- CP2 Internal Symmetry Space and RP1xS3 Spacetime
- Gauge Bosons as Pairs of Fermion Nearest-Neighbors
- SU(3) Color Force
- SU(2) Weak Force
- U(1) Electromagnetism
- Spin(2.3) MacDowell-Mansouri Gravity

Dimensional reduction reduces theoriginal nonassociative octonionic 8-dimensional Spacetime to aphysical associative 4-dimensional Spacetime and a coassociative4-dimensional Internal Symmetry Space. It also reducesthe 28 gauge bosons of Spin(8) to 16 generators of the Lie AlgebraU(4) = U(1)xSU(4) = U(1)xSpin(6) of Gravity and the Higgs Mechanismplus 12 generators of the Lie Algebra U(1)xSU(2)xSU(3) of theStandard Model.

The D4-D5-E6-E7-E8 VoDou Physicsmodel Lagrangian action, after dimensional reduction, is the sumof 4 terms:

Int Lgr dMgr + Int Lsu3 dMsu3 + Int Lsu2 dMsu2 + Int Lu1 dMu1

where Int stands for integration,

Lgr, Lsu3, Lsu2, and Lu1 are Lagrangian densities for the gravity,color, weak, and electromagnetic forces, and

dMgr, dMsu3, dMsu2, and dMu1 are the integration volume elementsfor the gravity, color, weak, and electromagnetic forces.

The integration Int is taken over a neighborhood M of4-dimensional spacetime. By Wick rotation, M can be seen as eitherEuclidean or Minkowski.

The Lagrangian terms for gravity, the SU(3) color force, the SU(2)weak force, and U(1) electromagnetism each determine the geometricforce strength constants for their respective forces. The gauge bosoncurvature part of the relative geometric force strengths (notincluding 1/mass squared terms for the Fermi weak constant and forthe Newton gravitational constant) can be calculated by looking atthe gauge boson curvature terms of the form F /\ *F for

Lgr, Lsu3, Lsu2, and Lu1,

and comparing the measures dMgr, dMsu3, dMsu2, and dMu1 ,

each measure being defined by the volume of a 4-dimensionalcompact symmetric space of constant curvature.

The compact spaces of constant curvature are defined by transitiveaction of each of the 4 forces on its respective 4-dimensionalSymmetry Space (for the SU(3) color, SU(2) weak, and U(1)electromagnetic forces, the coassociative Internal Symmetry Space;and for Gravity, Spacetime),

that is, by the way the gauge bosons "see" their SymmetrySpace:

Denote the 4 Symmetry Space components of the gauge boson by:

The Discrete HyperDiamond GeneralizedFeynman Checkerboard and ContinuousManifolds are related by QuantumSuperposition:

- Elements of a Discrete Clifford algebra correspond to Basis Elements of a Real Clifford algebra.
- General Elements of a Real Clifford algebra correspond to Superpositions of Basis Elements = Elements of the underlying Discrete Clifford algebra.
- Volumes of Spaces of Superpositions of some given Sets of Basis Elements correspond to Mass or Charge of Particles or Forces represented by those Basis Elements.
- Volumes of Spaces of Superpositions of other given Sets of Basis Elements correspond to Volume of Physical SpaceTime and Volume of Internal Symmetry Space represented by those Basis Elements.

Note that S1/Z2 can be described as an orbifold.

Dimensional reduction reduces theoriginal nonassociative octonionic 8-dimensional Spacetime to aphysical associative 4-dimensional Spacetime and a coassociative4-dimensional Internal Symmetry Space. It also reducesthe 28 gauge bosons of Spin(8) to 16 generators of the Lie AlgebraU(4) = U(1)xSU(4) = U(1)xSpin(6) of Gravity and the Higgs Mechanismplus 12 generators of the Lie Algebra U(1)xSU(2)xSU(3) of theStandard Model.

Corresponding to the way that **24of the 28 Gauge Bosons** of theD4-D5-E6-E7-E8 VoDou Physics model can be **representedby the vertices** of a 24-cell,**SpaceTime plus Internal SymmetrySpace**, and the **first-generationFermion Particles and anti-Particles**, can be **representedby the vertices of a ****dual24-cell**:

In the dual 24-cell, 8-dimensionalspacetime is represented by 8 vertices that correspond, afterdimensional reduction, to

- 4-dimensional Physical SpaceTime, whose corresponding vertices are denoted by
- 4-dimensional Internal Symmetry Space, whose corresponding vertices are denoted by

In this image

only the 8 vertices representing Physical SpaceTime and InternalSymmetry Space are marked.

