VisibLie_E8:

A Theory of Everything Visualizer, with:
 
({{  ({{  1) Math:                Chaos/Fibr/Fractal/Surface:  Navier Stokes/Hopf/MandelBulb/Klein}, {  2) Math:                Number Theory:                                     Mod 2-9 Pascal and Sierpinski Triangle }, {({{  3) Math:                Geometric Calculus:                         Octonion Fano Plane-Cubic Visualize}, {  4) Math:                Group Theory:                                       Dynkin Diagram Algebra Create}})}, {  5) Math:                Representation Theory:                   E8 Lie Algebra Subgroups Visualize}, {  6) Physics:         Quantum Elements:                              Fundamental Quantum Element Select}, {  7) Physics:         Particle Theory:                                CKM(q)-PMNS(ν) Mixing_CPT Unitarity}, {  8) Physics:         Hadronic Elements:                           Composite Quark-Gluon Select Decays}, { 18) CompSci:        Human/Machine Interfaces:          nD Human Machine Interface}})}})

By J Gregory Moxness
TheoryOfEverything.org

Details

This demonstration combines five original demonstrations by the author (#1-5 Fano, Pascal, Dynkin, E8, and Particle and an OpenCL based N-Body Universal history simulation #8). It combines and integrates many other author’s Wolfram demonstrations with significant original value added to each. All of these are in some way related to fundamental mathematical symmetries. Most are also theoretically related to and integrated with an  extended Standard Model (SM) originated by A.G. Lisi and modified by the author. It visualizes, among other things, the math and physics of “An Exceptionally Simple Theory of Everything“ as well as myToE as a basis for understanding the relationships between the fundamental constants, (C)harge, (T)ime, and space or spin (P)arity.

The resulting combination of 18 panes brings the user from the mathematical abstractions of geometry and algebra, through particle and nuclear physics, adds a universal history simulation engine, including quantum mechanics and general relativity of cosmological N-body gravitation and Big Bang physics, and goes on to visualize the chemistry of the atoms, molecules, and proteins, as well as speculation about the quantum brain-mind connection of neurons/axons as it relates to Orchestrated Objective Reductions (OrchOR). I’ve even thrown in a few panes investigating computational, musical, and theological links to math/science.

The #1 pane (Chaos) is taken largely from several great Wolfram Demonstrations of  Zeleny’s Five Mode Truncation Of The Navier-Stokes Equations, A Collection Of Chaotic Attractors and Hennegan's Rotating The Hopf Fibration. It now includes 2D and 3D visualizations of these equations. It also adds an OpenCL based visualization of 3D and 4D Fractals (e.g. the Menger Sponge and MandelBulb) and surfaces (e.g. the Boy surface, a rose and others from Paul Nylander, more from Zeleny, and Hanson’s 6D Calabi-Yau in String Theory).

Navier-Stokes is not only useful in Computational Fluid Dynamics (CFD) and weather prediction, these visualizations are useful in understanding alternate theories of Quantum Gravity's space-time structure. These OpenCL codes are only available with the Mathematica Source for this module (available by request). For those w/o OpenCL capability, I've integrated and extended Yu-Sung Chang's excellent interactive surface visualizations from ContoursOfAlgebraicSurfaces.

The #2 pane (Pascal) for visualizing the Pascal and Sierpinski triangles, along with the Fibonacci and Lucas numbers.  In addition to allowing the change in size of the triangle, it highlights the binomial functions and allows the changing the modulus of the numbers used. The number backgrounds are colorized in the selected gradient. This pane was created with ideas from  Peter S.Williams’ Mod 9 Pascal Triangle Physics http://naturalnumber.com/

The #3 pane (Fano) for visualizing the octonions, the Fano plane and its cubic. It also allows the manipulation of the 480 different permutations of the octonion basis (calculated from converted C source code from Donald Chesley of Davidson Laboratories, Stevens Institute of Technology). These are shown to be integrated 2:1 with the 240 vertices of E8 and its subgroups. Selection of the split octonion via triad number is a recently added feature. This demonstration also combines and extends Wolfram demonstrations from Ed Pegg Jr. and Oleksandr Pavlik. It also includes the generation of sedenions from the octonions by application of the Cayley-Dickson doubling procedure. As in the animated Fano Cube, the sedenion display includes the generation of an animated Fano Tesseract mnemonic visualization which steps through highlighting the vertices/edges of the 34 sedenion triads.

The #4 pane (Dynkin) allows for the creation of Dynkin diagrams and their corresponding Cartan matrices which generate Lie Algebras. This second pane drives the vertex content in the third pane. There is now a checkbox for showing the detail root vector data and Hasse visualizations instead of the interactive Dynkin pane, which is built from SuperLie 2.07 by P. Grozman.

The #5 pane (E8) provides 2D and 3D visualizations of E8 Lie Algebra and it's split real even sub group vertices. Each of these vertices are each assigned to fundamental physics particles. I’ve added a 2D Fourier Transform visualization sub-pane.

In the #6 pane (Particle), the vertices as fundamental physics particles are individually selectable by their quantum parameters. It is an interactive E8 linked demonstration similar to the idea behind the first pane of S.M. Blinder’s “Combining Quarks Into Hadrons”. I’ve also integreated Higgs theoretical & experimental mass particle decay branching ratios modified from Caique Ronqui’s BranchingRatiosForHiggsBosonDecays.

The #7 pane (CKM) is an improved form of Balázs Meszéna’s demonstration on Neutrino Oscillations which presents the Unitarity of CP=T violations by combining the  Lepton (Neutrino) Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS) with the Quark Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix calculations through the Quark–Lepton Complementarity (QLC) . The plots show the probability of finding a 1 GeV particle in the different flavors as a function of the distance between the source and the detector. It also demonstrates the hierarchy of the particle masses and the flavor coefficients in each eigenstate.

The #8 pane (Hadron) is modified from Hadron demonstrations from both Blinder and  Zeleny,which allows the visualization of the composite Quark particles and their decays. These demonstrations are extended here by allowing the selection of 2 Quark Mesons, 3 Quark Baryons and recently discovered 4 and 6 Quark Hadrons. This also drives the E8 sub group projection pane. The author has also added a query to show all experimentally discovered composite Meson/Baryon particles with the same quark content from the ParticleData Group curated data set (with decay modes shown with a button click).

The #18 pane (HMI) extends the Human Machine Interface (HMI) for the third pane. It has a tongue-in-cheek label referring to the user as a biological human life form. This UI provides for manipulation of all of the variables used to create beautiful E8 projections/animations.

References:

Chaos:

This demonstration combines several demonstrations related to chaos theory, fractals and Navier - Stokes computational fluid dynamics (CFD) as it relates to a theory of a Universe that has a superfluid space-time.

See http://demonstrations.wolfram.com/FiveModeTruncationOfTheNavierStokesEquations/

See http://demonstrations.wolfram.com/GeneratingFractalsWithRegularExpressions/

See http://demonstrations.wolfram.com/ACollectionOfChaoticAttractors

See http://demonstrations.wolfram.com/RotatingTheHopfFibration

See http://demonstrations.wolfram.com/CalabiYauSpace

From Andrew J.Hanson with Additional contributions by Jeff Bryant,
See http://demonstrations.wolfram.com/CalabiYauSpace

From Yu-Sung Chang,
See http://demonstrations.wolfram.com/ContoursOfAlgebraicSurfaces

Fano:

Octonions use a range of 1-7 numbered triplet subsets. A Fano plane is a collection of seven of these triplets (or triads) where no two share two numbers. A Fano plane is a configuration with 30 distinct numberings, when the numbers 1 to 7 (a flattened set of triads) are assigned to the seven points. For each of the 30 distinct numberings, there are eight possible reversals of triplet order, which are identified here as bitwise (hexadecimal) sign masks. These 240=30×8 triplets can also be inverted by inverting the eight sign masks, giving a total of 480 Fano planes. These are visualized here as either a directed 2D graph or a non-directed 3D cubic form. The product of two distinct units equals the third unit, such that the three form immediately connected vertices of the graph.

