The Arabic Divination system Ilm al Raml is based on 16
Elements.
Llull showed 120 lines connecting distinct Pairs of the
16 Elements, so Llull was looking at Pairs of the 16
Elements.
There are in all 16x16 = 256 Pairs of 16
Elements.
The Arabic Divination system Ilm al Raml came from
Africa, where the most Ancient Divination system IFA is based on 256
Elements seen as the result of casting 8 shells or coins to get 8
binary choices (face up or face down)
With a chain of 8 coins
there are
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
= 2^8 = 256 possible outcomes
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256
for 256 possibility Elements based on casting 8 shells or
coins.
The 8+8 = 16 Elements in Llull's A-wheel were two sets of
8 Elements each,
or two 8-Element sets of IFA Divination
Coins,
which
Llull showed in his X-wheel with two sets of 8 Elements
each.
Here one set of 8 Elements and their Pair-lines is shown
in Green
and the othe set of 8 Elements and their Pair-lines is
shown in Black.
To connect the Ancient Wisdom shown in Llull's wheels
with 21st Century Science means that it must be interpreted in terms
of the Language used by 21st Century Science, which is
Mathematics.
Llull's X-wheel shows that Llull's 16 Element Structure (
call it Llull(16) ), factors into two independent 8 Element
Structures ( call each of them Llull(8) ), so that by his X-wheel
Llull is saying, in Mathematical Language:
Llull(16) = Llull(8) (x) Llull(8)
If you follow Llull's idea to its logical conclusion, go
from 16 = 8 times 2 to 8 times N for any (maybe very large) number N
and get the factoring
Llull(8N) = Llull(8) (x) ...(N times)... (x)
Llull(8)
There is a Mathematical Structure with that factoring
property
that is based on the Binary Choice of Llull's wheels Y
and Z
Real Clifford Algebras with 8-Fold Periodicity
Factoring.
So, changing notation from Llull(16) to Cl(16) etc, because
mathematicians mostly use the term Clifford Algebra, named for
William Kingdon Clifford, who, over 100 years ago, wrote about such
Algebras, the factoring is
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
to get what might be called a generalized Hyperfinite II1
von Neumann Algebra factor
that represents a realistic Unified Physics Model
including Gravity and the Standard Model,
So, consider each set of 8 Elements as an 8-dimensional
Space.
The Simplest Geometric Objects in the 8-dimensional
Octonion Space are:
- 0-dimensional points (there is only
1 general type, involving none of the Basis Element directions)
- 1-dimensional line Subspaces (8
general types, one in each of the Basis Element directions 1, i,
j, k, E, I, J, K)
- 2-dimensional plane Subspaces (28
general types, each in a Pair of Basis Element
directions)
- 3-dimensional Subspaces (56 general
types, each in a Triple of Basis Element directions)
- 4-dimensional Subspaces (70 general
types, each in a Quadruple of Basis Element directions)
- 5-dimensional Subspaces (56 general
types, each in a Quintuple of Basis Element
directions)
- 6-dimensional Subspaces (28 general
types, each in a Sextuple of Basis Element directions)
- 7-dimensional Subspaces (8 general
types, each in a Septuple of Basis Element directions)
- 8-dimensional Space itself (there
is only 1, involving the Octuple of all the Basis Element
directions)
There are 1+8+28+56+70+56+28+8+1 = 256
of the Simplest Geometric Objects in 8-dimensional Space.
They fit together in the form of a Matrix Algebra that
is in fact the Clifford Algebra over 8-dimensional Space,
Cl(8).
Cl(8) is also called by some a Geometric Algebra
because it is an Algebra that describes the Simplest Geometric
Objects.
Each of the 8 Elements is a Basis Element of the
8-Dimensional Space, one of which is represented by the Real Number 1
and 7 of which are represented by the 7 Octonion Imaginary Numbers i,
j, k, E, I, J, K .
(The Octonion Imaginary Numbers i,j,k,E,I,J,K are
generalizations of the Complex Imaginary Number i.)