MattiPitkanen has suggested that the global structure of 4-dimensionalSpacetime and Internal Symmetry Space should be given by**8-dimensional SU(3)**, which decomposes by SU(3) / U(2) = CP2into CP2 base and U(2) fibre, both of which are 4-dimensional andhave Quaternionic structure.

**Associative 4-dimensional Spacetime**, in the Minkowskiversion with -+++ signature (1,3),** is topologically U(2) =SU(2)xU(1) = S3 x S1 **(while the Euclideanversion of Spacetime has topology S4),

which is consistent with the D4-D5-E6-E7-E8VoDou Phyiscs model Spacetime of RP1x S3. As MattiPitkanen has noted, the Quaternions have both a natural Minkowskimetric (given by the square root of the real part Re(ZZ) of Z times Zfor Quaternion Z) and a natural Euclidean metric (given by the squareroot of ZZ* of Z times Z conjugate for Quaternion Z).

**Coassociative 4-dimensional Internal Symmetry Space is CP2.**

What is the relationship between 8-dimensional SU(3) and8-dimensional Octonions?

- SU(3) has a 2-dim Cartan subalgebra - Octonions have {1,E}
- SU(3) has a 3-dim representation - Octonions have {i,j,k}
- SU(3) has a dual 3-dim representation - Octonions have {I,J,K}

However, 1^2 = 1 and E^2 = -1, so 1 and E are not isomorphic,sothe Octonion {1,E} is not isomorphic to the SU(3) Cartansubalgebra.

Also, Octonion {i,j,k} is not isomorphic to {I,J,K}, but SU(3)3-dim representation is isomorphic to the dual 3-dim rep.

So, SU(3) and Octonions have their differences.

Look at the automorphism group Aut(O) = G2 of the Octonions.

Unlike the Quaternion case, where Aut(Q) = SU(2) = Spin(3) = S3 =Imaginary Quaternions, Aut(O) = G2 is 14-dimensional, whereas S7 =Imaginary Octonions is 7-dim.

Also unlike the Quaternion case, where SU(2) = Spin(3)double-covers the rotation group in 3-space, Aut(O) = G2 is notSpin(7) and is not the double-cover of the rotation group in 7-space,but Spin(7) and G2 are related by Spin(7) / G2 = S7.

How does Aut(O) = G2 act on 7-space?

Look at the 6-sphere S6 in 7-space.

Aut(O) = G2 is transitive on S6, just as is the double-coverSpin(7) of the rotation group in 7-space, but the isotropy group ofthe double-cover Spin(7) of the rotation group in 7-space isSpin(6),

that is, Spin(7) / Spin(6) = S6,

whereas Spin(7) / G2 = S7 and Spin(6) / SU(3) = S7

and the isotropy group of G2 is SU(3), that is, G2 / SU(3) =S6.

The elements of SU(3) are automorphisms of the Octonions and SU(3)is a subgroup of Spin(6) = SU(4), by the fibration Spin(6) / U(3) =SU(4) / (SU(3)xU(1)) = SU(4) / U(3) = CP3.

What does SU(4) look like? Consider the exterior algebra gradedstructures:

1 + 4 + 6 + 4 + 1 SU(4) has 4 and 4 irreps 1 + 3 + 3 + 1 SU(3) has 3 and 3 irreps

The 8-dimensional adjoint rep of SU(3) is 3x3 - 1 = 9-1 = 8 dim,and can be mapped to a 1+3+3+1 = 8 dim representation of fullexterior algebra, and therefore can be mapped to the sum of the 4 and4 irreps of SU(4). (This map may not be an isomorphism but it is anice 1-1 homeomorphism of the representation vector spaces.)

How are the 4 and 4 irreps of SU(4) related to the Octonions? Lookat Spin(6) = SU(4). Spin(6) is based on the Clifford AlgebraCl(0,6):

1 + 6 + 15 + 20 + 15 + 6 + 1 = 2^6 = 64 = 8x8

rows------------> c x x x x x x x x o x x x x x x x x l x x x x x x x x u x x x x x x x x m n x x x x x x x x s x x x x x x x x | x x x x x x x x\ / x x x x x x x x

The 1x8 rows of the 8x8 matrices of Cl(0,6) are the full spinorminimal ideals, and each 1x4 and 1x4 half-row is an irreduciblehalf-spinor minimal ideal of Cl(0,6). Each 4-dim half-row half-spinorrep of Spin(6) is isomorphic to one of 4-dim reps of SU(4). Takentogether, the two 4-dim half-spinor reps of Spin(6) form its 8-dimfull spinor rep.