The octonions can be constructed from quaternions by means of the Cayley–Dickson construction. Like quaternions and the lower-dimensional complex numbers, octonions have one real number ((Subscript[ℯ, 0])=1). Octonions have seven imaginary units {(Subscript[ℯ, 1]),(Subscript[ℯ, 2]),(Subscript[ℯ, 3]),(Subscript[ℯ, 4]),(Subscript[ℯ, 5]),(Subscript[ℯ, 6]),(Subscript[ℯ, 7])}, or in IJKL format {i,𝕛,𝕜,𝕝,i 𝕝,𝕛 𝕝,𝕜 𝕝}, whose multiplication tables can be encoded using the Fano plane mnemonic, shown here in both numeric and symbolic form.

For any three octonions a, b, and c, the associator is (a∘b)∘c - a∘(b∘c). The associator measures the non-associativity of those three octonions. Select triples of octonionic units to compute their associator. Observe that their associator vanishes on immediately connected triples of octonionic units. Such triples, together with the unit element, form the quaternionic sub-algebras of octonions.

The odd-4 graph is constructed by taking the 35 triplets that share no values. The graph is the odd-4 graph (black edge lines). If a 1-7 point Fano plane is removed from the odd-4 graph, the result is the Coxeter graph (cyan edge lines). If connections to the fifteen Fano planes of a single color are added instead, the result is the Hoffman–Singleton graph (yellow edge lines).

Select any of the 480 permutations for the Fano basis using the same bitwise quantum particle selections (described in the ”Particle” section below, with the addition of the “flip” checkbox to select the second E8 linked octonion for the particle). Red or blue numbers in the flattened triads indicate it is one of 240 E8 related permutations needing those colored node positions flipped (or reversed) to create a proper Fano plane from it. Blue indicates it is a member of the 128-vertex C8 subgroup of E8 and red for subgroup D8 with 112 vertices. Node colors are changed from cyan to yellow for button bar selected associator dimensions {(Subscript[n, 1]),(Subscript[n, 2]),(Subscript[n, 3])}. A selection to view the symbolic multiplication tables in IJKL form is also included. Octonion math examples are shown below the numeric and symbolic multiplication tables.

Each new selection of the 480 different bases changes the octonion multiply and commutator results and verifies the conjugate, norm, and division results are the same. This verifies proper octonion operation as shown in the octonion math examples. Mouse over the Fano plane to show the detail particle information relating to E8, Cellular Automata and physics (as shown in the “Particle” pane 4).

The construction of the 480 octonion sets of 7 triples uses converted C source code from Donald Chesley of Davidson Laboratories, Stevens Institute of Technology.

The list of 30 Fano planes (made up of sets of 7 quaternion triads) can be determined algorithmically by brute force (checking each of the 35 possible unique triples against the 21 possible unique pairs and finding the 30 sets of 7 that don’t share more than one number).

The 16 sign mask (sm) sets can also then be algorithmically determined for each index into 30 sets of Fano plane quaternion triads (by taking each of the 30 Fano planes and brute force checking which of the 128=2^7 sign reversal bit patterns keep the associator=0). Each sign reversal constitutes an arrow direction reversal in the Fano plane mnemonic.

It turns out that all of the 128=(2^7) possible bit patterns for reversing the Fano plane arrows is used. These are grouped into 2 sets of 64 with an "anti" (or 0⟺1 bit reversal) operation. The 64 are found to be grouped into 8 sets of 8 which get associated (or indexed) with the 30 Fano planes. The tally of these associations between the 8 sign mask sets and the 30 Fano plane sets is {{1,2},{2,2},{3,7},{4,3},{5,3},{6,7},{7,3},{8,3}}. That is, there are 4 of the 8 that associate with 3 Fano planes each. There are 2 of the 8 that associate with 7 Fano planes and 2 that associate with 2 of them.

A split of the 30 triad sets into 15=7+8 (or D8+C8) provides the indicator on where to assign the “excluded” permutations. They get the 0 color and 0 generation assignment - the same logic applied to the excluded E8 vertices or particles. It becomes trivial to apply a naive sequential assignment of 30 Fano planes into two sets of 7/8 assigned to the 4 bits associated with generation and color in the Lisi model.

The binary patterns for the 8 sign mask groups are easily seen and can also be constructed (rather than brute force determined) using right/left symmetry patterns. Interestingly, this construction uses a Clifford algebra Cl(0,8) primitive idempotent (Subscript[e, {2,3,5,8}]) and its inverse (Subscript[e, {1,4,6,7}])  as a base. These sets are assigned (R)ight and (L)eft based on the pattern of adding Reverse@Range@3 to 1 (or subtracting Range@3 from 8) respectively. More interestingly, these are also used in assigning the octonions to the E8 particle assignments - specifically, they index to the 4 spins (up/down, left/right) with the set assigned by the pType (neutrino/electron lepton or up/down quark split). These two groups of 4 are also used to index the 15=7+8 (or D8+C8) split associated with generation and color bits. This is a common pattern between the split of octonions into the Cayley-Dickson quaternions (the first 4 a-p & spin bits of the first quaternion {(Subscript[ℯ, 0]),(Subscript[ℯ, 1]),(Subscript[ℯ, 2]),(Subscript[ℯ, 3])} and the second 4 color-generation bits of the second quaternion {(Subscript[ℯ, 4]),(Subscript[ℯ, 5]),(Subscript[ℯ, 6]),(Subscript[ℯ, 7])}).

The Fano plane indices also “pair up” when assigning the 480 octonion permutations into the 240 E8 permutations with D8Pairs={{1,2},{4,7},{8,10},{15,16},{17,19},{23,24},{25,26}} and C8Pairs={{3,5},{6,9},{11,12},{13,14},{18,20},{21,22},{27,29},{28,30}}. The pairs are always found together in the E8 particle assignment as “flipped” and “non-flpped” Fano planes (except the last of the C8 Pairs {28,30}, which have an anomalous flip pattern associated with the 3rd generation blue quarks).

To reach the doubling of the 512=480+32 "excluded" octonion permutations, I added what I call a 9th "flip" bit parameter. This correlates well with A Zee's approach to grand unification using a 9D SO(18) group theory. It turns out that the pattern required for flipping works in unison with the a and p bits in a binary logic relationship. As the demonstration clearly shows, the p bit flips the two sets of four sm bits. This is the key to a eureka moment where I created the notion of a 3rd spin type out of the pType bit (in/out or z-x or yaw intrinsic rotation or spin). Also note: Charge-Parity-Time (CPT) conservation law (symmetry) considerations also strongly link the a, p, and spin bits.

The binary logic relationship with the "flip" bit and the sm bits clearly create two 4 bit quaternion sets. The first quaternion is the anti(e0=1 Real) bit combined with 3 spin bits {e1,e2,e3} or {I,J,K=IJ}. The second quaternion is 2 generation bits {e4,e5} or {L,IL} combined with 2 color bits {e6,e7} or {JL,KL=IJL}. While this double set of quaternions was already obvious to me from the E8 construction (noted in http://theoryofeverything.org/TOE/JGM/DynkinParticleReductions.cdf), it was good to confirm it with octonion integration! It seems that this pair of quaternions could be visualized as Hopf Fibrations on 7-manifolds ( see http://www.nilesjohnson.net/seven-manifolds.html for an excellent review of this).

It correlates well with my theory relating time to octonions, charge to quaternions and 3-space to complex (imaginary) numbers. This idea also validates the dimensional considerations that started me down this path 15 years ago (documented in http://theoryofeverything.org/TOE/JGM/ToE.pdf).

A recent addition allows for the visualization of split octonions. The 3840=480*8 split octonion permutations with Fano planes and multiplication tables are now available.

There are 7 sets of split octonions for each of the 480 "parent" octonions. The 7 split octonions are identified by selecting a triad. The complement of {1,2,3,4,5,6,7} and the selected triad list leaves 4 elements which are the rows/columns corresponding to the negated elements in the multiplication table (highlighted with yellow background). The red arrows in the Fano Plane indicate the potential reversal due to this negation that defines the split octonions.

These allow for the simplification of Maxwell's four equations which define electromagnetism (aka.light) into a single equation.

