Voyage by Catamaran

Long-Distance Semantic Navigation, from Myth Logic to

Semantic Web, Can Be Effected by Infinite-Dimensional Zero-

Divisor Ensembles

“Long-Distance,” like the Polynesian crossings of the

South Pacific in catamarans…with “long” vs. “short” relating

to two distinct senses of distance:

“Twist products” vs “Edge traversals” in Zero-Divisor

navigating –

and “Keyword search” vs “Google-Earth neighborhoods”

in Web drill-downs

Same-colored edges of a Catamaran (one per each orthogonal Square in a Box-Kite’s Octahedron) “twist” to oppositely colored edges of the same Box-Kite … one with a different “Strut Constant” from its starting point

(which is where the “long-distance navigation” comes in).

But what’s the “twist”? Each vertex, recall, represents a plane (spanned by units B and b, say). Any point in one of its diagonals times any in the similarly (+) or oppositely (-) sloping diagonal in the plane of an edge-

joined vertex = ZERO. But you can also DO THE TWIST:(B + b)(D + d) = 0 (B + d)(D – b) = 0

Note that (B + b)(D + d) = 0 works on vertices; but (B + d)(D – b) = 0 works on edges. Also, that (B + d) and (D – b)

correspond to vertices in another box-kite, with opposite edge-sign.(OK, time for a little review…)

• Dr. Seuss’ Thing1 and Thing 2 (here, flying Box-Kites, I assume) inspired my “simple and stupid” reduction of Cayley-Dickson process to 2 rules:

• Rule 1: For imaginary unit with index a < G, its product with the next 2N-ion Generator has index (a+G), and is positive when ia is to the left of iG.

• Example: For Quaternion index-set (1, 2, 3), appending G with index 4 yields these 3 Octonion

triplets: (1, 4, 5); (2, 4, 6); (3, 4, 7)

• Rule 2: For imaginary units forming a triplet, written in cyclically positive order (a, b, c), appending a new generator to two of them yields a new triplet with the order of the two terms worked on reversed.

• Example: For Quaternion index-set (1, 2, 3), not appending G = 4 to 1, 2, and 3, yields these Octonion

triplets: (1, 7, 6); (2, 5, 7); (3, 6, 5)

• That’s it! Using these 2 rules, all triplets in all 2N-ions can be readily constructed; starting with n=4, one quickly sees that zero-divisor “collisions” between the two rules’ triplets cannot be avoided!

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=16 instead of 8. We are now in the Pathions – the on-ramp to the Metafractal Highway!

If you try to trace a sequence of ZDs around a Sail, you can keep going in a 6-cycle, including products of all

diagonals at both ends of each edge. With a Catamaran, you get two cycles of length 4, due to the even count of

edge-signs. Starting at the same vertex, the diagonals you pick will slope in sequence like this: ( / \ / \ ), else ( \ / \ / ).

Twist products, however, are even more exotic:

Here’s how to read this “Royal Hunt Diagram”: the colors relate to the Catamaran “mast” orthogonal to its prows –

that is, to the STRUT orthogonal to the pair of same which diagonally link its four corners, and about which one twists.

The edge and arrow colored in each square indicate the (always [-]) edge where flow reverses when traversing.

The strut constants of the 7 Sedenion box-kites, each with 3 color-coded Catamarans, have their mutual symmetries indicated in this “colorized”

version of the standard PSL(2,7) Triangle, which we call a “Twisted Sister Diagram”:

The triads these 7 lines each indicate reside at a meta-level, where nodes stand for Box-Kites. The cyclical

threading through 3 while rotating once can be taken as representing the core stratification of the Double Cusp – the model alleged to contain all “archetypal sentences” – called

the Umbilic Bracelet.

The above is a “coming attraction”: for the Double Cusp is the basis of Petitot’s model of Levi-Strauss’s “canonical law

of myths,” a sort of “air-traffic control tower” for myriad “Semiotic Squares” passing by like story fragments on the “song-lines” grid of the transcontinental “Matrix” of mythic

narrative.This same apparatus, I assert, will prove the basis for the

new mathematical support-structure the building of the Semantic Web will require.

The whole point of this presentation is to make the contents of this slide make sense! (And note, the “E6” or “symbolic umbilic” seam-line, doing the 3-to-1 suturing of the Bracelet

itself, “represents” the explosion implicit in one Box-Kite turning into three by the earlier discussed “boiling off”

process.)

Now, back in our Catamaran, the “3 + 1” structure of the flow orientation while traversing its square perimeter suggests the simplest non-cyclic

group, the Klein Group (which not only governs the symmetry of Spacetime, but is the quotient group of the Quaternions).