The points in the 8-dimensional Octonion Space can be
multiplied with each other
and the Octonion Multiplication Rules can be seen in
terms of Pair-lines between the 7 Octonion Imaginary
Elements
as shown by Llull in his V-wheel with two sets of 7
Elements and their Pair-lines.
Here one set of 7 Octonion Imaginary Elements and their
Pair-lines is shown in Red
and the other set of 7 Octonion Imaginary Elements and
their Pair-lines is shown in Black.
The Simplest Geometric Objects of the Clifford Algebra
Cl(8) are all Flat.
If you go beyond Flat to look at Curved
things,
the Simplest Curved Geometric Objects in the
8-dimensional Octonion Space are:
- Circles, which are 1-dimensional
Spheres S1
- 2-dimensional Spheres
S2
- 3-dimensional Spheres
S3
- 4-dimensional Spheres
S4
- 5-dimensional Spheres
S5
- 6-dimensional Spheres
S6
- 7-dimensional Spheres
S7
The Maximal Sphere in 8-dimensional Octonion Space is the
7-sphere S7.
In each of the two sets of 7 Elements in Llull's
V-wheel,
the 7 Elements represent the 7 dimensions of a 7-sphere
S7.
If you use the Octonion Multiplication Rules, you can
multiply points on the 7-sphere S7.
Some of the S7 Multiplications produce points already in
the 7-dimensional S7,
but others produce points in a 21-dimensional Space whose
21 dimensions correspond to the 21 Pair-lines shown by Llull in his
V-wheel, so
S7 Multiplication produces a 7+21 = 28-dimensional Curved
Geometric Object called a Lie Group,
denoted by Spin(8) since it describes rotations in
8-dimensional Octonion Space.
The 28-dimensional Lie Group of S7 Multiplication
corresponds to the 28 Pair-lines of Llull's X-wheel,
that is, the 2-dimensional plane Subspaces (28 general
types, each in a Pair of Basis Element directions),
thus connecting the 2-dimensional plane Subspaces of the
Clifford Algebra Cl(8)
with the Lie Group Spin(8) that describes rotations in
8-dimensional Octonion Space.
Spin(8) not only describes ordinary rotation in
8-dimensional Octonion Space,
it also describes the connection of the thing that is
rotating with its surroundings.
A 3-dimensional example of
Spin-Connection-With-Surroundings shows a cup held by a dancer in one
hand. Rotating the cup by 360 degrees gets the arm (which is
connected to the shoulder of the dancer) twisted, but turning the cup
another 360 degrees gets the arm back straight:
- picture 1 is the start,
- picture 2 is 180 degrees,
- picture 3 is 360 degrees (note how
the arm is twisted),
- picture 4 is 540 degrees,
- and picture 1 again is 720
degrees.
Since Spin-Connection-With-Surroundings action of Spin(8)
takes 720 degrees (2 full rotations) to get back where it started,
objects with Spin-Connection-With-Surroundings are called Spin 1/2
objects, or Spinors.
Since a Spinor must be rotated fully twice to get back
where it started, a Spinor can be seen as a Square Root of a
Rotation.
Mathematically in terms of the Clifford Algebra Cl(8),
there are 16 = SquareRoot(256) independent Spinors,
8 of which, because of the way their
Spin-Connection-With-Surroundings works, are called
+half-Spinors
and
the other 8 of which, because they look like mirror
images of the first 8, are called -half-Spinors.
In the case of the 8-dimensional Octonionic Clifford Algebra
Cl(8):
the 8 basis Elements 1,i,j,k,E,J,K that define
1-dimensional line Subspaces, called Cl(8) Vectors;
and
the 8 Cl(8) +half-Spinors
and
the 8 Cl(8) -half-Spinors
are functionally equivalent by what is called (since
there are 3 types of things that are equivalent) Spin(8)
Triality.