See http://en.wikipedia.org/wiki/Split-octonion

This demonstration combines and extends Wolfram demonstrations from Wolfram's Ed Pegg Jr. (Odd4GraphFanoPlanesAndTheCoxeterGraph) and Oleksandr Pavlik (OctonionsAndTheFanoPlaneMnemonic).

Dynkin:

This Demonstration lets you create and manipulate multiple Dynkin and Coxeter–Dynkin diagrams with some ability for topological pattern recognition of known Simple Lie Groups (up to rank 8) and its designated type, including finite, affine, hyperbolic, and very extended. You can use the drop down "Dynkin diagrams" to get various diagrams and geometric permutations.
It also provides for manipulating, recognizing, and naming of the (2^rank) (binary) Coxter-Dynkin geometric permutations on uniform polyhedron and their Wythoff construction operator naming. These permutations are indicated by filled nodes. When more than one diagram is created, they can be interpreted as a geometric Cartesian product, such as a DuoPrism. 

The Cartan matrix which defines the Lie algebra (along with the Coxeter, and Schlafli matrices) are calculated directly from the last diagram entered. If there are more than one diagrams, all with the same rank, the diagram matrices are dotted together.

Added controls for defining directed graphs, black and white vs. colored nodes, and left vs right affine (black) nodes used in recognizing the Dynkin topology. 
Please note, when affine and black and white the permutation name is always "snub" with no black filled nodes.

The main window is a clickable pane.
Clicking more than 1.5 cells away from all nodes creates a new diagram. 
Clicking less than 1.5 cells away from all nodes creates a node and a single line to its nearest neighboring node. 
Clicking on a node in the last diagram created toggles the node fill on/off and changes the geometric permutation (unless it is an already filled node that is adjacent to an unlinked node, which implies the creation of an affine loop). 
Clicking on a line between nodes in the last diagram created toggles the symmetry angle and/or the directionality between its nodes. There are a dozen line styles in a dropdown menu that allow creation of any valid (and many invalid ;-) diagrams, including custom pq Coxeter {begin,end} values. Some of these styles change based on the rank and setting of the “directed” checkbox.
Please note that when more than one diagram is in the clickable pane, only the last diagram is active. Clicking on previous diagrams will create a new overlapping diagram.

For more information on Dynkin Diagrams, see http://en.wikipedia.org/wiki/Dynkin_diagram 
For more information on Simple Lie Groups, see http://en.wikipedia.org/wiki/Simple_Lie_group 
For more information on Coxeter-Dynkin Diagrams, see http://en.wikipedia.org/wiki/Coxeter-Dynkin_diagram
For more information on Coxeter-Dynkin geometric permutations on uniform polyhedron and their Wythoff construction operator naming, see http://en.wikipedia.org/wiki/Uniform_polyhedron

E8:

The first drop down list provides for the selection of Coxeter (and other) projection symmetries. Most names indicate Simple Lie groups. The Bn and Cn groups are the same as the ones for Dn+1. (e.g. the menu label "D+1=BC3" applies to D4, B3, and C3 simple Lie algebras or groups  and will be projected to their Petrie projection with the basis vectors associated with this menu selection). The next set of projections indicate the number of perimeter vertices in parenthesis. The PASCAL# projections display the number of vertices in a column which is has the same number as that numbers row of the Pascal triangle.

The next two menus provide "Favorites" that are preset combinations of over 50 variables to produce beautiful projections of E8 and its subgroups. Their short names attempt to indicate what is in the resulting image. For example, the references to "n Cube Pascal" in menu 2 refers to a projection that creates columns of vertices (as in the PASCAL# in the Coxeter projection menu). The references to Petrie refer to the corresponding Petrie projection of that n dimensional cube. Favorite names using the word "binary" imply it uses a purely binary data set from the Pascal triangle instead of SRE E8.

The counts of selected vertices, edges, triality lines, and clicked vertex edge lines are shown.

The check boxes select major elements of what gets displayed in the output window:
* axes selects whether to show the 8 axes (and the frame bounding box)
* edges selects whether to show the edges between each vertex (subsets of edge lengths is selected in the HMI pane 8)
* physics shows the assigned physics particle shape/color/size (vs. dots)
* labels shows the particle labels
* dimLocs checkbox is used to activate the dimension locators in 2D (deactivates particle interactions).

Checking the "axes locator" box forces it into 2D mode with axes shown. Instead of being a clickable pane, you can now drag the locators on the basis vectors to show new projections on the horizontal and vertical 2D plane using "H" and "V". The controls to change the azimuthal (Z) basis vector are not available for manipulation in this Demonstration.

Viewing the static or animated 2D (or 3D) positions of selected vertices (or their theoretically assigned fundamental physics particle) is accomplished by projecting them using a dot product of 2 (or 3) 8D basis vectors. These can be manipulated with controls in the HMI pane 8. 3D displays use native Mathematica rotation and zoom options (cntrl or alt-click-drag)

The triality buttons "T1", "T1p", "T2", "T2p", "T4", and "T4p" select different sets of triality rotations using one or more of the selected rotation matrices. These are named for the number of sixths of a unit circle involved in selected hyperplanes of a higher-dimensional rotation. The suffix of "p" indicates that it uses the "physics" data set, which is a rotation of the SRE E8 vertices that let A. G. Lisi calculate particle charges on a single generation of fermion particles. The physics data set is activated in some "Favorite" selections.

Animations are generated by increasing the number of (time) steps and selecting the type of 8D "flight path" (Translational or Rotational and Spin - controlled in the HMI pane 8). 
It is interesting to note that the 8D vertex positions are always static. What is moving is the 2D (or 3D) perspective inside an 8D "information space". The "information space" in this case is made up of the highly symmetric vertex positions of the Split Real Even (SRE) E8 Lie Group assigned to fundamental physics particles based on their quantum numbers.  The specifics of each flight path are depend ant on the 3 basis vectors and dimensions selected for creating the hyper - planes about which they rotate. These are also manipulated with controls in the HMI pane 8.

Mouse over the shapes in the result to show the detail particle information relating to E8, Cellular Automata and physics (as shown in the “Particle” pane 4).

The output window is a clickable pane (if in 2D mode). If the physics box is not checked (showing vertex dots), clicking on or near a vertex calculates and shows the connecting vertex edges of the (nearest) clicked vertex. The edges are color coded by overlap count. If the physics box is checked (showing assigned particle shapes), clicking calculates triads of edges that include the particles whose E8 vertex values sum to the clicked particle. Clicking an already highlighted vertex now shows only that edge (or triad of edges).

The output window in 3D mode will save the rotation/zoom ViewPoint to the corresponding Slider2D(x,y) and z Slider controls in the HMI pane.

Edge calculation is done with a modified code snippet from  Eric W. Weisstein, "600-Cell." From MathWorld - A Wolfram Web Resource. http://mathworld.wolfram.com/600-Cell.html

If you have Mathematica and want to get a more fully interactive experience, simply set "localize=False" near the top of the initialization section. This activates many features not available in Demonstrations.

For the 2D Fourier Transform sub-pane, See John Holland’s http://demonstrations.wolfram.com/2DFourierTransforms

References

[1] A. G. Lisi. "An Exceptionally Simple Theory of Everything." (Aug 8, 2012) arxiv.org/abs/0711.0770.

Particle:

This demonstration creates an interactive pane for identifying fundamental particles in physics through the selection of quantum parameters. The information displayed in the output window is also displayed in the ToolTip on mouse-over of the particles in the E8 window.

It includes the well known Standard Model particles (leptons, quarks, W and Z bosons) as well as a small group of theoretically predicted particles related to a sector of Higgs boson(s). There are also included in this demonstration a very small group of predicted particles which are loosely based on an E8 Lie group extension of the Standard Model originated by A.G. Lisi.

The quantum parameters used to identify individual fundamental particles based on an 8 bit pattern are as follows:  
1 bit (a)nti-particle vs. the normal particle,
1 bit (p)article type identifying the Neutrino vs. Electron for Leptons and the Up vs. Down for 3 generations of Quarks,
2 bit (c)olor for the 4 colors w,r,g,b (where (w)hite is used to identify particles with no color), 
2 bit (s)pin for the 4 combinations of Up/Dn and Left/Right, 
2 bit (g)eneration for 4 generations (with 0 for Bosons and 1,2 and 3 for the {e, μ, τ} generations of Standard Model (SM) Fermions).