But this Catamaran also looks like the “double articulation” of another Square: the Semiotic Square formed by the four units

comprising any of a Box-Kite’s struts! (A bit of buffer refresh on this point is coming up next …)

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.

The Catamaran “double articulates” the Semiotic Square, because each of its diagonals is a strut (hence a Semiotic Square itself); and, because its corners are Box-Kite vertices (hence, pairs of units, not singletons!)

The Klein-Group Connection

• In the Semiotic Square’s upper left corner, replace a with 0 (the index of the reals); then the quartet of indices (0,S,G,X) form a ZD-free Quaternion copy – the 4 “hidden units” among the Sedenions’ 16 (with the 6 unit-pairings associated with the vertices yielding 12 “visible” dimenions in the Box-Kite representation). If we ignore sign-ing,we have the Klein group! (Which Greimas himself claimed was associated somehow with his Square.)

The Klein-Group Connection

• We can see how the {0,S,G,X} quartet form an “abstract class” underwriting the ZD structure of any Box-Kite when we consider the minimal manner of repre-senting the latter’s Sail and Strut structure: eliminate the empty spaces by collapsing the Octahedron to a Tetrahedron, and associating opposite edges with strut-opposite vertices.

The Bicycle Chain: a Box-Kite lanyard, like Sails and Cata-marans, which threads through all six vertices. The name is suggested by an analogy to shifting gears on a speed-bike in the “Finale” to the fourth and last volume of his Introduction to a Science of Mythology; the comparison is with the man-ner in which hundreds of Klein Groups get chained together when one studies mythic systems in the large. (Which is how Box-Kites get chained, as well, in higher-dimensional ensembles!)

Levi-Strauss, in his own words:

The Canonical Law of Myths has confused two generations of interpreters (it was first announced 50 years ago). Its author is still with us, and here it is as

he has used it in Vol. 2 of his Mythologiques, From Honey to Ashes:

The “pseudocode” given in the formula baffled Harvard’s Howard Gardner, who had the honesty to admit he couldn’t

make the least bit of sense out of it. But in fact, as with Monsieur Pangloss speaking prose, it is something we do

every day without knowing it!

• Fx(a):Fy(b) :: Fx(b):F[a-1](y) is just short-hand for this rhetorical form we find in ads, glib movie reviews, and celebrity bon-mots everywhere: “X makes Y look like the opposite of what you thought, until now, Y epitomized so well!”

Here’s how a classical rhetorician sees it:

Examples:

• 70s blue-movie ad for long-forgotten skin flick that compared itself to the then-reign-ing cause celebre of the genre:

• Hot Lust makes The Devil in Miss Jones look like a PTA meeting.

Examples:

• Celebrity bon-mot: Frank Lloyd Wright said of his final creation, built in close viewing distance of its rival art palace, that

• “The Guggeheim makes the Museum of Modern Art look like a Protestant barn.”

Examples

• Movie Review: Boston critic/radio host David Brudnoy said of a movie featuring do-gooder WASPs trying to improve the lot of some local yokels in the boonies, that the targets of their charity “made the Beverly Hillbillies look like members of the Myopia Hunt Club.”

Examples

• Ad touting online employment service:

• “Our database makes the Taj Mahal look like a second-floor walkup.”

Interpretation

• What makes such formulations candidates for concrete organizing principles given oversight of hundreds of intertwined themes? They convert dual cusp setups (two things start out in pan-balance equili-brium) and turn them into competitors (the standard cusp), a transformation requiring at least E6, plus the “factitive” operator (Capt. Jean-Luc Picard’s “Make it so!” – i.e., the tell-tale splice-phrase “looks like”)

Now let’s put all the above into some Semantic Web Q&A:

• {1Q.} In his May 12 Plenary address at the W3C conclave in Banff, Sir Tim Berners-Lee noted how we’ve learned in the last few years that the Web has rich (and quite surprising) built-in features, yet none of our modes of description incorporate yet: in particular, it’s a “scale-free” network, hence implicitly fractal.

• {1A.} Ergo, bring in Zero-Divisors as minimal descriptive tool! (Time to review…)

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

-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

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!

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).

