 -23 -20 -21 45 44 47 46 41 43 -42 -37 -36 -39 -38 -33 32 -35 -61 -60 -63 -62 -57 -59 53 52 55 54 49 5111 -10 -9 -15 -14 -13 -12 2 1 7 6 5 4 26 25 -24 31 30 29 28 19 -18 -17 -23 -22 -21 -20 44 45 46 47 41 42 -43 -36 -37 -38 -39 32 -33 -34 -60 -61 -62 -63 -57 -58 52 53 54 55 49 5012 -13 -14 -15 -9 -10 -11 5 6 7 1 2 3 29 30 31 -24 25 26 27 20 -21 -22 -23 -17 -18 -19 43 42 41 47 46 45 -44 -35 -34 -33 32 -39 -38 -37 -59 -58 -57 -63 -62 -61 51 50 49 55 54 5313 -12 -15 -14 -9 -11 -10 4 7 6 1 3 2 28 31 30 25 -24 27 26 21 -20 -23 -22 -17 -19 -18 42 43 41 46 47 44 -45 -34 -35 32 -33 -38 -39 -36 -58 -59 -57 -62 -63 -60 50 51 49 54 55 5214 -15 -12 -13 -10 -11 -9 7 4 5 2 3 1 31 28 29 26 27 -24 25 22 -23 -20 -21 -18 -19 -17 41 43 42 45 44 47 -46 -33 32 -35 -34 -37 -36 -39 -57 -59 -58 -61 -60 -63 49 51 50 53 52 5515 -14 -13 -12 -11 -10 -9 6 5 4 3 2 1 30 29 28 27 26 25 -24 23 -22 -21 -20 -19 -18 -17 41 42 43 44 45 46 -47 32 -33 -34 -35 -36 -37 -38 -57 -58 -59 -60 -61 -62 49 50 51 52 53 5416 -24 8 -48 3217 -19 -18 -21 -20 -23 -22 -24 27 26 29 28 31 30 3 2 5 4 7 6 -9 -11 -10 -13 -12 -15 -14 54 55 52 53 50 51 49 -62 -63 -60 -61 -58 -59 -38 -39 -36 -37 -34 -35 32 46 47 44 45 42 4318 -19 -17 -22 -23 -20 -21 27 -24 25 30 31 28 29 3 1 6 7 4 5 -10 -11 -9 -14 -15 -12 -13 53 52 55 54 49 51 50 -61 -60 -63 -62 -57 -59 -37 -36 -39 -38 -33 32 -35 45 44 47 46 41 4319 -18 -17 -23 -22 -21 -20 26 25 -24 31 30 29 28 2 1 7 6 5 4 -11 -10 -9 -15 -14 -13 -12 52 53 54 55 49 50 51 -60 -61 -62 -63 -57 -58 -36 -37 -38 -39 32 -33 -34 44 45 46 47 41 4220 -21 -22 -23 -17 -18 -19 29 30 31 -24 25 26 27 5 6 7 1 2 3 -12 -13 -14 -15 -9 -10 -11 51 50 49 55 54 53 52 -59 -58 -57 -63 -62 -61 -35 -34 -33 32 -39 -38 -37 43 42 41 47 46 4521 -20 -23 -22 -17 -19 -18 28 31 30 25 -24 27 26 4 7 6 1 3 2 -13 -12 -15 -14 -9 -11 -10 50 51 49 54 55 52 53 -58 -59 -57 -62 -63 -60 -34 -35 32 -33 -38 -39 -36 42 43 41 46 47 4422 -23 -20 -21 -18 -19 -17 31 28 29 26 27 -24 25 7 4 5 2 3 1 -14 -15 -12 -13 -10 -11 -9 49 51 50 53 52 55 54 -57 -59 -58 -61 -60 -63 -33 32 -35 -34 -37 -36 -39 41 43 42 45 44 4723 -22 -21 -20 -19 -18 -17 30 29 28 27 26 25 -24 6 5 4 3 2 1 -15 -14 -13 -12 -11 -10 -9 49 50 51 52 53 54 55 -57 -58 -59 -60 -61 -62 32 -33 -34 -35 -36 -37 -38 41 42 43 44 45 4624 -25 -26 -27 -28 -29 -30 -31 -16 17 18 19 20 21 22 23 8 -9 -10 -11 -12 -13 -14 -15 1 2 3 4 5 6 7 -63 -62 -61 -60 -59 -58 -57 55 54 53 52 51 50 49 -48 -47 -46 -45 -44 -43 -42 -41 40 39 38 37 36 35 34 3325 -24 27 26 29 28 31 30 -19 -18 -21 -20 -23 -22 -11 -10 -13 -12 -15 -14 1 3 2 5 4 7 6 -62 -63 -60 -61 -58 -59 -57 54 55 52 53 50 51 46 47 44 45 42 43 -38 -39 -36 -37 -34 -35 3226 27 -24 25 30 31 28 29 -19 -17 -22 -23 -20 -21 -11 -9 -14 -15 -12 -13 2 3 1 6 7 4 5 -61 -60 -63 -62 -57 -59 -58 53 52 55 54 49 51 45 44 47 46 41 43 -37 -36 -39 -38 -33 32 -3527 26 25 -24 31 30 29 28 -18 -17 -23 -22 -21 -20 -10 -9 -15 -14 -13 -12 3 2 1 7 6 5 4 -60 -61 -62 -63 -57 -58 -59 52 53 54 55 49 50 44 45 46 47 41 42 -36 -37 -38 -39 32 -33 -3428 29 30 31 -24 25 26 27 -21 -22 -23 -17 -18 -19 -13 -14 -15 -9 -10 -11 4 5 6 7 1 2 3 -59 -58 -57 -63 -62 -61 -60 51 50 49 55 54 53 43 42 41 47 46 45 -35 -34 -33 32 -39 -38 -3729 28 31 30 25 -24 27 26 -20 -23 -22 -17 -19 -18 -12 -15 -14 -9 -11 -10 5 4 7 6 1 3 2 -58 -59 -57 -62 -63 -60 -61 50 51 49 54 55 52 42 43 41 46 47 44 -34 -35 32 -33 -38 -39 -3630 31 28 29 26 27 -24 25 -23 -20 -21 -18 -19 -17 -15 -12 -13 -10 -11 -9 6 7 4 5 2 3 1 -57 -59 -58 -61 -60 -63 -62 49 51 50 53 52 55 41 43 42 45 44 47 -33 32 -35 -34 -37 -36 -3931 30 29 28 27 26 25 -24 -22 -21 -20 -19 -18 -17 -14 -13 -12 -11 -10 -9 7 6 5 4 3 2 1 -57 -58 -59 -60 -61 -62 -63 49 50 51 52 53 54 41 42 43 44 45 46 32 -33 -34 -35 -36 -37 -3839 -38 -37 -36 -35 -34 -33 32 46 45 44 43 42 41 54 53 52 51 50 49 -63 -62 -61 -60 -59 -58 -57 1 2 3 4 5 6 7 -9 -10 -11 -12 -13 -14 -17 -18 -19 -20 -21 -22 -24 25 26 27 28 29 3038 -39 -36 -37 -34 -35 32 -33 47 44 45 42 43 41 55 52 53 50 51 49 -62 -63 -60 -61 -58 -59 -57 1 3 2 5 4 7 6 -9 -11 -10 -13 -12 -15 -17 -19 -18 -21 -20 -23 25 -24 27 26 29 28 3137 -36 -39 -38 -33 32 -35 -34 44 47 46 41 43 42 52 55 54 49 51 50 -61 -60 -63 -62 -57 -59 -58 2 3 1 6 7 4 5 -10 -11 -9 -14 -15 -12 -18 -19 -17 -22 -23 -20 26 27 -24 25 30 31 2836 -37 -38 -39 32 -33 -34 -35 45 46 47 41 42 43 53 54 55 49 50 51 -60 -61 -62 -63 -57 -58 -59 3 2 1 7 6 5 4 -11 -10 -9 -15 -14 -13 -19 -18 -17 -23 -22 -21 27 26 25 -24 31 30 2935 -34 -33 32 -39 -38 -37 -36 42 41 47 46 45 44 50 49 55 54 53 52 -59 -58 -57 -63 -62 -61 -60 4 5 6 7 1 2 3 -12 -13 -14 -15 -9 -10 -20 -21 -22 -23 -17 -18 28 29 30 31 -24 25 2634 -35 32 -33 -38 -39 -36 -37 43 41 46 47 44 45 51 49 54 55 52 53 -58 -59 -57 -62 -63 -60 -61 5 4 7 6 1 3 2 -13 -12 -15 -14 -9 -11 -21 -20 -23 -22 -17 -19 29 28 31 30 25 -24 2733 32 -35 -34 -37 -36 -39 -38 43 42 45 44 47 46 51 50 53 52 55 54 -57 -59 -58 -61 -60 -63 -62 6 7 4 5 2 3 1 -14 -15 -12 -13 -10 -11 -22 -23 -20 -21 -18 -19 30 31 28 29 26 27 -2432 33 34 35 36 37 38 39 40 -41 -42 -43 -44 -45 -46 -47 -48 49 50 51 52 53 54 55 -57 -58 -59 -60 -61 -62 -63 7 6 5 4 3 2 1 -15 -14 -13 -12 -11 -10 -9 8 23 22 21 20 19 18 17 -16 -31 -30 -29 -28 -27 -26 -2547 46 45 44 43 42 41 -38 -37 -36 -35 -34 -33 32 -62 -61 -60 -59 -58 -57 55 54 53 52 51 50 49 -9 -10 -11 -12 -13 -14 -15 1 2 3 4 5 6 -24 25 26 27 28 29 30 -17 -18 -19 -20 -21 -2246 47 44 45 42 43 41 -39 -36 -37 -34 -35 32 -33 -63 -60 -61 -58 -59 -57 54 55 52 53 50 51 49 -9 -11 -10 -13 -12 -15 -14 1 3 2 5 4 7 25 -24 27 26 29 28 31 -17 -19 -18 -21 -20 -2345 44 47 46 41 43 42 -36 -39 -38 -33 32 -35 -34 -60 -63 -62 -57 -59 -58 53 52 55 54 49 51 50 -10 -11 -9 -14 -15 -12 -13 2 3 1 6 7 4 26 27 -24 25 30 31 28 -18 -19 -17 -22 -23 -2044 45 46 47 41 42 43 -37 -38 -39 32 -33 -34 -35 -61 -62 -63 -57 -58 -59 52 53 54 55 49 50 51 -11 -10 -9 -15 -14 -13 -12 3 2 1 7 6 5 27 26 25 -24 31 30 29 -19 -18 -17 -23 -22 -2143 42 41 47 46 45 44 -34 -33 32 -39 -38 -37 -36 -58 -57 -63 -62 -61 -60 51 50 49 55 54 53 52 -12 -13 -14 -15 -9 -10 -11 4 5 6 7 1 2 28 29 30 31 -24 25 26 -20 -21 -22 -23 -17 -1842 43 41 46 47 44 45 -35 32 -33 -38 -39 -36 -37 -59 -57 -62 -63 -60 -61 50 51 49 54 55 52 53 -13 -12 -15 -14 -9 -11 -10 5 4 7 6 1 3 29 28 31 30 25 -24 27 -21 -20 -23 -22 -17 -1941 43 42 45 44 47 46 32 -35 -34 -37 -36 -39 -38 -59 -58 -61 -60 -63 -62 49 51 50 53 52 55 54 -14 -15 -12 -13 -10 -11 -9 6 7 4 5 2 3 30 31 28 29 26 27 -24 -22 -23 -20 -21 -18 -1940 32 -48 8 -2455 54 53 52 51 50 49 -62 -61 -60 -59 -58 -57 -38 -37 -36 -35 -34 -33 32 -47 46 45 44 43 42 41 -17 -18 -19 -20 -21 -22 23 -24 25 26 27 28 29 30 1 2 3 4 5 6 -9 -10 -11 -12 -13 -1454 55 52 53 50 51 49 -63 -60 -61 -58 -59 -57 -39 -36 -37 -34 -35 32 -33 -46 47 44 45 42 43 41 -17 -19 -18 -21 -20 -23 22 25 -24 27 26 29 28 31 1 3 2 5 4 7 -9 -11 -10 -13 -12 -1553 52 55 54 49 51 50 -60 -63 -62 -57 -59 -58 -36 -39 -38 -33 32 -35 -34 -45 44 47 46 41 43 42 -18 -19 -17 -22 -23 -20 21 26 27 -24 25 30 31 28 2 3 1 6 7 4 -10 -11 -9 -14 -15 -1252 53 54 55 49 50 51 -61 -62 -63 -57 -58 -59 -37 -38 -39 32 -33 -34 -35 -44 45 46 47 41 42 43 -19 -18 -17 -23 -22 -21 20 27 26 25 -24 31 30 29 3 2 1 7 6 5 -11 -10 -9 -15 -14 -1351 50 49 55 54 53 52 -58 -57 -63 -62 -61 -60 -34 -33 32 -39 -38 -37 -36 -43 42 41 47 46 45 44 -20 -21 -22 -23 -17 -18 19 28 29 30 31 -24 25 26 4 5 6 7 1 2 -12 -13 -14 -15 -9 -1050 51 49 54 55 52 53 -59 -57 -62 -63 -60 -61 -35 32 -33 -38 -39 -36 -37 -42 43 41 46 47 44 45 -21 -20 -23 -22 -17 -19 18 29 28 31 30 25 -24 27 5 4 7 6 1 3 -13 -12 -15 -14 -9 -1149 51 50 53 52 55 54 -59 -58 -61 -60 -63 -62 32 -35 -34 -37 -36 -39 -38 -41 43 42 45 44 47 46 -22 -23 -20 -21 -18 -19 17 30 31 28 29 26 27 -24 6 7 4 5 2 3 -14 -15 -12 -13 -10 -1148 32 40 -16 -2463 -62 -61 -60 -59 -58 -57 54 53 52 51 50 49 46 45 44 43 42 41 39 -38 -37 -36 -35 -34 -33 32 -24 25 26 27 28 29 30 -31 -17 -18 -19 -20 -21 -22 -9 -10 -11 -12 -13 -14 1 2 3 4 5 662 -63 -60 -61 -58 -59 -57 55 52 53 50 51 49 47 44 45 42 43 41 38 -39 -36 -37 -34 -35 32 -33 25 -24 27 26 29 28 31 -30 -17 -19 -18 -21 -20 -23 -9 -11 -10 -13 -12 -15 1 3 2 5 4 761 -60 -63 -62 -57 -59 -58 52 55 54 49 51 50 44 47 46 41 43 42 37 -36 -39 -38 -33 32 -35 -34 26 27 -24 25 30 31 28 -29 -18 -19 -17 -22 -23 -20 -10 -11 -9 -14 -15 -12 2 3 1 6 7 460 -61 -62 -63 -57 -58 -59 53 54 55 49 50 51 45 46 47 41 42 43 36 -37 -38 -39 32 -33 -34 -35 27 26 25 -24 31 30 29 -28 -19 -18 -17 -23 -22 -21 -11 -10 -9 -15 -14 -13 3 2 1 7 6 559 -58 -57 -63 -62 -61 -60 50 49 55 54 53 52 42 41 47 46 45 44 35 -34 -33 32 -39 -38 -37 -36 28 29 30 31 -24 25 26 -27 -20 -21 -22 -23 -17 -18 -12 -13 -14 -15 -9 -10 4 5 6 7 1 258 -59 -57 -62 -63 -60 -61 51 49 54 55 52 53 43 41 46 47 44 45 34 -35 32 -33 -38 -39 -36 -37 29 28 31 30 25 -24 27 -26 -21 -20 -23 -22 -17 -19 -13 -12 -15 -14 -9 -11 5 4 7 6 1 357 -59 -58 -61 -60 -63 -62 51 50 53 52 55 54 43 42 45 44 47 46 33 32 -35 -34 -37 -36 -39 -38 30 31 28 29 26 27 -24 -25 -22 -23 -20 -21 -18 -19 -14 -15 -12 -13 -10 -11 6 7 4 5 2 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 39 38 37 36 35 34 33 32 47 46 45 44 43 42 41 40 55 54 53 52 51 50 49 48 63 62 61 60 59 58 571 3 2 5 4 7 6 -11 -10 -13 -12 -15 -14 -19 -18 -21 -20 -23 -22 -25 -24 27 26 29 28 31 30 -38 -39 -36 -37 -34 -35 32 33 46 47 44 45 42 43 54 55 52 53 50 51 -62 -63 -60 -61 -58 -592 3 1 6 7 4 5 -11 -9 -14 -15 -12 -13 -19 -17 -22 -23 -20 -21 -26 27 -24 25 30 31 28 29 -37 -36 -39 -38 -33 32 -35 34 45 44 47 46 41 43 53 52 55 54 49 51 -61 -60 -63 -62 -57 -593 2 1 7 6 5 4 -10 -9 -15 -14 -13 -12 -18 -17 -23 -22 -21 -20 -27 26 25 -24 31 30 29 28 -36 -37 -38 -39 32 -33 -34 35 44 45 46 47 41 42 52 53 54 55 49 50 -60 -61 -62 -63 -57 -584 5 6 7 1 2 3 -13 -14 -15 -9 -10 -11 -21 -22 -23 -17 -18 -19 -28 29 30 31 -24 25 26 27 -35 -34 -33 32 -39 -38 -37 36 43 42 41 47 46 45 51 50 49 55 54 53 -59 -58 -57 -63 -62 -615 4 7 6 1 3 2 -12 -15 -14 -9 -11 -10 -20 -23 -22 -17 -19 -18 -29 28 31 30 25 -24 27 26 -34 -35 32 -33 -38 -39 -36 37 42 43 41 46 47 44 50 51 49 54 55 52 -58 -59 -57 -62 -63 -606 7 4 5 2 3 1 -15 -12 -13 -10 -11 -9 -23 -20 -21 -18 -19 -17 -30 31 28 29 26 27 -24 25 -33 32 -35 -34 -37 -36 -39 38 41 43 42 45 44 47 49 51 50 53 52 55 -57 -59 -58 -61 -60 -637 6 5 4 3 2 1 -14 -13 -12 -11 -10 -9 -22 -21 -20 -19 -18 -17 -31 30 29 28 27 26 25 -24 32 -33 -34 -35 -36 -37 -38 39 41 42 43 44 45 46 49 50 51 52 53 54 -57 -58 -59 -60 -61 -628 -24 -16 40 329 -11 -10 -13 -12 -15 -14 3 2 5 4 7 6 -24 27 26 29 28 31 30 17 -19 -18 -21 -20 -23 -22 46 47 44 45 42 43 -41 -38 -39 -36 -37 -34 -35 32 -62 -63 -60 -61 -58 -59 54 55 52 53 50 51

10 -11 -9 -14 -15 -12 -13 3 1 6 7 4 5 27 -24 25 30 31 28 29 18 -19 -17 -22 -23 -20 -21 45 44 47 46 41 43 -42 -37 -36 -39 -38 -33 32 -35 -61 -60 -63 -62 -57 -59 53 52 55 54 49 5111 -10 -9 -15 -14 -13 -12 2 1 7 6 5 4 26 25 -24 31 30 29 28 19 -18 -17 -23 -22 -21 -20 44 45 46 47 41 42 -43 -36 -37 -38 -39 32 -33 -34 -60 -61 -62 -63 -57 -58 52 53 54 55 49 5012 -13 -14 -15 -9 -10 -11 5 6 7 1 2 3 29 30 31 -24 25 26 27 20 -21 -22 -23 -17 -18 -19 43 42 41 47 46 45 -44 -35 -34 -33 32 -39 -38 -37 -59 -58 -57 -63 -62 -61 51 50 49 55 54 5313 -12 -15 -14 -9 -11 -10 4 7 6 1 3 2 28 31 30 25 -24 27 26 21 -20 -23 -22 -17 -19 -18 42 43 41 46 47 44 -45 -34 -35 32 -33 -38 -39 -36 -58 -59 -57 -62 -63 -60 50 51 49 54 55 5214 -15 -12 -13 -10 -11 -9 7 4 5 2 3 1 31 28 29 26 27 -24 25 22 -23 -20 -21 -18 -19 -17 41 43 42 45 44 47 -46 -33 32 -35 -34 -37 -36 -39 -57 -59 -58 -61 -60 -63 49 51 50 53 52 5515 -14 -13 -12 -11 -10 -9 6 5 4 3 2 1 30 29 28 27 26 25 -24 23 -22 -21 -20 -19 -18 -17 41 42 43 44 45 46 -47 32 -33 -34 -35 -36 -37 -38 -57 -58 -59 -60 -61 -62 49 50 51 52 53 5416 -24 8 -48 3217 -19 -18 -21 -20 -23 -22 -24 27 26 29 28 31 30 3 2 5 4 7 6 -9 -11 -10 -13 -12 -15 -14 54 55 52 53 50 51 49 -62 -63 -60 -61 -58 -59 -38 -39 -36 -37 -34 -35 32 46 47 44 45 42 4318 -19 -17 -22 -23 -20 -21 27 -24 25 30 31 28 29 3 1 6 7 4 5 -10 -11 -9 -14 -15 -12 -13 53 52 55 54 49 51 50 -61 -60 -63 -62 -57 -59 -37 -36 -39 -38 -33 32 -35 45 44 47 46 41 4319 -18 -17 -23 -22 -21 -20 26 25 -24 31 30 29 28 2 1 7 6 5 4 -11 -10 -9 -15 -14 -13 -12 52 53 54 55 49 50 51 -60 -61 -62 -63 -57 -58 -36 -37 -38 -39 32 -33 -34 44 45 46 47 41 4220 -21 -22 -23 -17 -18 -19 29 30 31 -24 25 26 27 5 6 7 1 2 3 -12 -13 -14 -15 -9 -10 -11 51 50 49 55 54 53 52 -59 -58 -57 -63 -62 -61 -35 -34 -33 32 -39 -38 -37 43 42 41 47 46 4521 -20 -23 -22 -17 -19 -18 28 31 30 25 -24 27 26 4 7 6 1 3 2 -13 -12 -15 -14 -9 -11 -10 50 51 49 54 55 52 53 -58 -59 -57 -62 -63 -60 -34 -35 32 -33 -38 -39 -36 42 43 41 46 47 4422 -23 -20 -21 -18 -19 -17 31 28 29 26 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