This unique 8 bit pattern for each particle is ordered as {a, p, c2, c1, s2, s1, g2, g1} into a base 2 number from 0 to 255. It is also used to generate the display for Steven Wolfram’s New Kind of Science (NKS) Cellular Automata (CA) patterns, which may be related to the theory. The (2^8)=256 particles identify the known or predicted SM Fermions (192), Bosons (30 with 18 in the "Higgs Sector"), and the 8 dimension generator vertices (plus 8 excluded "anti-dimension" vertices). There are 18 remaining (Boson) particles added which are identified using an approach loosely connected with the theoretical work of A.G. Lisi in arxiv.org/pdf/0711.0770.

This approach to particle assignments allows for them to be associated with the 240 vertices of the split real even (SRE) E8 Lie group plus the 8 Orthoplex (dimensional) generator vertices (and 8 anti-dimensional generator vertices typically excluded from E8). The lexicographic ordering of the SRE E8 is consistent with a Cl(8) Clifford algebra, which has a familiar sequence of row 9 of the Pascal Triangle.

The particle families are split into 5 rows {two rows being the familiar 192 Fermion (l)eptons and (q)uarks, the 48 Bosons in the next two rows ωg and ϕΦ, and a row for the 16 (Ex)ception dimension generator particles}. This demonstration applies a rigorous particle reference label for symbolic pattern matching to facilitate this identification. Each particle is given a unique 2D and 3D reference symbol. The symbol’s size, shape, color, and shade are used to uniquely identify the bitwise pattern.  There are 2D particle shapes {circle, square, triangle} and their corresponding anti-shapes {pentagon, diamond, inverted triangle} and 3D shapes {sphere, cube, tetrahedron} and anti-3D shapes {dodecahedron, icosahedron, octahedron}. The particle’s SM generation alters the size in accordance to the tendency for mass to increase with increasing generation. The particle mass and lifetime (if known), are given by the ParticleData PDG curated data set for particles. The colors are, of course, used to identify the color content. The shade is modified with spin.

See http://demonstrations.wolfram.com/BranchingRatiosForHiggsBosonDecays

CKM:

See http://demonstrations.wolfram.com/NeutrinoOscillations/

There are added capabilities to view both the PNMN and CKM unitary triangle matrices, print and reference my ToE Neutrino mass predictions, which now accomodate the Koide relationships in particle masses.

Hadron:

See http://demonstrations.wolfram.com/CombiningQuarksIntoHadrons/

There are added capabilities to view the 4 and 6 Quark Hadrons as well as list all composite particles made up of the same quark content, including buttons to access their decay modes. To this list is a menu selectable Tooltip display of any particle property, and with decay modes shown with a button click.

See http://demonstrations.wolfram.com/HadronDecays/

This integrates with particle selections.

HMI:

Added user controls for manipulating the third pane (VisibLie_E8) Lie Algebra particle projection viewer are provided in the top section, including:

* The artPrint checkbox allowing for higher resolution 2D images and cylinders (vs. lines) for edges/trialities/clicked vertex relationships.
* The dataSet dropdown menu allows for selection of multiple sets of coordinates, including binary, E8, Physics (rotated E8), and the 600 Cell.
* The limit checkbox allows the vertices to be shown at the edges of the frame bounding box rather than be off the page (enabling a "dimensionally rigorous" hyperbolic viewing).
* The pSize dropdown menu selects the variable size for the boson particles (tiny, small, medium large, huge).
* The zoom slider is an 10^+/-24 zoom capability (for non-E8 data sets).
* The particle scale slider allows all particles to be scaled (0 to 1).
* The ticks determines how many tick marks to show on each axis (0 to 50) given the frame size.
* The frame slider allows a fine grained zoom (0 to 2) determines the  X,Y,Z bounding box (frame) dimension.
* The background color slider allows selection of any of none, black, white or one of 24 colors determined by the selected particle gradient.
* The VisibLie_E8 also has checkboxes for the display of Symbols, Coordinates, Mass/Lifetimes,

An extensive set of controls in this main window include:

If working with the .NB version with full Mathematica license, several added controls are available:
File In (read) / Out (save) - Similar to ECE, the "plus" in the upper right corner allows the temporary save of selected configuration to a list of Bookmarks (pre-loaded with MetaFavorites). In addition with Player Pro or Mathematica, buttons to select a saved inFile can be used. The "out to file" checkbox activates output (the noAnim8 checkbox is for very large animations that need too much memory to display). The Dir button & file (input field + 2 digit random #) and type dropdown menu establish output files. This also produces a < file > .m to record the configuration changes.

Checkboxes for optional displaying of:

* particle vertices (pVertices)
* clicked vertex edge relationships (clickVerts)
* parametric 3D surfaces (surfaces) currently set to display electron probability densities from the Stowe-Janet-Scerri periodic table E8 root assignments
* triangle surface meshes (polygons) with surface area and volume calculations on the hull of selected edges
* enabling perspective projection using the nD camera viewpoint slider below (perspective)
* vectors from {0,0,0} to a vertex (pVectors)
* vertex color coding based on how many are overlapping that projected position (pLocations)
* measured particle mass and lifetime (pMassLife)
* 2D circle or 3D sphere unit boundary (perimeter)

Sliders for controlling of:

* n-Dimensional camera location (+/- 5) (nD camera)
* rectifications of particles (0 to 5) (rectify)
* change the number of dimensions used for edge calculations, e.g. for n-cube polytopes (1 to 8 ) (dim)
* selecting the gradient color set used for color coding the particles (0 to 51). Allows selection from the 50 Mathematica color gradients or 0=Lisi’s original or 51=my new 24 color gradients.
Note: pOverlaps overrides the colors.

Miscellaneous controls:

* clear clickVerts will clear the clicked vertices
* 2D Face select allows switching which of 3 unique (of 6 cubic) faces to display. This also impacts which H, V, Z projection vectors the dimension locators modify (they operate on the first two in the list).
* frame center and view point are two sets of 2D sliders plus a z-slider which control a 3D position, buttons for small and large incremental movement are also provided.
** frame center controls where the center of the frame is relative to the origin (the 3D camera points to this location) 
** view point controls where the 3D camera position is in space. These controls can also be updated by using the rotation/zoom mouse control in the 3D mode of the output window of the E8 pane (#3).
* Norm’d edges ListPicker menu allows the selection of multiple sets of edges based on norm’d edge lengths {Value, Qty}. These change based on datasets and the number of edge dimensions (vertical dim slider).
* eFrames  and eWindow sliders create animations of edges (the number of movie frames and the number of edges in each window - each calculated based on how many edges are selected for display - starting with shortest projected inner edges).
* EdgeListAnimate checkbox will force an animation of all sets of Norm’d edges.
* The inner filter slider will filter out 0 to 100% of the selected edges (starting with shortest projected inner edges).
* eColorPos checkbox colors the edges based on their position in the projection as opposed to assignment based on their sequence in the edge list.
* eColors is a slider (like pColor) that selects the edge color gradient.
* scale surface is a slider to select how big the parametric surface objects are relative to the frame.
* surface color selects whether to apply edge color selections to the surfaces (vs. the default surface).
* eRadius is a slider (.001 to .1) that determines the diameter of the edge cylinder (3D w/artprint) or line (2D or w/o artprint).

Controls related to the particle filters:

Binary or bitwise particle filters:
* clear filters resets all 256 particles "optional" display property to "?".
* OR / AND Logic radio button selects the type of binary logic to be applied to the selected binary filters. OR uses button-bars (allows combining of multiple elements of a binary set). AND uses radio buttons (only 1 selection per binary set)
* Row Filter filters particle sets by the assigned rows in the ToE model (including Leptons, quarks, bosons (1 & 2), and/or excluded particles).
* Anti-Type-Color-Spin-Gen Filters are 1 or 2 bit binary sets associated with each particle parameter (anti, pType, color, spin, gen).
Note: Each of these sets defines the shape/color/shade/size of every particle.

pLists dropdown menu for selecting subsets of particles. Based on their anti-type-color-spin-generation properties, or Group Theory (S0{10},SO(8), E7,F4,G2,A2.”Favorite” projections.

