LULL: from 256 Odu of IFA to 256 Cellular Automata

Ramon Llull Connected Ancient Wisdom to 21st Century Science


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


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

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)


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


each Cl(8) has 8 +half-spinors and 8 -half-spinors


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


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).


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)


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:


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.


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 


as well as ratios of force strength constants:

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


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'hamsmith/ or mirror site at 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). ..."


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)



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'hamsmith/ or mirror site at the contents of which are hereby incorporated herein by this parenthetical reference )