When you embed Cl(0,6) in Cl(0,7), the 8-dim reducible full spinorrep of Spin(6) is mapped to the 8-dim irreducible spinor rep ofSpin(7), which is isomorphic to the 8-dim Octonions.

All this does not give an isomorphism of the algebra of Octonionswith the global group structure of SU(3), but it does give ahomeomorphism between the 8-dim vector space of the Octonions and the8-dim vector space of the SU(3) Lie Algebra (NOT the global SU(3) LieGroup).

The 8-dim spacetime Octonion vector space reduces to 4-dimphysical Spacetime and 4-dim Internal Symmetry Space, and the vectorspace identification with SU(3) indicates that

**the topology of the ****Minkowskiversion**** of Spacetime should be S1 x S3 (the same topologyas RP1 x S3)** (while the Euclideanversion of Spacetime has topology S4),

and

**the topology of Internal Symmetry Space should be CP2. **

Since RP1xS3 is the Shilov boundary of the bounded complex homogeneous domain corresponding to the symmetric space Spin(2,4) / Spin(2)xSpin(4), where Spin(2,4) is the Conformal group of 4-dim Minkowski spacetime, if you regard Physical SpaceTime as the 6-dimensional vector space of Spin(2,4), and Internal Symmetry Space as 5-dimensional CP2, then the total space is 6+4=10-dimensional with signature (2,8). With respect to the D4-D5-E6-E7-E8 VoDou Physics model, that 10-dim space corresponds:to the10-dim vector space of theD5Lie Algebra Spin(2,8); andto the

10-dim element of the decomposition of the 27-dim representation of theE6Lie Algebra into 10 + 16 + 1under its D5 subalgebra (see, for example, Lie Algebras in Particle Physics, 2nd edition, by Howard Georgi, Perseus Books (1999), page 308).

The (1,7)-dimensional RP1 x S7 = S1 x S7 = U(1) x S7 spacetime ofthe D4-D5-E6-E7-E8 VoDou Physics modelprior to dimensional reduction can be represented by QuaternionicProjective 2-space QP2.

As Atiyah and Berndt say in theirpaper Projective Planes, Serveri Varieties, and Spheres,math.DG/0206135,the S1 x S7 considered as QP2 breaks down into two parts:

- an S1 bundle over the S3 that is the complement S7 \ CP2, which plays the role of (1,3) RP1 x S3 Physical Minkowski Spacetime after dimensional reduction in the D4-D5-E6-E7-E8 VoDou Physics model.
- a CP2 acted on by an SU(3), which plays the role of Internal Symmetry Space after dimensional reduction in the D4-D5-E6-E7-E8 VoDou Physics model; and

Here are some excerpts from the paper by Atiyahand Berndt entitled ProjectivePlanes, Serveri Varieties, and Spheres,math.DG/0206135,where they denote Quaternions by H rather than the Q that I usuallyuse: "... There are natural diffeomorphisms

- CP2 / O(1) = S4 ... S0 = O(1) ...
HP2 / U(1) = S7... S1 = U(1) ...- OP2 / Sp(1) = S13 ... S3 = Sp(1) ...
... The maps from the projective planes to the spheres ... are fibrations outside the "branch locus" given by the preceding projective plane. ... The embeddings of the branch loci

RP2 in S4 , CP2 in S7 , HP2 in S13 ... If we denote the projective planes by Pn (n=0,1,2,3), so that dim Pn = 2^(n+1), then their complexifications Pn(C) are complex algebraic varieties of complex dimension 2^(n+1). ... The compact complex manifolds Pn(C) are ...

- P0(C) = SU(3) / S(U(1)xU(2)) = CP2
- P1(C) = SU(3)^2 / S(U(1)xU(2))^2 = CP2 x CP2
- P2(C) = SU(6) / S(U(2)xU(4)) = Gr2(C6)
- P3(C) = E6 / Spin(10)xU(1)
... The isometry group of Pn extends to an action of its complexification on Pn(C) leaving invariant a complex hypersurface Pn(oo). Moreover, Pn(C) has a "real structure", i.e. a complex conjugation. The real points are just Pn , and Pn(oo) inherits a real structure with no real points. ...