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 59 581 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 -573 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 -574 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 -625 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 -636 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 -607 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 -618 -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 4911 -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 4912 -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 5413 -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 5514 -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 5215 -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 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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 38 39 36 37 34 35 32 33 46 47 44 45 42 43 40 41 54 55 52 53 50 51 48 49 62 63 60 61 58 59 561 -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 -39 38 37 -36 35 -34 -33 32 -47 46 45 -44 43 -42 55 -54 -53 52 -51 50 63 -62 -61 60 -59 582 -3 -1 10 8 19 17 -26 -27 -24 -25 32 33 34 35 -40 -42 -49 -51 56 583 2 -1 -11 8 -18 17 27 26 -25 -24 33 32 -35 -34 -40 43 -49 50 56 -594 -5 -1 12 8 21 17 -28 -29 -24 -25 32 33 36 37 -40 -44 -49 -53 56 605 4 -1 -13 8 -20 17 29 28 -25 -24 33 32 -37 -36 -40 45 -49 52 56 -616 7 -1 -14 8 -23 17 30 31 -24 -25 32 33 -38 -39 -40 46 -49 55 56 -627 -6 -1 15 8 22 17 -31 -30 -25 -24 33 32 39 38 -40 -47 -49 -54 56 638 10 -11 12 -13 -14 15 2 -3 4 -5 -6 7 -25 -26 27 -28 29 30 -31 -17 -18 19 -20 21 22 -23 46 -47 -44 45 -42 43 40 38 -39 -36 37 -34 35 32 -62 63 60 -61 58 -59 -54 55 52 -53 50 -519 -25 -16 41 32

