Ramon Llull (ca 1232 to 1315) developed his
16-vertex A-wheel by studying Arabic language
and culture.
The Arabic divination system Ilm al Raml is based on 16
elements.
Llull showed 120 lines connecting distinct pairs of the
16 vertices.
They were all the distinct pairs ( 16x16 - 16 = 256 - 16
= 240 ) but considering that order does not matter ( line from A to B
is considered the same as line B to A ) so that
the total number of distinct pairs (disrearding order) is
240/2 = 120 .
The 16 vertex elements and 120 line pairs correspond to
the 16 vectors and 120 bivectors of the real Clifford Algebra
Cl(16).
Cl(16) = Cl(8 (x) Cl(8)
each Cl(8) is 16x16 = 256-dimensional
and
since any very large Clifford Algebra Cl(8N) can be
factored into the tensor product
Cl(8N) = Cl(8) ... (x) ... (N times tensor product) ...
(x) ... Cl(8)
you can take the completion of the union of all the
tensor products and get a generalized Hyperfinite II1 von Neumann
Algebra factor that represents a realistic Unified Physics Model
including Gravity and the Standard Model,
so, effectively,
Llull's A-wheel is a Seed that grows into a realistic
Unified Physics Model including Gravity and the Standard
Model.
Where did Llull's A-wheel Seed come from?
Llull probably developed his A-wheel from the 16-element
Arabic divination system Ilm al Raml.
Ilm al Raml came to Arabs from African IFA.
IFA has 2^8 = 16x16 = 256 Odu
based on 16 Orishas
- 0 Ogbe (light of creation) - Orunmila (wisdom
Tao)
- 1 Osa (creativity) - Oya (wind)
- 2 Otura (unity of everything) - Osain (forest
spirits)
- 3 Owonrin (rain) - Oshun (rivers) (He
Xiangu)
- 4 Irete (fate) - Babalu Aye (healer) (TieGuai
Li)
- 5 Ofun (taboo) - Eshu Legba (trickster)
- 6 Edi (womb) - Obatala (hermaphrodite father
of humanity)
- 7 Okanran (lightning) - Chango
(thunder)
- 8 Ogunda (sword) - Ogun (iron) (Lu
Dongbin)
- 9 Iwori (consciousness) - Eshu Legba
(messenger)
- 10 Ose (victim of abuse) - Osun (guardian
angel)
- 11 Oturopon (trap) - Ochosi (hunter)
- 12 Irosun (fire) - Aganyu
(volcano)
- 13 Ika (forest land) - Yemaya
(ocean)
- 14 Obara (rainbow) - Inle (two intertwined
snakes, medicine)
- 15 Oyeku (dark of earth) - Ibeyi (twins
Yin-Yang)
Pairs of the 16 Orishas give 16x16 = 256
Odu
What are the structural relationships among the 256
Odu,
corresponding to each of the two 256-dimensional Cl(8) of
Llull's Cl(16) = Cl(8) (x) Cl(8) ?
Since the graded structure of each Cl(8) is 256 = 1 + 8
+ 28 + 56 + 70 + 56 + 28 + 8 + 1 we have
Cl(8) (x) Cl(8) = ( 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 ) (x) (
1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 ) =
= 1 + 16 + 120 + 560 + 1,820 + 4,368 + 8,008 + 11,440 + 12,870 +
11,440 + 8,008 + 4,368 + 1,820 + 560 + 120 + 16 + 1 = 65,536 =
Cl(16)
and
for vectors, 1x8 + 8x1 = 8 + 8 = 16
for bivectors, 1x28 + 8x8 + 28x1 = 28 + 64 + 28 = 120
for spinors, sqrt(65,536) = sqrt(256x256) = sqrt(256) x sqrt(256)
= 16 x 16
for half-spinors, Cl(16) has 128 + 128 and Cl(8) has 8 + 8
so
each Cl(8) has 8 +half-spinors and 8 -half-spinors
and
by Triality for Cl(8)
the 8 Cl(8) vectors = the 8 Cl(8) +half-spinors = the 8
Cl(8) -half-spinors
Llull's X-wheel has 8 vertices
(green) for vectors of one
Cl(8)
and 8 vertices (black) for vectors of the other
Cl(8)
For each Cl(8), Llull shows 28 lines between distinct
pairs of vertices (disregarding order)
corresponding to the 28 bivectors of that
Cl(8).
The 8 dimensions of each Cl(8) vector space correspond to
the 8-dimensional Octonions with basis { 1 , i , j , k , E , I , J ,
K } .
The 7 imaginary Octonion basis elements { i , j , k , E ,
I , J , K } correspond to the 7 dimensions of the unit 7-sphere S7 in
Octonionic space.