... when n = 1 ... P1(oo) ... with fibre L1(oo) ... is the twistor fibration for CP2 regarded as a 4-manifold with self-dual metric ... For n = 2,3 the fibration is a partial twistor fibration ...

- P2(oo) = Sp(3) / Sp(1)U(2) -> Sp(3) / Sp(1)Sp(2) = HP2 ... fibre ... L2(oo) = Sp(2) / U(2) ...
- P3(oo) = F4 / Spin(7)U(1) -> F4 / Spin(9) = OP2 ... fibre ... L3(oo) = Spin(9) / Spin(7)U(1) ...
... For each n = 0,1,2,3 we have a natural map

PSn : Pn(C) -> S(d(n)) , d(n) = 3 x 2^n + 1 ...[i.e.

- PS0 : P0(C) -> S4
- PS1 : P1(C) -> S7
- PS2 : Pn(C) -> S13
- PS3 : P3(C) -> S25
]... which is a fibration outside the branch locus Pn and the hypersurface Pn(oo). The fibres are the spheres S0, S1, S3, S7. ... the complexification of CP2 is just the product CP2 x CP2, while ... the complexification of HP2 turns out to be the complex Grassmannnian Gr2(C6) of 2-planes in C6 ... and ... the complexification of OP2 is the exceptional Hermitian symmetric space E6 / Spin(10)xU(1) ...

... denote by Gn the groups of isometries of Pn ... namely

- G0 = SO(3)
- G1 = SU(3)
- G2 = Sp(3)
- G3 = F4
... The space Hn of Hermitian ..[3x3]... matrices is a representation of Gn , which splits off a trivial factor (corresponding to the trace). ... Pn is contained in Hn , the space of Hermitian 3x3 matrices over the division algebra An . It is the orbit of the diagonal matrix Diag(1,0,0) under the compact Lie group Gn ... Note that Pn is contained in Hn(1) , the affine subspace of matrices of trace one. ... the real projective space of the vector space Hn ...[has dimension]... 3 x 2^n +2 ... the generic orbit of the action of Gn on Pn(C) has codimension one ... the generic orbits are

- n = 0 : SO(3) / O(1)
- n = 1 : SU(3) / U(1)
- n = 2 : Sp(3) / Sp(1) x Sp(1)
- n = 3 : F4 / Spin(7) ... this follows from ... S15 = Spin(9) / Spin(7) ...
...

consider ... the action of SU(3) on HP2via the embedding SU(3) in Sp(3) , and onS7 as the unit sphere in the vector space of Hermitian 3x3 complex matrices of trace zeroequipped with its usual norm.For S7 we find two copies of CP2 as special orbitsand the complex flag manifold of all full flags in C3 as generic orbit ...For HP2 the two special orbits are CP2 and a circle bundle over the dual CP2 which is a 5-dimensional sphere S5(as one sees by using all quaternion lines, i.e. 4-spheres, determined by complex lines). ...[There is]... a map HP2 -> S7 compatible with the two SU(3) actions. It identifies the CP2 in HP2 with one of the two CP2 in S7 , and on the complement HP2 \ CP2 it is an S1 bundle over the complement S7 \ CP2. The SU(3) determines a maximal subgroup U(3) of Sp(3) , and the central U(1) in this U(3) acts trivially on the CP2 and gives the fibres on HP2 \ CP2 . ...... Wallach proved that the only simply connected closed smooth manifolds admitting a homogeneous Riemannian metric with positive sectional curvature are ...

- ... the even-dimensional spheres
- ... the projective spaces CPm , HPm , and OP2 (m
>2)- ... the flag manifolds

- SU(3) / U(1)^2 ... of all full flags in CP2 ...
- Sp(3) / Sp(1)^3 ... of all full flags in ... HP2 ...
- F4 / Spin(8) ... of all full flags in ... OP2 ...
However, we shold point out that the positive curvature metrics on these flag manifolds are not the induced metrics from the spheres with their standard metrics. ...

... Now we shall extend these results ... to the non-compact groups ... of projectivities of Pn. ... we have a cubic polynomial det in Hn whose vanishing defines a hypersurface Zn ...[in].... the real projective space associated with Hn ... Consider ... the projective lines of Pn ...[to get a]... picture of the geometry of Zn in relation to Pn. ... any line in Pn ... is a sphere of dimension 2^n ... For a fixed line in Pn the geometry of its interior is just hyperbolic geometry and the subgroup of ... projectivities ... preserving the line is the conformal group of the sphere or the isometry group of its interior. ... ".

Since Gauge Bosons can be represented as AntiSymmetric Pairs ofFermion Nearest-Neighbors, consistently with

Since n (n-1) / 2 is the dimension of grade-2 Gauge BosonBivectors of a Clifford Algebra Cl(n), and

since, by Triality, for the Cl(8) Clifford Algebra of theD4-D5-E6-E7-E8-E8 VoDou Physics model the dimension 8 of thehalf-spinor representation space of the Fermion Particles, and hencethe number 8 of the Fermion Particles prior to dimensional reduction,is equal to the number 8 of Fermion AntiParticles, and to thedimension 8 of unreduced vector SpaceTime:

Since gluons can carry {+r,-r;+g,-g;+b,-b} charge, a gluon canemit three pairs: {+r,-r};{+g,-g};{+b,-b}.

As a gluon can emit three pairs, it can enlarge its path oftransitive action in Internal Symmetry Space to 4 dimensions.Therefore, gluon boundary conditions are the same all around theboundary. Since SU(3) is complex, the corresponding compact space ofconstant curvature is complex projective 2-space CP2:

CP2 Internal Symmetry Space is a nice spaceon which SU(3) acts globally,

( Global Group Structure of the Standard Model is discussedhere.)

The global nature of the color force SU(3) action on CP2 isrelated to the nature of agluon as a link with both spacetime and color internal symmetry spacecharacteristics, which requires gluonconfinement within protons,pions, and other particlescontaining them.

Since weak bosons can carry {+1,-1} charge, a weak boson can emita {+1,-1} pair.

As a weak boson can emit a pair, it can enlarge its path oftransitive action in Internal Symmetry Space to 2 dimensions.Therefore, weak boson boundary conditions form two 2-spheres:

Since CPn = Cn u CP(n-1) (where u = union, by putting a CP(n-1) atinfinity),

CP2 = C2 u C1 u C0 = C2 u CP1 = C2 u S2.

Therefore, CP2 Internal Symmetry Spacecontains a 2-sphere S2 on which SU(2) can act globally,

since SU(2) / U(1) = S2.

A second copy of S2 is necessary to get a 4-dimensional manifoldS2xS2.

The second copy of S2 can live in the C2 of CP2 = C2 u C1 u C0 =C2 u S2.

In the C2 of CP2 = C2 u S2, the C2 contains a copy of S2 that islinearly independent of the S2.

The local nature of the weak force SU(2) action on CP2 is relatedto the nature of weak bosonsas having only internal symmetry space characteristics.

The mass factor for the weak forcehas a visualization (arising from e-mail discussion with DickAndersen). Gauge bosons are visualized asgoing from a source through a medium to a target. The weak force massfactor is related to the Higgs mechanism. The Higgs scalar fieldabsorbs some of the weak bosons as they go through the medium, thusweakening the weak force and producing the weaker effective weakforce that is observed by experiments;

Since all photons have no charge, a photon can only move along itspath of transitive action in Internal Symmetry Space.

As a photon cannot emit another photon, it cannot enlarge its pathof transitive action in Internal Symmetry Space beyond the 1dimension of the path of transitive action in Internal SymmetrySpace. Therefore, photon boundary conditions form a 4-torus, or four1-spheres:

Since CPn = Cn u CP(n-1) (where u = union, by putting a CP(n-1) atinfinity), and CP0 = C0 = {point},

CP2 = C2 u C1 u C0.

Since C1 contains the unit circle S1 = U(1),

U(1) has a natural action on a part of CP2Internal Symmetry Space.

Three more copies of S1 are necessary to get a 4-dimensionalmanifold S1xS1xS1xS1 = T4.

In the C2 of CP2 = C2 u C1 u C0, the C2 contains three copies ofS1 that are linearly independent of each other and of the S1 that isthe unit circle in the C1.

The local nature of the electromagnetic U(1) action on CP2 isrelated to the nature ofphotons as having only internal symmetry space characteristics.

Since gravitons can carry {+1,-1;+r,-r;+g,-g;+b,-b} charge, agraviton can emit four pairs: {+1,-1};{+r,-r};{+g,-g};{+b,-b}.

As a graviton can emit at least three pairs, it can enlarge itspath of transitive action in its Symmetry Space to 4 dimensions.Therefore, graviton boundary conditions are the same all around theboundary. Spin(5), a compact Lie group corresponding to Spin(2,3),acts globally on a compact space of constant curvature, a 4-sphere= S4 = Spin(5)/Spin(4), theEuclidean version of Spacetime (the non-compact Minkowskiversion on which Spin(2,3) acts being RP1xS3 ):

Unlike the other 3 forces that act transitively on thecoassociative Internal Symmetry Space, gravity comes from the U(4)subgroup of Spin(8) and therefore acts transitively on theassociative physical spacetime.

By the emission of four pairs, a graviton can describe thestructure of a 4-dimensional spacetime.

A 4-pair graviton emission is effectively creation of a virtual1-vertex spacetime Planck mass blackhole.

The mass factor for gravitation has a visualization (arising from e-mail discussion with Dick Andersen). Gauge bosons are visualized as going from a source through a medium to a target. The graviton by itself is long-range and massless, but virtual Planck-mass black holes in spacetime absorb some of the gravitons as they go through the spacetime medium, thus weakening the gravitational force and producing the weaker effective gravitational force that is observed by experiments;

If the space and time axes of the 1-vertex black hole becomeconnected with the space and time axes of the original spacetime,then the virtual Planck-mass black hole acts to provide the massfactor (1/MPlanck^2) for the force strength of low-energy effectivegravitation in the D4-D5-E6-E7-E8model.

4-pair graviton interactions are important in the ManyWorlds Quantum Theory of the D4-D5-E6-E7-E8 model.

Virtual Gravitons might allowcommunication between a Parent Universe and a New Universecreated at a vertex of the Parent Universe by a CosmologicalInstanton.

Quantum fluctuations of new spacetime creation may interchangetime and space axes, creating a new spacetime with lightcones tiltedwith respect to the original spacetime. The new tilted spacetime thenhas its own lightcone and conformal symmetry. For an illustration oftilted lightcones, see the lightcones in Duchamp'sThe Large Glass.

The tilted lightcone spacetimes probably constitute a very smallpart of any quantum superposition describing physical spacetime, sothat conformal symmetry and preservation of lightcone structure isprobably a good enough approximation to use quantum conformalfluctuations to describe cosmology using theD4-D5-E6-E7-E8 model.

However, even if only a very small fraction of the spacetimegeometries in the superposition contain closed timelike loopsresulting from tilted lightcones, non-computableoperations could be performed by a human-brain quantumcomputer.

The mass factor for gravitationhas a visualization (arising from e-mail discussion with DickAndersen). Gauge bosons are visualized asgoing from a source through a medium to a target. The graviton byitself is long-range and massless, but virtualPlanck-mass black holes in spacetime absorb some of the gravitonsas they go through the spacetime medium, thus weakening thegravitational force and producing the weaker effective gravitationalforce that is observed by experiments;

According to week120 of This Week's Finds in Mathematical Physics by JohnBaez: , **the Regge Calculus formalism of GravitationalCurvature was "... invented by Regge in 1961**. Regge came up witha discrete analog of the usual formula for the action in classicalgeneral relativity. His formula applies to a triangulated 4-manifoldwhose edges have specified lengths. In this situation, each trianglehas an "angle deficit" associated to it. It's easier to visualizethis two dimensions down, ...[as in this figure

adapted from the book Gravitation, by Misner,Thorne, and Wheeler (Freeman 1972)]... where each vertexin a triangulated 2-manifold has an angle deficit given by adding upangles for all the triangles having it as a corner, and thensubtracting 2 pi. No angle deficit means no curvature: the trianglessit flat in a plane.

The idea works similarly in 4 dimensions. Here's **Regge'sformula for the action: take each triangle in your triangulated4-manifold, take its area, multiply it by its angle deficit, and thensum over all the triangles**. Simple, huh? **In the continuumlimit, Regge's action approaches the integral of the Ricci scalarcurvature - the usual action in general relativity**. For more see... T. Regge, General relativity without coordinates, Nuovo Cimento19 (1961), 558-571. ...".

Note that the Regge Calculus of angle deficits only describes theCurvature part of Gravitation, and does not describe TorsionGravitational effects. To see the Torsion part of Gravitation, theRegge Calculus must be given addtional structure, such as describingthe edges of the 4-simplices by spinors in units of hbar instead ofonly by length magnitudes.

Since **the Regge Calculus formulation of GravitationalCurvature**

- is based on a decompostion of 4-dim spacetime into 4-simplices, and
- each 4-simplex can be regarded as an elementary "local place", and
- each 4-simplex is characteristically defined by the lengths of its 10 edges, and
- if you take one of the vertices of the 4-simplex as its "center", then the 10 edges have the characteristics of
- 4 vectors from center to outer vertices plus
- 6 outer edges, each having 2 end-points that correspond to 2 of the 4 vectors, (i.e, 6 bivectors),

it **has a correspondence with the ****MacDowell-Mansouriformulation**** of Gravitation**

which is based on a local Spin(2,3) gauge Liegroup (anti-deSitter) with 10 generators, having thecharacteristics of

- 4 "big" rotations (like vector translations in 4-dim spacetime) plus
- 6 Lorentz transformations.

Spin(2,3) comes from the Clifford algebraCl(2,3) with graded structure **1**+5+**10**+10+**5**+1.

The even subalgebra of Cl(2,3) is1+10+5= 16-dimensional, and is the Clifford algebra Cl(2,2) with graded structure 1+4+6+4+1 and dimensionality 16 = 2^4 = 1+4+6+4+1 = 4x4 = 4^2.The graded structure 1+4+6+4+1 of the even subalgebra of Cl(2,3) can be given a product so that it forms the 16-dimensional Lie algebra U(2,2) = U(1) x SU(2,2), with the 15-dimensional SU(2,2) corresponding to the

10+5gradcd elements of Cl(2,3).

Note that in the D4-D5-E6-E7-E8 VoDouPhysics model Gravitation is constructed, not just fromanti-deSitter Spin(2,3) but from the larger Conformal Group Spin(2,4)of which Spin(2,3) is a subgroup.

Spin(2,4) comes from the Clifford algebra Cl(2,4) with graded structure 1+6+15+20+15+6+1.The

15-dimensional bivectors of Cl(2,4) can be given a product so that they form the 15-dimensional Conformal Lie algebra Spin(2,4) = SU(2,2), which corresponds to the10+5gradcd elements of Cl(2,3).

The Future Light Cone at a vertex of **the ****4-dimHyperDiamond lattice**** of ****D4-D5-E6-E7-E8VoDou Physics** is a 4-simplex

with the same graded structure **1**+5+**10**+10+**5**+1as the Cl(2,3) Clifford algebra:

**1 empty set**,- 5 vertices,
**10 edges**(the**10**-dimensional grade-2 Spin(2,3) bivectors of Cl(2,3),**4****edges****from the given vertex to 4 Future Light Cone vertices**plus**6****edges****connecting the 4 Future Light Cone vertices, forming an outer tetrahedron**,

- 10 triangles,
**5 tetrahedra**(the**5**-dimensional grade-4 elements of Cl(2,3)**1****outer****tetrahedron**(the Conformal Dilatation) plus**4 tetrahedra with 3****edges****from the given vertex**(the Special Conformal Transformations) ,

- 1 simplex itself.

When you add the Past Light Cone, you have two 4-simplices:

- 8 edges from the given vertex to 8 Light Cone vertices plus
- 12 edges forming two outer tetrahedra;

When you put in 12 furtherconnections between Past and FutureLight Cones, you have a total of six 4-simplices:

- 8 edges from the given vertex to 8 Light Cone vertices plus
- 24 edges forming four more outer tetrahedra,

so that there are a total of six outer tetrahcdra.

The 24 outeredges ofthe six outer tetrahedra correspond to the 24 vertices of a 24-cell:

The 24-cell is the vertex figure of theD4 lattice. The D4lattice is the evensublattice of the 4-dim HyperDiamondlattice of D4-D5-E6-E7-E8 VoDouPhysics. That is, the D4 lattice can bedescribed either as

- a minimum lattice distance of 2 (as a sublattice of the 4-dim HyperDiamond lattice) or
- as a minimum lattice distance of 1 (as the lattice of integral quaternions):

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