10 -11 8 2 -1 27 -24 -25 -18 -19 17 -40 42 43 32 33 -34 56 -59 -49 5011 10 8 -3 -1 -26 -25 -24 19 18 17 -40 -43 -42 33 32 35 56 58 -49 -5112 -13 8 4 -1 29 -24 -25 -20 -21 17 -40 44 45 32 33 -36 56 -61 -49 5213 12 8 -5 -1 -28 -25 -24 21 20 17 -40 -45 -44 33 32 37 56 60 -49 -5314 15 8 -6 -1 -31 -24 -25 22 23 17 -40 -46 -47 32 33 38 56 63 -49 -5415 -14 8 7 -1 30 -25 -24 -23 -22 17 -40 47 46 33 32 -39 56 -62 -49 5516 -25 9 -48 3217 19 -18 21 -20 -23 22 -25 27 -26 29 -28 -31 30 -3 2 -5 4 7 -6 8 -11 10 -13 12 15 -14 55 -54 -53 52 -51 50 -49 63 -62 -61 60 -59 58 -39 38 37 -36 35 -34 32 -47 46 45 -44 43 -4218 19 17 -26 -24 -25 -3 -1 10 -11 8 -49 50 -51 56 58 32 33 35 -40 -4219 -18 17 27 -25 -24 2 -1 -11 10 8 -49 -51 50 56 -59 33 32 -34 -40 4320 21 17 -28 -24 -25 -5 -1 12 -13 8 -49 52 -53 56 60 32 33 37 -40 -4421 -20 17 29 -25 -24 4 -1 -13 12 8 -49 -53 52 56 -61 33 32 -36 -40 4522 -23 17 30 -24 -25 7 -1 -14 15 8 -49 -54 55 56 -62 32 33 -39 -40 4623 22 17 -31 -25 -24 -6 -1 15 -14 8 -49 55 -54 56 63 33 32 38 -40 -4724 -25 -26 27 -28 29 30 -31 -18 19 -20 21 22 -23 10 -11 12 -13 -14 15 -1 2 -3 4 -5 -6 7 -62 63 60 -61 58 -59 56 -54 55 52 -53 50 -51 46 -47 -44 45 -42 43 38 -39 -36 37 -34 35 3225 24 -27 26 -29 28 31 -30 -17 -16 -19 18 -21 20 23 -22 9 8 -11 10 -13 12 15 -14 -1 -3 2 -5 4 7 -6 63 -62 -61 60 -59 58 56 55 -54 -53 52 -51 50 -49 -48 47 -46 -45 44 -43 42 41 40 39 -38 -37 36 -35 34 -3326 27 -24 -25 -18 17 -11 8 2 -3 -1 56 58 -59 -49 50 -40 43 32 33 -3427 -26 -25 -24 19 17 10 8 -3 2 -1 56 -59 58 -49 -51 -40 -42 33 32 3528 29 -24 -25 -20 17 -13 8 4 -5 -1 56 60 -61 -49 52 -40 45 32 33 -3629 -28 -25 -24 21 17 12 8 -5 4 -1 56 -61 60 -49 -53 -40 -44 33 32 3730 -31 -24 -25 22 17 15 8 -6 7 -1 56 -62 63 -49 -54 -40 -47 32 33 3831 30 -25 -24 -23 17 -14 8 7 -6 -1 56 63 -62 -49 55 -40 46 33 32 -3938 -39 32 33 46 -40 55 -49 -62 63 56 -1 -6 7 8 -14 17 -23 -24 -25 3039 38 33 32 -47 -40 -54 -49 63 -62 56 -1 7 -6 8 15 17 22 -25 -24 -3136 37 32 33 -44 -40 -53 -49 60 -61 56 -1 4 -5 8 12 17 21 -24 -25 -2837 -36 33 32 45 -40 52 -49 -61 60 56 -1 -5 4 8 -13 17 -20 -25 -24 2934 35 32 33 -42 -40 -51 -49 58 -59 56 -1 2 -3 8 10 17 19 -24 -25 -2635 -34 33 32 43 -40 50 -49 -59 58 56 -1 -3 2 8 -11 17 -18 -25 -24 2732 -33 34 -35 36 -37 -38 39 40 41 42 -43 44 -45 -46 47 -48 -49 50 -51 52 -53 -54 55 56 58 -59 60 -61 -62 63 -6 7 4 -5 2 -3 -1 -14 15 12 -13 10 -11 8 9 -22 23 20 -21 18 -19 -16 -17 -30 31 28 -29 26 -27 2433 32 35 -34 37 -36 -39 38 43 -42 45 -44 -47 46 -51 50 -53 52 55 -54 56 -59 58 -61 60 63 -62 7 -6 -5 4 -3 2 -1 15 -14 -13 12 -11 10 -23 22 21 -20 19 -18 -31 30 29 -28 27 -26 -2546 -47 -40 38 32 33 63 56 -54 55 -49 8 -14 15 -1 -6 -24 -25 -31 17 2247 46 -40 -39 33 32 -62 56 55 -54 -49 8 15 -14 -1 7 -25 -24 30 17 -2344 45 -40 -36 32 33 -61 56 52 -53 -49 8 12 -13 -1 4 -24 -25 29 17 -2045 -44 -40 37 33 32 60 56 -53 52 -49 8 -13 12 -1 -5 -25 -24 -28 17 2142 43 -40 -34 32 33 -59 56 50 -51 -49 8 10 -11 -1 2 -24 -25 27 17 -1843 -42 -40 35 33 32 58 56 -51 50 -49 8 -11 10 -1 -3 -25 -24 -26 17 1940 -42 43 -44 45 46 -47 32 -34 35 -36 37 38 -39 58 -59 60 -61 -62 63 -49 50 -51 52 -53 -54 55 -14 15 12 -13 10 -11 8 -6 7 4 -5 2 -3 30 -31 -28 29 -26 27 -25 22 -23 -20 21 -18 1941 32 -48 9 -2554 55 -49 -62 56 -39 32 33 46 47 -40 17 -22 -23 -24 -25 30 -1 7 8 -1455 -54 -49 63 56 38 33 32 -47 -46 -40 17 23 22 -25 -24 -31 -1 -6 8 1552 -53 -49 60 56 37 32 33 -44 -45 -40 17 20 21 -24 -25 -28 -1 -5 8 1253 52 -49 -61 56 -36 33 32 45 44 -40 17 -21 -20 -25 -24 29 -1 4 8 -1350 -51 -49 58 56 35 32 33 -42 -43 -40 17 18 19 -24 -25 -26 -1 -3 8 1051 50 -49 -59 56 -34 33 32 43 42 -40 17 -19 -18 -25 -24 27 -1 2 8 -1148 32 41 -16 -2549 -51 50 -53 52 55 -54 -59 58 -61 60 63 -62 32 35 -34 37 -36 -39 38 40 43 -42 45 -44 -47 46 -23 22 21 -20 19 -18 -17 -31 30 29 -28 27 -26 -25 7 -6 -5 4 -3 2 15 -14 -13 12 -11 1062 63 56 -54 -49 -47 -40 38 39 32 33 -24 -25 -30 -31 17 22 8 15 -1 -663 -62 56 55 -49 46 -40 -39 -38 33 32 -25 -24 31 30 17 -23 8 -14 -1 760 -61 56 52 -49 45 -40 -36 -37 32 33 -24 -25 28 29 17 -20 8 -13 -1 461 60 56 -53 -49 -44 -40 37 36 33 32 -25 -24 -29 -28 17 21 8 12 -1 -558 -59 56 50 -49 43 -40 -34 -35 32 33 -24 -25 26 27 17 -18 8 -11 -1 259 58 56 -51 -49 -42 -40 35 34 33 32 -25 -24 -27 -26 17 19 8 10 -1 -356 58 -59 60 -61 -62 63 50 -51 52 -53 -54 55 -42 43 -44 45 46 -47 32 -33 -34 35 -36 37 38 -39 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

And now for some more Semantic Web Q&A:

• {2Q.} The Semantic Web, as currently implemented and thought about, isn’t!

• {2A.} It’s all just syntax, with semantic content deferred indefinitely (let the end-user incorporate it implicitly in his OWL and RDF tinkerings!) But 1-word search, a decade ago, is no longer ade-quate: the typical search uses over 3 key-words now. As data to be mined grows ever more humongous, we’ll need archetypal sentences – that is, explicit semantics!! – in our searches.

(Well, at least ELVIS is in the building ;)

Here’s what I’m talking about:

The Double Cusp, the local tool of choice, contains all these:

ZDs as representation theory

• All the Double Cusp’s strata (hence, all the “valence theory” of Tesniere and the like, which resides “under the radar” of the blithe recursiveness of Chomsky-ism) can be represented as traversal-patterns on Box-Kites and/or ensembles of same. We’ll just look very quickly at a few of these, the most basic being the already alluded-to “Dual Cusp.”

The Dual Cusp is just the usual Cusp, with its behavioral UI turned upside down: but the two equal and opposite copies of the 3rd Assessor

“emanated” by traversing the edge joining its two “Sailing Partners” is perhaps the archetypal for-

instance here: these two, instead of competing, are coordinating their actions in “pan-balance” fashion. With 3 such “Dual Cusps” tracing a Sail, we get the

Umbilics, with “trip synch” action determining sendings and receivings. (See “Umbilic Bracelet”

slide above. And see Web 2.0 “for-instance” next!)

The ESP Game (Setup)

• Luis von Ahn, a recent PhD snapped up by Princeton as a first-round draft pick, considered that people waste 6 billion hours a year playing Solitaire on their computers. Yet the Empire State building only required 20 million man-hours to erect.

• Meanwhile, the human-computer interface has been horribly parasitic from the get-go: but “catchas” (those little boxes with the weirdly distorted font styles and colors whose contents you have to reproduce in a textbox to prove you’re not a robot) show, by a “reverse Turing Test” kind of logic, that humans are super-good at things that currently are deemed “hard AI.”

• Perhaps inspired by the SETI project, von Ahn designed this game: two players, mutually anonymized except for Internet handles, are given an image to describe. If they both hit upon the same description, they both get points.

The ESP Game (Punch Line)

• So what, you say? Well, the game gets addictive, and now has thousands of players forsaking all those billions of hours of Solitaire for it … and, within a few months’ time, the entire inventory of Google Images gets tagged with highly dependable search-term labels! The Dual Cusp (forget about its geometry) thereby underwrites this prototypical pattern of “Hard AI” problem-solving by human/computer sybiosis.

• Then von Ahn invented another game, to take component parts of images (users click on spots, and use expandable viewing windows).

And now for some final Semantic Web Q&A:

• {3Q/A}: Adding in the semantics will require taking adaptive parsing seriously (a shameless plug for the Meta-S Grammar Forge of my business partner, Quinn Taylor Jackson, of Thothic Technology Partners)… and exploiting the expertise of people embroiled in radically disparate ontologies. Since so many instances of the Canonical Law cause catastrophic joke responses (the “haha effect”), a popular feature in The Metro (the free daily rag found on subway platforms) suggests itself for “von Ahn treatment.”

Will the Semantic Web kill the blogosphere star?(Will Michael Heim’s prognostications in “The Metaphysics of Virtual Reality” come to pass – will Hermann Hesse’s “Glass Bead Game”

arrive among us at last, and be played in The Matrix?)

Coda (I couldn’t help myself)In each of successive volume of his 4-tome Mythologique, Levi-Strauss unearths a deeper layer of logique beneath broader domains of myth-data, covering retraversed terrain with a new depth (and much broader system of inputs). The 3x3 grid at the basis of Sky recur-sion has its centermost box ever the same, regardless of “spreadsheet size” … yet the same shown cell-values are generated by an ever-widening backdrop of interactions.

I am not the first to point out that the scheme he unveils smells like Vico’s “Four Ages,” that hoary old pousse café of Providence’s working from above. Yet the bit-twiddling logic of

Box-Kites and Skies makes it clear that there’s an NKS burbling from beneath whose ultra-simple No-Mind algorithmics must be exceeding rich (and which remains to be explored). I

don’t know about God or Devil, but I do believe in the Baroness Orczy’s scarlet hero:

They seek him here,they seek him there:

those Frenchies seek him everywhere!Is he in Heaven?Is he in (ahem!)?

That d**d elusive Pimpernel!*

*(The version with Leslie Howard, Merle Oberon, Raymond Massey and Nigel Bruce is the only one worth watching: don’t be fooled!)

Voyage by Catamaran

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Transcript of Voyage by Catamaran