At this point, using the A-wheel, X-wheel, and
V-wheel
Llull has described the basic ingredients for a
high-energy (around Planck Energy) physics model with:
Clifford Algebra Cl(8) = 16x16 Real Matrix
Algebra
8 Cl(8) +half-Spinors representing the first-generation
Fermion Particles
8 Cl(8) -half-Spinors representing the first-generation
Fermion Antiparticles
8 Cl(8) Vectors representing the 8-dimensional spacetime
that looks like S1 x S7
28 Spin(8) Pair-line Elements representing 28 Gauge
Bosons
The half-Spinor parts of the Llull Model already
look realistic, by the correspondences:
- 1 =
Neutrino
- i = Red Up Quark
- j = Green Up Quark
- k = Blue Up
Quark
- E = Electron
- I = Red Down Quark
- J = Green Down
Quark
- K = Blue Down
Quark
However, the Spacetime we see is 4-dimensional that looks
like S1 x S3
and
our Standard Model for the SU(3) Color Force, the SU(2)
Weak Force, and U(1) Electromagnetism has 8+3+1 = 12 Gauge
Bosons
and
Gravity can be seen, through a generalized MacDowell
Mansouri mechanism, as due to 15 Gauge Bosons of the Conformal Group
SU(2,2) = Spin(2,4).
Llull's S-wheel and T-wheel show how to get our Spacetime
and the Standard Model and Gravity.
Llull's S-wheel has a central S-square whose 4 corners
represent { 1, i, j, k, }, which are 4 of the 8 Cl(8) basis
Elements.
{ 1, i, j, k } are basis Elements for a 4-dimensional
Quaternionic Subspace that Freezes Out of high-energy
8-dimensional spacetime at lower (with respect to Planck Energy)
energies.
That 4-dimensional Quaternionic Subspace is our
4-dimensional Physical Spacetime that looks like S1 x S3. The other 4
of 8 dimensions become a CP2 Internal Symmetry Space.
When that Quaternionic Structure is introduced, the 16x16
Real Matrix Algebra of Cl(8) is transformed into the 8x8 Quaternionic
Matrix Algebra of Cl(2,6).
Since the 28-dimensional gauge group Spin(8) no longer
has a unified 8-dimensional Spacetime on which to act, its 28
generators break down into 28 generators capable of acting
on
4-dimensional Physical Spacetime and 4-dimensional CP2
Internal Symmetry Space
and on the Fermion Particles and Antiparticles, which now
come in 3 types, or generations:
- 1 - living in S1xS3 Physical
Spacetime
- 2 - living in S1xS3 Physical
Spacetime and CP2 Internal Symmetry Space
- 3 - living in CP2 Internal Symmetry
Space
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
the 4 U(2) ElectroWeak Gauge Bosons ( U(2) = SU(2) x U(1)
for 3 Weak Bosons and 1 Electromagnetic Photon) are represented by
the 4 corners of one of the three back squares of Llull's
S-wheel
the 8 SU(3) Color Gluon Gauge Bosons are represented by
the 8 corners of the two remaining back squares of Llull's
S-wheel.
To see the Gravity and its 15 Gauge Bosons, look at
Llull's T-wheel.
The 3 corners of the front T-triangle in Llull's T-wheel
represent a 3-dimensional Cartan subalgebra of the 15-dimensional
Conformal Group SU(2,2) = Spin(2,4).
The 12 corners of the other 4 triangles in Llull's
T-wheel represent the 12 vertices of the Cuboctahedron
Root Vector Polytope of the Conformal Group SU(2,2) =
Spin(2,4)
The 3+12 = 15 SU(2,2) = Spin(2,4) Conformal Group
GraviPhoton Gauge Bosons act on 4-dimensional Physical Spacetime
by:
- 4
Translations,
- 6 Lorentz
Transformations,
- 4 Special Conformal Transformations
and
- 1
Dilation.
They produce Gravity through a generalized
MacDowell-Mansouri mechanism.
Using the Llullian Structures, you can construct 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 Ancient Wisdom 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 Elements 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 )
......