Controls related to the vertex projections:
Viewing the static positions of particles by flying through the 8D space is done by projecting them (dot product) into a 2D or 3D view.

* reset Path resets all path settings back to original “Favorite” (aka. numeric other projections) selection.
* Translational/Rotational/Trans-Rot/Spin3D radio buttons select whether the chosen sliders will calculate an 8D rotational or translational path. Spin3D will simply rotate the 3D image around the Z axis.
* Identity/ptCoords radio buttons that select the type of beginning and ending flight path endpoint calculations:
** The 3 selected H,V,Z Identity dimensions are applied to the initial path point. Final path point is the current H,V,Z location indicators below.
** The 3 selected ±H,V,Z ptCoord coordinate dimension indicators are applied to the initial and final identity dimensions path points.

* Favorites (aka. other projections) dropdown menu is used for selecting a growing set of favorite 8D H, V, Z projection vectors (by number, annotated in the full .CDF and .NB files).
* H,V,Z ptCoord Indicators/Adjustors - for each pair of H,V,Z Dim & Projection vector rows:
** Toggler Buttons select which of the 8 dimensions can be used in calculating rotational/translational paths and/or surface calculations. 
** Vertical slider bars and numeric text fields can be used to adjust the value of the projections and endpoints for paths.

Other Supporting material:

Binary/Clifford Algebra to E8 assignments via the Pascal triangle

Bitwise particle assignments

Anti(p(OverBar[p]))(Subscript[2, a]) (pType(0  1)(Subscript[2, p]) (({{Gen , ---------->, ({{0 , Generations}}), <---------, (Subscript[3, g])Generations, (1e,2μ,3τ)}, {Spin, (Subscript[4, s]) Spin((Overscript[L, ⋀⋁])(Overscript[R, ⋀⋁])), (Subscript[3, s]) Spin((Overscript[L, ⋀⋁])(Overscript[R, ⋀])),   1Spin((Overscript[R, ⋁])), -----> (Subscript[4, s]) Spin, ((Overscript[L, ⋀⋁])(Overscript[R, ⋀⋁]))<-----}, {Color, 0Color(w), ----------->, (Subscript[3, c])Color(rgb), <------------, 0Color(w)}, { Row, 5, 4, 3, 2, 1}, {Count , (Subscript[4, s]), (Subscript[3, s])(Subscript[x3, c]), (Subscript[3, c]), (Subscript[3, g])(Subscript[x3, c])x (Subscript[4, s]), (Subscript[3, g])(Subscript[x4, s])}})))

Anti (p(OverBar[p]))(Subscript[2, a]) (({{Gen , ---------->, ({{0 , Generations}}), <---------, (Subscript[3, g])Generations, (1e,2μ,3τ)}, {Spin, (Subscript[4, s]) Spin((Overscript[L, ⋀⋁])(Overscript[R, ⋀⋁])), (Subscript[3, s]) Spin((Overscript[L, ⋀⋁])(Overscript[R, ⋀])),   1Spin((Overscript[R, ⋁])), -----> (Subscript[4, s]) Spin, ((Overscript[L, ⋀⋁])(Overscript[R, ⋀⋁]))<-----}, {Color, 0Color(w), ----------->, (Subscript[3, c])Color(rgb), <------------, 0Color(w)}, { Row,  5
Ortho8=, 4, 3, 2, 1}, {pType=1, {(Subscript[Ex, 5-8]), F4={D4={{{(Subscript[e, S])ϕ,(Subscript[e, T])ϕ},B}, G2=A2=D3RL={{D2={(Subscript[ω, L]),(Subscript[ω, R])},W}}, {u,c,t}, {(Subscript[ν, e]),(Subscript[ν, μ]),(Subscript[ν, τ])}}, {pType=0, (Subscript[Ex, 1-4])}, (F4^S)={{(Subscript[x, 1])Φ,(Subscript[x, 2])Φ,(Subscript[x, 3])Φ}, (G2^S)=A2RL={(g^(g(OverBar[b]))),(g^(r(OverBar[b]))),(g^(r(OverBar[g])))}}}, {d,s,b}, {e,(Subscript[e, μ]),(Subscript[e, τ])}}}))

Projection Favorites

(* rotating to Lisi Physics *)
rotPhysics@in_:=FullSimplify[rot3.rot3a\[Transpose].#]&/@in;

H1a=Range@4;
V1a=Range[6,8];
Z1a={5};

H2a={1,5,8};
V2a={2,4,7};
Z2a={3,6,1};

(* Original *)
H3a=(*)First@adrt N@**){0,-Zt1,0,1};
V3a=(*)First@adrt N@**){Zt5,0,Re@cω@11,0};
Z3a=(*)First@adrt N@**){0,1,0,Zt1};

H3a={0,-Φ 2Sin@((2π)/60),0,1};
V3a={ Φ 2 Sin@((2 π)/30),0,2 Sin@((2 π)/15),0};
Z3a={0,1,0,Φ 2 Sin@((2π)/60)};

(* Petrie Projections #9-11 *)
doCube@n_:={
H=Join[Table[Cos[(i-1) π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2,
V=Join[Table[Sin[(i-1) π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2,
Z=Join[-Table[(((-1))^(i+1))Cos[(n-i) π/n]+(((-1))^i)Sin[(n-i) π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2};

(* "Pascal Triangle" projections #27-29 *)
doCubePascal@n_:={
H=Normalize@Join[Array[1&,n],Array[0&,8-n]],
V=Normalize@Join[Table[i,{i,-(n-1),n-1,2}],Array[0&,8-n]],
Z=Normalize@Join[Table[(((-1))^i)i/2,{i,-(n-1),n-1,2}],Array[0&,8-n]]};

(* Creates the Ruen A5 style *)
do2PiCube@n_:={
H=Join[Table[Cos[(i-1) 2π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2,
V=Join[Table[Sin[(i-1) 2π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2,
Z=Join[-Table[(((-1))^(i+1))Cos[(n-i) 2π/n]+(((-1))^i)Sin[(n-i) 2 π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2};

(* Creates the Ruen A7 style #3 - plus offsets *)
do2PiCubeAlt[n_,m_,offset_]:={
H=Join[Table[Cos[(i-1) 2π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2+offset;H[[m+4]]=H[[m]];H,
V=Join[Table[Sin[(i-1) 2π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2-offset;V[[m+4]]=V[[m]];V,
Z=Join[-Table[(((-1))^(i+1))Cos[(n-i) 2π/n]+(((-1))^i)Sin[(n-i) 2 π/n],{i,Min[dims,n]}],Array[0&,Max[0,dims-n]]]/2-offset;Z[[m+4]]=Z[[m]];Z};

(* E8(12, 24) projections *)
doProj@n_:={
H=Flatten[Join[Table[{a=Cos[(i-1) 2π/n],a/2},{i,j=Ceiling[n/2]}],Array[0&,dims-2 j]]]/2;H[[2]]=-H[[2]];H,
V=Flatten[Join[Table[{a=Sin[(i-1) 2 π/n],a/2},{i,j=Ceiling[n/2]}],Array[0&,dims-2 j]]]/2,
Z=Flatten[Join[-Table[{a=(((-1))^(i+1))Cos[(n-i) 2 π/n]+(((-1))^i)Sin[(n-i) 2 π/n],a/2},{i,j=Ceiling[n/2]}],Array[0&,dims-2 j]]]/2};

