In CERN-CDS-EXT-2004-031,E6, Strings, Branes, and theStandard Model, the 5-graded Graded LieAlgebra of the 78-dimensional exceptional Lie algebra

= 8-dim + 16-dim + ( Spin(8) + R + R ) + 16-dim + 8-dim =

= 8-complex-dim + 16-complex-dim + ( Spin(8) + R + R)

provides a physical interpretation of the 26 dimensions of StringTheory.

Of the 26 dimensions of String Theory, 2 correspond to complexstructure of E6, 8 correspond to 4-dimensional SpaceTime and4-dimensional Internal Symmetry Space, and 16 = 8+8 represent 8first-generation fermion particles and 8 first-generationantiparticles.

Although the 8 SpaceTime and Internal Symmetry Space dimensionscan be thought of as, at least approximately at scales much largerthan the Planck length, being continuous,

the 16 = 8+8 fermion particles and antiparticles are clearly 16 =8+8 discrete things, and their representation as such requiresdiscretization of the corresponding 16 dimensions of 26-dimensionalString Theory, which is accomplished by Orbifolding. The purpose ofthis paper is to describe the geometry of such Orbifolding.

Begin by looking at the root vectorstructure of E6:

the 6-dimensional polytope formed by the root vectors of the Liealgebra E6 has 72 vertices. A 2-dimensional projection, by PeterMcMullen,

has 72 vertices in sets of 12 on 4 concentric circles, 2 of which(white centers) are double (one vertex behind another). The6-dimensional 72-vertex Root Vector Diagram of E6 can beconstructed from the 4-dimensional 24-vertex 24-cell Root VectorDiagram of D4 by adding to it the following:

- a 5th dimension, with an 8-vertex 4-dim hyperoctahedron cross polytope above in the 5th dimension, and another 8-vertex 4-dim hyperoctahedron cross polytope below in the 5th dimension, to get a 40-vertex Root Vector Diagram of D5; and
- a 6th dimension, with, instead of adding to it hyperoctahedra cross polytopes, adding to it a 16-vertex 5-dim hemi-hypercube (half of the 2^5 = 32 vertices of a 5-dim hypercube) above in the 6th dimension, and a 16-vertex 5-dim hemi-hypercube (the other half of the 2^5 = 32 vertices of a 5-dim hypercube) below in the 6th dimension.

This construction is the first three stages of the 5-stageconstruction of the E8 Root Vector Diagram,

that is, proceeding from the left in the above image, the D424-cell, the two D5 hyperoctahedra cross polytopes, and the 5-dimhypercube.

The 32 vertices of the 5-dim hypercube correspond to the 16complex (32 real) dimensions of the symmetricspace E6 / D5xU(1), corresponding to a bounded complex domainwhose Shilov boundary is an 8complex (16 real) dimensional non-tube-type domain.

Here (adapated from animage on a johnh web page) is an image of a 5-dim hypercube

in which you can see the graded structure of its vertices - 1 (topyellow), 5 (blue), 10 (yellow), 10 (blue), 5 (yellow), and 1 (bottomblue).

Divide the 32 vertices into two sets of 16

each of which corresponds to a 16-vertex 4-dimensional hypercube,whose vertices have graded structure 1, 4, 6, 4, 1.

Now look at one of the 4-dimensional hypercubes (call it the+hypercube)

and notice that it is made up of two 3-dimensional cubes (oneshown with circle vertices, the other shown with square vertices.

In CERN-CDS-EXT-2004-031,E6, Strings, Branes, and theStandard Model, orbifolding to discretize fermion particles andantiparticles corresponding to the 16-vertex +hypercube is done thisway:

"... To use Urs Schreiber's idea to get rid of the world-brane scalars corresponding to the Octonionic O+ space, orbifold it by the 16-element discrete multiplicative group Oct16 = {+/-1,+/-i,+/-j,+/-k,+/-E,+/-I,+/-J,+/-K} to reduce O+ to 16 singular points {-1,-i,-j,-k,-E,-I,-J,-K,+1,+i,+j,+k,+E,+I,+J,+K}.Let the 8 O+ singular points {-1,-i,-j,-k,-E,-I,-J,-K} correspond to the fundamental fermion particles {neutrino, red up quark, green up quark, blue up quark, electron, red down quark, green down quark, blue down quark} located on the past D7 layer of D8. [These 8 can be taken to be the cube

with 8 circle vertices.]Let the 8 O+ singular points {+1,+i,+j,+k,+E,+I,+J,+K} correspond to the fundamental fermion particles {neutrino, red up quark, green up quark, blue up quark, electron, red down quark, green down quark, blue down quark} located on the future D7 layer of D8. [These 8 can be taken to be the cube with 8 circle vertices.] ...".

A geometrical picture of the Orbifolded **8-dimensional fermionicfirst-generation particle representation space** as 8 of the 26dimensions of String Theory comes from looking at the **organizationof the 16 vertices of the 4-dimensional +hypercube**. Each of itstwo 8-vertex 3-dimensional cubes can be seen as two 3-dimensionaloctahedron cross-polytopes

or also as one 4-dimensional hyperoctahedroncross-polytope

Therefore, the **16 vertices of the 4-dimensional +hypercube,**made up of the two 8-vertex 3-dimensional cubes taken together, canbe seen as a 4-dimensional hyperoctahedron cross-polytope in one4-dimensional space plus a second 4-dimensional hyperoctahedroncross-polytope in a second 4-dimensional space,

or also as **one 16-vertex 8-dimensional hyperoctahedroncross-polytope**.

Therefore:

**The geometry related to the other 16 vertices of the5-dimensional 32-vertex hypercube, the 4-dimensional -hypercube, andthe first-generation fermion antiparticles, is similar.**

Note that:

**each + vertex (say, +i) is the singularity representative ofthe segment [0,+i]**

**which in turn represents the +i component part of all of theorbifold with nonzero +i coordinate**

**which in turn by extension to the full line (-oo, -i, 0, +i,oo) and by triality automorphism among the three octonionic spaces O+representing first-generation fermion particles, O- representingfirst-generation fermion antiparticles, and Ov 8-dimensionalspacetime, has a global connection with all of 8-dimensionalspacetime**

which is consistent with the fact that

**Fermions - quaternions****- have ****a Dirac belttrick**** type connection with global spacetime.**

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