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Voyage by Catamaran

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

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

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

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Same-colored edges of a Catamaran (one per each orthogonal Square in a Box-Kites 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).

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But whats 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

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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…)

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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:

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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 i a is to the left of i G. 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)

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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)

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Thats 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!

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

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Vents, Sails, and Box-Kites The strut constant (S) is the missing Octonion: in the 16-D Sedenions, where Box-Kites first show up, the vertices each take 2 integers, L less than the CDP generator (G) of the Sedenions from the Octonions (2 3 = 8), and U greater than it (and <> G + L). There being but 6 vertices, one Octonion must go AWOL, in one of 7 ways. Hence, there are 7 Box-Kites in the Sedenions. But 7 * 6 = 42 Assessors (the planes whose diagonals are ZDs!)

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

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

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A different view, with numbers too! Note the edges of the Zigzag and the Vent opposite it are red, while the other 6 edges are blue. If the edge is red, then the ZDs joined by it make zero by multiplying / with \: for S=1, in the Zigzag Sail ABC, the first product of its 6-cyle {/ \ / \ / \} is (i 3 + i 10 )*(i 6 – i 15 ) = (i 3 – i 10 )*(i 6 + i 15 ) = A*B = {+ C – C} = 0 For a blue edge, /*/ or \*\ make 0 instead: again for S=1, in Trefoil Sail ADE, the first product of its 6-cycle { / / / \ \ \ } is (i 3 + i 10 )*(i 4 + i 13 ) = (i 3 – i 10 )*(i 4 – i 13 ) = A*D = {+ E – E} = 0

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

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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:

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Heres 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.

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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:

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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.

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The above is a coming attraction: for the Double Cusp is the basis of Petitots model of Levi-Strausss 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.)

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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).

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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-Kites struts! (A bit of buffer refresh on this point is coming up next …)

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Strut Opposites and Semiotic Squares René Thoms disciple, Jean Petitot, has been translating the structures of literary and mythic theory – Algirdas Greimas Semiotic Square, Lévi-Strauss Canonical Law of Myth – into Catastrophe Theory models; here, we translate these into Box-Kite strut-opposite logic: ZD representation theory as semiotics. 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!)

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The Klein-Group Connection In the Semiotic Squares 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.)

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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 latters Sail and Strut structure: eliminate the empty spaces by collapsing the Octahedron to a Tetrahedron, and associating opposite edges with strut- opposite vertices.

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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!)

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Levi-Strauss, in his own words:

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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:

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The pseudocode given in the formula baffled Harvards Howard Gardner, who had the honesty to admit he couldnt 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!

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Heres how a classical rhetorician sees it:

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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.

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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.

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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.

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

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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 Picards Make it so! – i.e., the tell-tale splice-phrase looks like)

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Now lets 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 weve 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, its a scale-free network, hence implicitly fractal. {1A.} Ergo, bring in Zero-Divisors as minimal descriptive tool! (Time to review…)

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The Simplest (Sedenion) Emanation Tables For S=1 Box-Kite, put L-indices of the 6 vertices as labels of Rows and Columns of a ZD multiplication table, entering them in left-right (top-down) order, with smallest first, and its strut-opposite in the mirror-opposite slot: 2 xor 3 = 4 xor 5 = 6 xor 7 = 1 = S. If R and C dont mutually zero-divide, leave cell (R,C) blank. Otherwise, enter the L-index of their emanation (the 3 rd Assessor in their common Sail). (Oh, yeah: ignore the minus signs.)

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

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

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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!

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

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

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And now for some more Semantic Web Q&A: {2Q.} The Semantic Web, as currently implemented and thought about, isnt! {2A.} Its 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, well need archetypal sentences – that is, explicit semantics!! – in our searches.

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

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Heres what Im talking about:

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

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ZDs as representation theory All the Double Cusps 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. Well just look very quickly at a few of these, the most basic being the already alluded-to Dual Cusp.

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

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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 youre 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.

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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).

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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.

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Will the Semantic Web kill the blogosphere star? (Will Michael Heims prognostications in The Metaphysics of Virtual Reality come to pass – will Hermann Hesses Glass Bead Game arrive among us at last, and be played in The Matrix?)

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Coda (I couldnt 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 Vicos Four Ages, that hoary old pousse café of Providences working from above. Yet the bit-twiddling logic of Box-Kites and Skies makes it clear that theres an NKS burbling from beneath whose ultra- simple No-Mind algorithmics must be exceeding rich (and which remains to be explored). I dont know about God or Devil, but I do believe in the Baroness Orczys scarlet hero:

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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: dont be fooled!)

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