(* Coxeter and other projections *)
prjListName=Join[
{"OrthoXYZ"},
catStr@{"BC(n)=D(n+1)",2,8},
catStr@{"Pascal",2,7},
{"G2(6)","A5(6)","A7(8)","E6(12)","E7(18)",
"E8(6)","E8(12x5)","E8(20)","E8(24)","E8(30)",
"H4(OrthoXYZ)","H4(5)","H4(30)","E8->H4",
"F4(12)"},
catStr@{"Triality",1,3},
catStr@{"Physics",1,5},
catStr@{"BinSq(4)",1,3},
catStr@{"Madore",1,2}];

prjList=Join[
(* OrthoXYZ *)
{{dla@{1},dla@{2},dla@{3}}},
(* BC(n)=D(n+1) *)
doCube/@Range[2,8],
(* Pascal *)
doCubePascal/@Range[2,7],
(* G2(6) *)
{rotPhysics@N[Join[#,{0,0,0,0}(Sqrt[2])]&/@{{-bst,bst,(Sqrt[1-2(bst^2)]),0},{bst,bst,0,(Sqrt[1-2(bst^2)])},{1-bst,bst-1,bst,0}}],
(* A5(6),A7(8), E6(12) *)
do2PiCube@6,do2PiCubeAlt[8,1,0],doProj@6,
(* E7(18) *)
FullSimplify@#[[{1,8,3,4,5,6,7,2}]]{-1,-1,1,1,1,1,1,1}2/15&/@{H18,V18,Reverse@H18},
(* E8(6) *)
Join[#,{0,0,0,0}]/2&/@{{-(Sqrt[3])+1,0,1,1},{0,(Sqrt[3])-1,-1,1},{2,0,(Sqrt[3])-1/2,(Sqrt[3])-1/2}},
(* E8(12x5) *)
rotPhysics@{{0,0,1,0,0,0,0,1}(Sqrt[3]),{1,1,0,1,0,0,(Sqrt[3]),0},{1,0,1,,0,(Sqrt[3]),0,0}}/(Sqrt[6]),
 (* E8(20) *)
{H20,V20,Reverse@H20}/30,
(* E8(24) *)
doProj@8,
(* E8(30) *)
FullSimplify[Inverse[C600].Join[{0,0,0,0},#[[{2,1,4,3}]]]]&/@{H3a,V3a,Z3a},
(* H4(OrthoXYZ) *)
{dla@{1,5},dla@{2,6},dla@{3,7}}/2,
(* H4(5) *)
Join[#,{0,0,0,0,0}]/2&/@{Normalize@{8,-13,0},{0,0,1},{1,0,-1}},
(* H4(30) *)
Join[#,{0,0,0,0}]/2&/@{H3a,V3a,Z3a},
(* E8->H4 *)
Join[#,{0,0}]&/@C600[[2;;4,{2,3,4,6,7,8}]],
(* F4(12) *)
rotPhysics@{{0,0,(Sqrt[2]) Ht8,0,0,-Ht6,0,(Sqrt[2]) Ht8},{1/(Sqrt[6]),1/(Sqrt[6]),0,bst,-Ht6,0,(Sqrt[2]) Ht8,0},{bst,(Sqrt[2]) Ht8,0,bst,0,0,0,0}}(Sqrt[2]),
(* F4(Triality)1/3 *)
rotPhysics@{{1-(2/(Sqrt[3])),-1+(2/(Sqrt[3])),((-1+(Sqrt[3]))/(Sqrt[3])),0,0,0,1,0},{-1+(2/(Sqrt[3])),-1+(2/(Sqrt[3])),0,1-(1/(Sqrt[3])),0,0,0,-1},{1-(2/(Sqrt[3])),1-(2/(Sqrt[3])),-1+(1/(Sqrt[3])),0,0,0,-(1/2),-((Sqrt[3])/2)}}(Sqrt[2]),
{{0,Ht2,Ht6,-Ht6,Ht5,Ht4,0,Ht3},{Vt7,0,Vt6,Vt6,0,Vt3,Vt1,Vt3},{0,Vt7,Vt6,Vt6,0,Vt3,Vt1,Vt3}},
dlXs=dls@{7};dlYs=dls@{8};dlZs=dls@{6};setIFpath;setPTo[5°],
(* Physics1/5 *)
rot3.#&/@{-{0,Ht5,Ht6,Ht8,Ht2,Ht3,Ht4,0},{Vt7,0,Vt6,Vt8,0,Vt3,Vt4,Vt1},-{0,Ht2,-Ht8,Ht8,Ht5,Ht3,-Ht3,0}},
{{0,0,0,0,1,0,0,0},{0,0,0,0,0,1,0,0},{0,0,-(1/2),-(1/2),0,0,(1/2),(1/3)}},
rot3.#&/@{{0,Ht5,Ht6,Ht8,Ht2,Ht3,Ht4,0},{Vt7,0,Vt6,Vt8,0,Vt3,Vt4,Vt1},{0,-Ht2,Ht8,-Ht8,-Ht5,-Ht3,Ht3,0}},
{{-Vt7/(Sqrt[2]),-Vt7/(Sqrt[2]),0,(Sqrt[2]) Ht8,Ht5,0,-Ht6,bst},{(Sqrt[2]) Ht8,-(Sqrt[2]) Ht8,-Ht6,0,0,-bst,0,0},{0,(Sqrt[2]) Ht8,-(Sqrt[2])Ht8,-Ht6,-bst,0,0,0}},
rotPhysics@(dlXs=dls@{5};dlYs=dls@{6};dlZs=dls@{7};setIFpath;setHVZ@initial),
(* BinSq(4)1/3 *)
Join[#,{0,0,0,0}]/(Sqrt[7])&/@{{0,0,(Sqrt[7]),0},{(Sqrt[2]),0,0,(Sqrt[5])},{0,(Sqrt[2]),0,(Sqrt[5])}},
Flatten[{#,Reverse@#,RotateRight@RotateRight@#}&/@{Join[Array[0&,4],(2^(#-4))&/@Range@4]},1],
dla@#&/@{H2a,V2a,Z2a}/(Sqrt[3]),
(* Madore1/2 #5/6*)
N@Normalize@#&/@{{73,67,53,17,-17,-53,-67,-73},{17,53,67,73,73,67,53,17},{17,53,67,73,73,67,53,17}},
N@Normalize@#&/@{{14,8,7,-1,-2,-4,-11,-13},{2,4,11,13,14,8,7,-1},Reverse@{2,4,11,13,14,8,7,-1}}}];

Lie Groups

n-Ortho (4 & 8)

8 Orthoplex(16)=E8 Dim/Anti-Dim=Excluded

nOrtho@8

ortho8=e8bFind[nOrtho,8]
pLbl@%

nOrtho@8==Sort[exception2⋃exception8]

4 Orthoplex(8)=Cell16(8) [used in cell24={4-Cube (16) 8-Cell aka. Tesseract}+{Cell16(8)=4Orthoplex}]

{dual-mc5?(16){eϕ}+2B (half of D2-md3)} & {18 Dual B/C3{6 (B3^S)+12 (C3^S)} xΦ} are converted through 
Dual 24 cells D4(24) & (F4^S)(24) 
and the standard 24 Cell {4-Cube (16) 8-Cell aka. Tesseract}+{Cell16(8)=4Orthoplex}

e8cell16=Sort@e8bFind[nOrtho,4]
pLbl@%

B8(128)=SO(17) & C=Sp(8, C)=Sp(16)=8 DemiCube(DemiOctaract)=7-Cube=Heptaract= {16 (B8^S)+112 (C8^S)}=128 gen 1&3 fermions}=E8 Half Integers

c8=cube7=sp16=b8=so17=e8HalfInteger=e8bFind[cn,8][[;;(2^7)]]
Length@%

n-Cubes 2-6

cube6=e8bFind[cn,7][[;;(2^6)]]
Length@%
pLbl@%%

cube6=cubeFind@6
Length@%
pLbl@%%

cubeFindRL@6
Length@%
pLbl@%%

cube5=e8bFind[cn,6][[;;(2^5)]]
Length@%
pLbl@%%

cube5=cubeFind@5
Length@%
pLbl@%%

cube4=e8bFind[cn,5][[;;(2^4)]]
pLbl@%
Length@%
pE8/@cube4+1/2

cube4=e8cell8=cubeFind@4
Length@%
pLbl@%%

cubeFindRL@4
Length@%
pLbl@%%

cube3=e8bFind[cn,4][[;;(2^3)]]
Length@%
pLbl@%%

pLbl@e8bFindRL[cn,4][[;;(2^3)]]