Llull's V-wheel has 7 vertices
(red) for the vector space 7-sphere S7
of one Cl(8)
and 7 vertices (black) for the vector space 7-sphere S7
of the other Cl(8)
For each Cl(8), Llull shows 21 lines between distinct
pairs of vertices (disregarding order)
corresponding to 21 new independent dimensions created by
taking the Lie Group Product of points on S7.
Unlike the 3-sphere S3 in Quaternionic space with basis {
1 , i , j , k } for which the Lie Group Product of any two points is
also a point on S3,
the Lie Group Product of any two points on S7 is NOT
necessarily also a point on S7, but may lie in any of 21 newly formed
independent dimensions,
so that
the Lie Group Product for S7 produces the 7+21 =
28-dimensional Lie Group Spin(8) of the bivectors for each
Cl(8).
At this point ,
Llull has described the basic ingredients for a
high-energy (around Planck Energy) physics model with:
Clifford Algebra Cl(8) = Cl(1,7) = 16x16 Real Matrix
Algebra
8 first-generation fermion particles
8 first-generation fermion antiparticles
8-dimensional spacetime RP1 x S7
28 Spin(8) = Spin(1,7) gauge bosons
a Lagrangian integrated over the 8-dimensional spacetime
with fermion and gauge boson terms
a Many-Worlds Sum-Over-Histories Path Integral Quantum
Theory.
To visualize this 16x16 = 256 element structure, use the
correspondence with the 256 Cellular Automata
with
blue for the 8 spacetime
dimensions
cyan for the 8 +half-spinor
first generation fermion particles
yellow for the 8
-half-spinor first generation antiparticles
magenta for the 28 Spin(8)
gauge bosons.
The 256 Cellular Automata, in turn, correspond to the 256
Odu of IFA, so, with the same color coding, the structure of Llull's
8-dimensional Science can be seen in terms of the 256 Odu of
IFA:
(click here or on the image to see
a larger scale rotated 90 degrees)
How is Llull's 8-dimensional Science Structure related to
our 4-dimensional Physical Spacetime ?
At low (compared to Planck) energies, a 4-dimensional
Quaternionic subspace Freezes Out of high-energy 8-dimensional
spacetime. Its Quaternionic basis { 1 , i , j , k } corresponds to
the 4 vertices of the S-square of Llull's S-wheel.
When the Quaternionic Structure is introduced, the 16x16
Real Matrix Algebra of Cl(8) = Cl(1,7) is transformed into the 8x8
Quaternionic Matrix Algebra of Cl(2,6).
Full Spinors of the 8x8 Quaternionic Matrix Algebra of
Cl(2,6) have 8 Quaternion dimensions or 32 Real
dimensions,
and so are incompatible with the 16-Real-dimensional Full
Spinors of the 16x16 Real Matrix Algebra of Cl(1,7) =
Cl(8).
To get a Quaternionic Clifford Algebra with
16-Real-dimensional Full Spinors, in the form of 4 Quaternion
dimensions, reduce
from Cl(2,6) = 8x8 Quaternionic Matrix
Algebra
to the 4x4 Quaternionic Matrix Algebra of
Cl(2,4).
with
blue for the 4 spacetime
dimensions and the 4 internal symmetry space dimensions
cyan for the 8 +half-spinor
first generation +half-spinor fermion particles
yellow for the 8
-half-spinor first generation -half-spinor fermion
antiparticles.
The fixed Quaternionic subspace corresponds to
4-dimensional Physical Spacetime with structure RP1xS3, with the
remaining 4 dimensions of 8-dimensional Spacetime corresponding to a
4-dimensional Internal Symmetry Space with structure
CP2.
Since the 28-dimensional gauge group Spin(1,7) no longer
has a unified 8-dimensional RP1xS7 Spacetime on which to act, its 28
generators break down into 28 generators capable of acting on
4-dimensional RP1xS3 Physical Spacetime and 4-dimensional CP2
Internal Symmetry Space:
magenta for the U(1)
propagator phase that is defined with respect to the fixed
Quaternionic 4-dimensional spacetime subspace corresponding to the
S-square of Lull's S-wheel
gold for the 4 U(2)
electroweak gauge bosons
green for the 8 SU(3) color
gluon gauge bosons
red for the 15 SU(2,2) =
Spin(2,4) Conformal Group graviphoton gauge bosons that act on
4-dimensional Physical Spacetime by:
4 Translations,
6 Lorentz Transformations,
4 Special Conformal Transformations and1
Dilation.
