Voyage by Catamaran

1. Voyage by Catamaran

2. Long-Distance Semantic Navigation, from Myth Logic to Semantic Web, Can Be Effected by Infinite-Dimensional Zero-Divisor Ensembles

3. “Long-Distance,” like the Polynesian crossings of the South Pacific in catamarans…with “long” vs. “short” relating to two distinct senses of distance:

4. “Twist products” vs “Edge traversals” in Zero-Divisor navigating – and “Keyword search” vs “Google-Earth neighborhoods” in Web drill-downs

5. 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).

6. 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

7. 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…)

8. 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:

9. 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)

10. 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)

11. 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!

12. 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.

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

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

15. 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.

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

17. A different view, with numbers too! One surprisingly deep aspect among many in this simple structure: the route to fractals is already in evidence! The 4 Q-copies in a Sail split into 1 “pure” Octonion triple and 3 “mixed” triples of 1 Octonion + 2 Sedenions; the 4 Sails also split: into one with 3 “red” edges, and 3 with 1 “red,” 2 “blue.” Implication: the Box-Kite’s structure can graph the substructure of a Sail’s Q-copies – which is not an empty execise! Why? Take the Zigzag’s (A,a); (B,b); (C,c) Assessors and imagine them agitated or “boiled” until they split apart. Send L and U terms to strut-opposite positions, then let them “catch” higher 32-D terms, with a higher-order G=16 instead of 8. We are now in the Pathions – the on-ramp to the Metafractal Highway!

18. 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:

19. 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.

20. 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”:

21. 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.

22. 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.)

23. 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).

24. 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 …)

25. 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!)

26. 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.)

27. 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.

29. 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!)

30. Levi-Strauss, in his own words:

31. 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:

32. 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!”

33. Here’s how a classical rhetorician sees it:

34. 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.

35. 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.”

36. 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.”

37. Examples • Ad touting online employment service: • “Our database makes the Taj Mahal look like a second-floor walkup.”

38. 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”)

39. 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…)

40. 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.)

41. 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

42. 2nd Nested “Sky-Box” emerges in 64-D:30 “blue-sky” border cells, one per each newAssessor 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:

44. 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!

45. 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).

46. If S, asstring, 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. Canonical Récipés