cube3=cubeFind@3
Length@%
pLbl@%%

cube2=e8bFind[cn,3][[;;(2^2)]]
Length@%
pLbl@%%

cube2=cubeFind@2
Length@%
pLbl@%%

n-Cube Orthogonal subgroups (DemiCubes) 2-7

mc6=mc@6
Length@%
pLbl@%%

(* DemiPenteract *)
mc5=mc@5
Length@%
pLbl@%%

(* DemiTesseract 16Cell (which is also from a 4Ortho)*)
mc4=mc@4
pLbl@%
Length@%
pE8/@mc4+1/2

mc3=mc@3
Length@%
pLbl@%%

mc2=mc@2
Length@%
pLbl@%%

D8(112)=SO(16)= 
all 48 (row 3 & 4) bosons F4=24 D4{{dual - mc5?(16)eϕ}+2B}, 6A2 {{4 D2 ω}+2W}+(F4^S){{18 Dual B/C3{6 (B3^S)+12(C3^S)} xΦ}, 6 (G2^S) (g) luons}
& 64 {(Subscript[e, μ]),c,s} gen 2 fermions}=SU(8)?=Schläfli (Subscript[t, 1]){(3^(1,1,1,1,1,1,1,1))}

d8=so16=e8Integer=e8bFind[dn,8]
Length@%
pLbl@%%

E8(240)=all fundamental particles={2^8 nodes minus the 8 Orthoplex {dimensions+anti-dimensions}}=Schläfli{(3^(4,2,1))}

cn@7==Union[fermion1,fermion3,fermion5,fermion7,fermion9][[All,;;7]]

dn@8==boson4⋃boson6

e8=cube7⋃d8;
Length@%

e8all=Union[c8,d8,ortho8];
Length@%

e8all==Range@(2^8)

E1-7=Schläfli{(3^(n-3,2,1))}

(* 126 E7 are 72 E6 + (48 red quarks + 6 red (RowDefault[(Subscript[x, 1]), Φ])) 
70+32+24=(72-2B)+(2B+6 red (RowDefault[(Subscript[x, 1]), Φ])+24 antiquarks)+24 quarks *)
e7=mE@7
Length@%
pLbl@%%

(* 72 E6 are 48 Leptons + (RowDefault[18(Subscript[e, S/T]), Φ]) + 2B/2W/4ω 
42+30=(24neutrino+(RowDefault[18(Subscript[e, S/T]), Φ]) )+(6W/ω +24electron) *)
e6=mE@6
Length@%
pLbl@%%

e5=mE@5
Length@%
pLbl@%%

e4=mE@4
Length@%
pLbl@%%

e3=mE@3
Length@%
pLbl@%%

mc4
Complement[mc4,e3]
Length@%
pLbl@%%

e2=mE@2
Length@%
pLbl@%%

e1=mE@1
Length@%
pLbl@%%

e1==e2==mc2

D2-7=SO(2n)=SU(n)=Schläfli (Subscript[t, 1]){(3^(n 1s))}

d7=so14=su7=e8bFind[dn,7]
Length@%
pLbl@%%

d6=so12=su6=e8bFind[dn,6]
Length@%
pLbl@%%

d5=so10=su5=e8bFind[dn,5]
Length@%
pLbl@%%

(* supposed to be same as e5 *)
e5
Complement[e5,d5]
Length@%
pLbl@%%

d4=so8=su4=e8bFind[dn,4]
pLbl@%
Length@%
pE8/@%%%

dynkinD4={{-1,-1,0,0},{-1,0,-1,0},{0,-1,-1,0},{-1,0,0,-1},{-1,0,0,1},{0,-1,0,-1},{0,-1,0,1},{-1,0,1,0},{0,0,-1,-1},{0,0,-1,1},{0,-1,1,0},{-1,1,0,0},{1,-1,0,0},{0,1,-1,0},{0,0,1,-1},{0,0,1,1},{1,0,-1,0},{0,1,0,-1},{0,1,0,1},{1,0,0,-1},{1,0,0,1},{0,1,1,0},{1,0,1,0},{1,1,0,0}};
Length@%

Sort@dynkinD4==Sort[pE8[#][[;;4]]&/@d4]

d4==Sort[position[e8b[[All,2]],#]&/@(Join[#,Array[0&,4]]&/@dynkinD4)]

d3=so6=su6=e8bFind[dn,3]
Length@%
pLbl@%%

(* supposed to be same as e3 *)
e3
Complement[e3,d3]
Length@%
pLbl@%%

d2=so4=su2=e8bFind[dn,2]
Length@%
pLbl@%%

D3-7 Orthogonal subgroups

md7=mD@7
Length@%
pLbl@%%

md6=mD@6
Length@%
pLbl@%%

md5=mD@5
Length@%
pLbl@%%

md4=mD@4
Length@%
pLbl@%%

(* D2 *)
md3=mD@3
Length@%
pLbl@%%

A1-7=SL(n+1)

a7=sl8=mA@7
Length@%
pLbl@%%

a6=sl7=mA@6
Length@%
pLbl@%%

a5=sl6=mA@5
Length@%
pLbl@%%

a4=sl5=mA@4
Length@%
pLbl@%%

(* supposed to be same as e4 *)
e4
Complement[e4,a4]
Length@%
pLbl@%%

a3=sl4=mA@3
Length@%
pLbl@%%

(* supposed to be same as e3 and d3 *)
e3
Complement[e3,a3]
Length@%
pLbl@%%

pE8/@a3

Sort[%[[All,;;3]]]//MatrixForm

(* supposed to be same as e3 and d3 *)
d3
Complement[d3,a3]
Length@%
pLbl@%%

a2=sl3=mA@2
pLbl@%
Length@%
a2==d2

pE8/@a2
%[[All,;;2]]//MatrixForm

a1=sl2=so3=sp2=mA@1
Length@%
pLbl@%%

HammingCode (16 up-right (Overscript[R, ⋀]) and down-left(Overscript[ L, ⋁]) gen 1 fermions - e1 & 14 of the 70 fermion5 Perms@{1,1,1,1,0,0,0,0}, leaving 56 for correspondence to Perms@{1,0,0,0,0,0,0-1}})

H841={"00","0F","33","3C","55","5A","66","69"};

H84=Join[H841,ToUpperCase@IntegerString[pAntiBit@FromDigits[#,16]-2,16]&/@Reverse@H841]

H84hex=IntegerDigits[FromDigits[#,16],2,8]&/@H84

e8Bdec=FromDigits[#,2]&/@e8bin

MatrixForm@PaddedForm[FromDigits@#,8,NumberPadding->{"0","0"}]&/@H84hex

HammingDistance[zeros,#]&/@e8bin

E8H84a=FromDigits[#,16]&/@H84

E8H84b=position[e8Bdec,#]&/@E8H84a

Clifford Algebra Primitive Idempotents (also has e1 & 14 of the 70 fermion5 Perms@{1,1,1,1,0,0,0,0}, leaving 56 for correspondence to Perms@{1,0,0,0,0,0,0-1}})

clToList@(Subscript[Cl, a_]):=a
clToList@(Subscript[Cl, a_,b_]):=b
sigToNum:=clToList@GetSignature

toArray1@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~"}"]
toArray2@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~", "~~x2_~~"}"]
toArray3@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~", "~~x2_~~", "~~x3_~~"}"]
toArray4@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~", "~~x2_~~", "~~x3_~~", "~~x4_~~"}"]
toArray5@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~", "~~x2_~~", "~~x3_~~", "~~x4_~~", "~~x5_~~"}"]
toArray6@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~", "~~x2_~~", "~~x3_~~", "~~x4_~~", "~~x5_~~", "~~x6_~~"}"]
toArray7@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~", "~~x2_~~", "~~x3_~~", "~~x4_~~", "~~x5_~~", "~~x6_~~", "~~x7_~~"}"]
toArray8@a_:=ToExpression@StringCases[ToString@a,"{"~~x1_~~", "~~x2_~~", "~~x3_~~", "~~x4_~~", "~~x5_~~", "~~x6_~~", "~~x7_~~", "~~x8_~~"}"]

toNum4@a_:=(array=toArray4@a;
num=Array[Array[0&,ls=sigToNum]&,Length@array];
Do[(num[[ii,ls+1-#]]=1)&/@array[[ii]],{ii,Length@array}];num)

rightSM
leftSM

(* Primitive Idempotents *)
SetSignature[0,8];
x1=(1/2)(1+(Subscript[e, {1,2,4,8}]));
x2=(1/2)(1+(Subscript[e, {2,3,5,8}]));
x3=(1/2)(1+(Subscript[e, {3,4,6,8}]));
x4=(1/2)(1+(Subscript[e, {4,5,7,8}]));
f=x1⊙x2⊙x3⊙x4/.slRep//ClExpand

Reverse/@toNum4@f

Sort[position[pBin/@Range@256,#]&/@toNum4@f]

getNumLen@a_:=
If[Length@toArray1@a==1,1,
If[Length@toArray2@a==1,2,
If[Length@toArray3@a==1,3,
If[Length@toArray4@a==1,4,
If[Length@toArray5@a==1,5,
If[Length@toArray6@a==1,6,
If[Length@toArray7@a==1,7,
If[Length@toArray8@a==1,8,0]]]]]]]]

toNum8@a_:=(array=First[ToExpression["toArray"<>ToString@getNumLen@a][a]];
num=Array[0&,sigToNum];
(num[[sigToNum+1-#]]=1)&/@array;num)

ClBasis

Sort[position[pBin/@Range@256,#]&/@toNum8/@ClBasis]==Range@256

(*)
SetSignature[0,2];
ClBasis

(*)
SetSignature[3,1];
ClBasis

getNumLen/@ClBasis//Tally
%[[All,2]]==permPset

short Bs 1-8

These are the 16 that make up the complement between the long 128 B/C 1/2 integer and the 112 D integers -positionally equivalent for D-1/2 or B/C+1/2

(* Clifford Primitive Indempotents *)
(*) bS8=e1⋃Sort[position[pBin@Range@256,#]&/@toNum4@f] **)
bS8={1,97,99,107,112,113,122,127,130,135,144,145,150,158,160,256}
pLbl@%
Length@%
pE8/@bS8+1/2

E8H84b=Sort[position[e8bin,#]&/@H84hex]

(* Hamming Codes (*)
bS8=E8H84b
pLbl@%
Length@%
pE8/@bS8+1/2

bS7=bS@7
Length@%
pLbl@%%

bS6=bS@6
Length@%
pLbl@%%

bS5=bS@5
Length@%
pLbl@%%

bS4=bS@4
Length@%
pLbl@%%

bS3=bS@3
Length@%
pLbl@%%

bS2=bS@2
Length@%
pLbl@%%

bS1=bS@1
Length@%
pLbl@%%

short Cs 1-8

Maps the 128 C8 1/2 integer fermion permutations to the 112 integer boson permutations by removing the 16 bS8 and aligning 28 {1,1,0,0...} & 28 {0,0...-1,-1}.
This leaves the 56 1/2 integers to be mapped to the integer boson6={1,0,0...,-1}by removing the ends bS1{1,256} and 14 others in the middle 70 
(both Hamming code and Primitive idempotents include bS1 and 14 others in the middle 70).

cS8=cS@8
Length@%
pLbl@%%

cS7=cS@7
Length@%
pLbl@%%

cS6=cS@6
Length@%
pLbl@%%

cS5=cS@5
Length@%
pLbl@%%

cS4=cS@4
Length@%
pLbl@%%

cS3=cS@3
Length@%
pLbl@%%

cS2=cS@2
Length@%
pLbl@%%

B1-7=SO(2n+1) & C=Sp(n, C)=Sp(2n)=Schläfli{n,(3^2)} (need to determine B & C group difference)

C7(98)=B7={14 (B7^S)+84 (C7^S)}=Schläfli{4,(3^5)}

c7=so15=b7=sp14=bS7⋃cS7
Length@%
pLbl@%%

C6(72)=B6={12 (B6^S)+60 (C6^S)}=(Subscript[S, 12]^(+))=Sp(8, C)=Schläfli{4,(3^4)}

c6=so13=b6=sp12=bS6⋃cS6
Length@%
pLbl@%%

C5(50)=B5={10 (B5^S)+40 (C5^S)}=(Subscript[S, 10]^(+))=Sp(6, C)=Schläfli{4,(3^3)}

c5=so11=b5=sp10=bS5⋃cS5
Length@%
pLbl@%%

C4(32)=B4={8 (B4^S)+24 (C4^S)}=5-Cube=Pentaract=(Subscript[S, 8]^(+))=Sp(4, C)=Schläfli{4,(3^2)}

c4=so9=b4=sp8=bS4⋃cS4
pLbl@%
Length@%
pE8/@%%%+1/2

dynkinB4={{-1,-1,0,0},{-1,0,-1,0},{0,-1,-1,0},{-1,0,0,-1},{-1,0,0,0},{0,-1,0,-1},{0,0,-1,-1},{0,-1,0,0},{-1,0,0,1},{0,0,-1,0},{0,-1,0,1},{-1,0,1,0},{0,0,0,-1},{0,0,-1,1},{0,-1,1,0},{-1,1,0,0},{1,-1,0,0},{0,1,-1,0},{0,0,1,-1},{0,0,0,1},{1,0,-1,0},{0,1,0,-1},{0,0,1,0},{1,0,0,-1},{0,1,0,0},{0,0,1,1},{0,1,0,1},{1,0,0,0},{1,0,0,1},{0,1,1,0},{1,0,1,0},{1,1,0,0}};
Length@%

Complement[dynkinB4,dynkinD4]

Complement[c4,cube5]
Length@%
pLbl@%%

C3(18)=B3={6 (B3^S)+12 (C3^S)}=(Subscript[S, 6]^(+))=Sp(3 C)

c3=so7=b3=sp6=bS3⋃cS3
Length@%
pLbl@%%

C2(8)={(Subscript[ν, e]),e}=B2={4 (B2^S)+4 (C2^S)}=half of mc5=3-cube=Cube=SO(3)=(Subscript[S, 4]^(+))=Sp(2, C)=Schläfli{4,3}

c2=so5=b2=sp4=bS2⋃cS2
Length@%
pLbl@%%

Complement[c2,mc4]
Length@%
pLbl@%%

Complement[c2,cube3]
Length@%
pLbl@%%

C1(2){2 (B1^S)+0 (C1^S)}=B1=E1=H1=mc2=E2?(2)=mc2=(Subscript[ν, e(Overscript[R, ⋀])]) =line=SO(2)?=SU(1)?=U(1)?

A2-RL(6)=gluons=(G2^S) ={g}

(* A2 RL *)
gluons=a2rl=gS2=Select[e8bFindRL[dn,3],Total@pE8@#==0&]
pLbl@%
Length@%
pE8/@gluons
%[[All,7;;]]//MatrixForm

G2(12)=A2(6)={{D2 ω}+2W}+(6)(G2^S)gluons {g}

g2=a2⋃gluons
Length@%
pLbl@%%

Dual C3(18)= B3={6 (B3^S)+12 (C3^S)}={xΦ}=SO(7)=Schläfli  (Subscript[t, 1]){3,4}?

dc3=db3=Sort@Select[Range@256,pRow@#==4&&MemberQ[pls[[position[plNames,If[ZLM,"!pType","pType"]]]],#]&];
Length@%
pLbl@%%

(F4^S)(24)={{18 Dual B/C3{6 (B3^S)+12(C3^S)} xΦ},6(G2^S)(g)luons}=row 4 pType 1(0 if ZLM)+A2=cell24dual=Schläfli {3,4,3}

row 3 & 4 bosons of pType=1(0 if ZLM) also found by taking d4 right-to-left (vs. left-to-right)

fS4=d4rl=e8cell24dual=e8bFindRL[dn,4]
Length@%
pLbl@%%

fS4==Sort[gluons⋃dc3]

but (F4^S) is not intersecting w/E6

fS4⋂e6

bosons(48) (row 3 & 4)
F4=24 D4{dual-mc5?(16)eϕ}+2B}, 6A2 {{4 D2 ω}+2W}+(F4^S){18  Dual B/C3 xