Voyage by Catamaran

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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 catamaranswith 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 lt G, its
  product with the next 2N-ion Generator has index
  (aG), 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)

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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 n4, 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 (23 8), and U
  greater than it (and ltgt 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 S1,
• in the Zigzag Sail ABC, the first product of its
  6-cyle / \ / \ / \ is
• (i3 i10)(i6 i15) (i3 i10)(i6 i15)
  AB C C 0
• For a blue edge, // or \\ make 0
  instead again for S1,
• in Trefoil Sail ADE, the first product of its
  6-cycle / / / \ \ \ is
• (i3 i10)(i4 i13) (i3 i10)(i4 i13)
  AD 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 G16
  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.)
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).
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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)Fa-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 QA

• 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 S1 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 3rd Assessor in their common Sail). (Oh,
  yeah ignore the minus signs.)

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S15 Sky emerges in 32-D Pathions

• 6 Sedenion Assessor-dyads of S 7 Box-Kite SPLIT
  UP
• L (index lt 8) and U (index gt 8) units all become
  L-units
• (index lt 16 new G) in Pathion 3-BK ensemble
  with S 15 ( 87 ),
• along with prior G (8) S (7), which capture
  U-units (index gt G)
• from ambient turbulence, resulting in 14 Pathion
  Assessors

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2nd Nested Sky-Box emerges in 64-D30
blue-sky border cells, one per each
newAssessor in the 26-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 ( 2N-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 S15 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 gt 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 gt 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 b1,b2,,bk in L-to-R
  positions from 2H to 2L (L gt 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 ( RCP
  (S0) ) 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.

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And now for some more Semantic Web QA

• 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
57
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.

59
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!)
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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 QA

• 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 dd elusive
Pimpernel!(The version with Leslie Howard,
Merle Oberon, Raymond Massey and Nigel Bruce is
the only one worth watching dont be fooled!)