They produce Gravity through a generalized
MacDowell-Mansouri mechanism,
and are shown by Llull in his T-wheel
The 3 vertices of the T-triangle in Llull's T-wheel
correspond to a 3-dimensional Cartan subalgebra of the 15-dimensional
Conformal Group SU(2,2) = Spin(2,4) and 3 Quaternionic Real (blank)
Cellular Automata.
The other 4 triangles in Llull's T-wheel correspond to 4
sets of associative triples ( / | \ ) of Cellular Automata and to the
12 vertices of the Cuboctahedron Root Vector Polytope of the
Conformal Group SU(2,2) = Spin(2,4)
and
the other 4 triangles (red, blue, green, and magenta in
the figure below) have 4x3 = 12 vertices corresponding to the 12
vertices of the cuboctahedron root vector polytope of the Conformal
Group SU(2,2) = Spin(2,4)
Returning to Llull's S-wheel
the remaining 3 squares of Llulls' S-wheel form the
SU(3)xSU(2)xU(1) Standard Model:
Two of the 3 squares can be considered as the vertices of
a cube:
Now look at the cube along a diagonal axis, and project
all 8 vertices onto a plane perpendicular to that axis, giving the
root vector diagram of SU(3)
Since each gluon links 4-dim spacetime to color internal
symmetry space, the gauge group SU(3) acts globally on CP2 Internal
Symmetry Space, as can be seen by the fibration CP2 = SU(3) /
U(2)
The third of the remaining squares, that is the
final square, corresponds to the 4 Cellular Automata
that correspond to the 3 SU(2) weak bosons and the U(1)
electromagnetic photon.
Since SU(2)xU(1) = U(2), and since CP2 = SU(3) / U(2),
they act locally on CP2 Internal Symmetry Space.
If you start with the Lagrangian over 8-dimensional
Spacetime, and freeze out a Quaternionic subspace as described
above,
you get a Lagrangian over 4-dimensional Physical
Spacetime that allows you to calculate:
- Me-neutrino = Mmu-neutrino =
Mtau-neutrino = 0 (tree-level)
- Higher-order corrections give:
Neutrino mixing matrix:
nu_1 nu_2 nu_3
nu_e 0.87 0.50 0
nu_m -0.35 0.61 0.71
nu_t 0.35 -0.61 0.71
- Me = 0.5110 MeV
- Md = Mu = 312.8 MeV (constituent quark
mass)
- Mmu = 104.8 MeV
- Ms = 625 MeV (constituent quark
mass)
- Mc = 2.09 GeV (constituent quark
mass)
- Mtau = 1.88 GeV
- Mb = 5.63 GeV (constituent quark
mass)
- Mt = 130 GeV (constituent Truth Quark
mass) as the ground state of a 3-state T-quark - Higgs -
Triviality Boundary System
NJL
state with Tquark mass = 130 GeV and Higgs mass = 146
GeV in the stable region far from the triviality and
vacuum stability bounding curves and therefore closely
related to other quarks in the stable region and therefore
to single-Tquark events involving such things as T-Bbar
events
8-dim Kaluza-Klein state with
Tquark mass 172-175 GeV = and Higgs mass = 176-188
GeV, which state is closely related to the Higgs and
T-Tbar condensates, and hence to T-Tbar events
BHL state with Tquark mass = 218
+/- 3 GeV and Higgs mass = 239 +/- 3 GeV at the
Triviality Bound - Vacuum Stability Critical Point.
and
Kobayashi-Maskawa
matrix:
d s b
u 0.975 0.222 0.00249 -0.00388i
c -0.222 -0.000161i 0.974 -0.0000365i 0.0423
t 0.00698 -0.00378i -0.0418 -0.00086i 0.999
and
- W+ mass = W- mass = 80.326
GeV
- Z0 mass = 91.862 GeV
- Higgs mass = 145.8 GeV
- weak force - Higgs VEV = 252.5
GeV (assumed, since ratios are calculated)
as well as ratios of force strength constants:
- Gravitational G = (Ggravity)(Mproton)^2
= 5 x 10^(-39) (assumed, since ratios are calculated)
- electromagnetic fine structure constant
= 1/137.03608
- Gfermi = (Gweak)(Mproton)^2 = 1.02 x
10^(-5)
- color force strength = 0.6286 (at 0.245
GeV) - perturbative QCD running gives
- color force strength = 0.167 (at 5.3
GeV)
- color force strength = 0.121 (at 34
GeV)
- color force strength = 0.106 (at 91
GeV)
If Nonperturbative QCD and other things are taken into account,
then the color force strength = 0.123 (at 91 GeV).
Dark Energy : Dark Matter : Ordinary Matter
Ratio
0.75 : 0.21 : 0.04
( for details of the physics model and the calculations, and for
references and source material, see my web site at
www.valdostamuseum.org/'hamsmith/ or mirror site at
www.tony5m17h.net/ the contents of which are hereby incorporated
herein by this parenthetical reference )
So, following Llull's path leads to a Unified Physics
Model that meets Einstein's Criterion:
"... a theorem which at present can not be based upon
anything more than upon a faith in the simplicity, i.e.,
intelligibility, of nature:
there are no arbitrary constants ...
that is to say,
nature is so constituted that it is possible logically to
lay down such strongly determined laws that within these laws only
rationally completely determined constants occur
(not constants, therefore, whose numerical value could be
changed without destroying the theory). ..."
and
Ramon Llull was a Nexus between the Ancient Wisdom of the
256 Odu of IFA and a Unified Model of 21st Century
Physics.
His Lullian Wheels were only one of the ways by which he
sought to transmit Ancient Wisdom to the People of his
Future.
Within a century or two after Llull's death, his
followers in what is now known as Italy produced the 78-card Tarot
Deck of Cards.
Just as the practices of Divination had preserved in
Africa
(and in the Mediterranean through 16-element Ilm al Raml, and
in China through the 64-element I Ching, and in Japan through
128-element Futomani Book of Shinto Divination, and in India through
the 24+192 = 240-element First Sukt of the Rig Veda)
the Ancient Wisdom of the 256 Odu of IFA,
the 78-Card Tarot Game/Divination
(and the common 52-Card Game Deck that descended from Tarot,
which 52 Cards correspond to the 52-dimensional exceptional Lie
Algebra F4 of which the 78-dimensional exceptional Lie Algebra E6 is
a complexification)
would spread throughout the Global Society that began to
be formed in the 1400-1500s and so preserve Fundamental Details of
the Ancient Wisdom seen by Ramon Llull.
In this Tarot spread
( s=swords, w=wands, p=pentacles, c= cups and k=knave(page),
j=Knight(Jack), K=King, Q=Queen, and
0=Fool, 1=Magician, 2=Popess, 3=Empress, 4=Emperor, 5=Pope,
6=Lovers, 7=Chariot, 8=Justice, 9=Hermit, 10=WheelofFate,
11=Strength, 12=HangedMan, 13=Death, 14=Temperance, 15=Devil,
16=Tower, 17=Star, 18=Moon, 19=Sun, 20=Judgment, 21=World)
- The 28 magenta 28 are the 28
Spin(8) adjoint bivectors of Cl(8).
- The 16 blue are the 8 vectors of
Spin(8) and Cl(8) and their 8 dual/conjugates.
- The 32 red are the 16 spinors (8
+halfspinors and 8 -halfspinors) of Spin(8) and Cl(8) and their 16
dual/ conjugates.
- The 2 black are diagonal degrees of freedom in the 26-dim
traceless J3(O)o part of the J3(O) Jordan algebra.
Considered all together, the 78 Tarot Cards correspond to
the 78 elements of the exceptional Lie Algebra E6, which has a
5-graded structure with dimensionalities
8 + 16 + (28+1+1) + 16 + 8
that represents the Llullian Realistic Unified Physics
Model.
If you regard strings as world-lines of particles in the
Quantum Path Integral Sum-Over-Histories in the Many-Worlds, then an
E6 String Theory produces a generalized Bohm Quantum Potential, with
Sarfatti-type Back-Reaction, that is useful in describing
Penrose-Hameroff Quantum Consciousness.
In acting as a Nexus connecting Ancient Wisdom with 21st
Century Physics,
Ramon Llull expressed what Einstein (in
the New York Times Magazine on November 9, 1930 pp 1-4) called
cosmic religious feeling:
"... It is very difficult to elucidate ... cosmic
religious feeling ... to anyone who is entirely without it
The individual feels the futility of human desires and
aims and the sublimity and marvelous order which reveal themselves
both in nature and in the world of thought.
Individual existence impresses him as a sort of prison
and he wants to experience the universe as a single significant
whole.
... the cosmic religious feeling is the strongest and
noblest motive for scientific research.
What a deep conviction of the rationality of the universe
and what a yearning to understand
Kepler and Newton must have
had to enable them to spend years of solitary labor in disentangling
the principles of celestial mechanics!
Those whose acquaintance with scientific research is
derived chiefly from its practical results easily develop a
completely false notion of the mentality of the men who, surrounded
by a skeptical world, have shown the way to kindred spirits scattered
wide through the world and through the centuries.
It is cosmic religious feeling that gives a man such
strength.
".
Frank Dodd (Tony) Smith, Jr.
April 2007
( for further material and details, and for references and source
material, see my web site at www.valdostamuseum.org/'hamsmith/ or
mirror site at www.tony5m17h.net/ the contents of which are hereby
incorporated herein by this parenthetical reference )
......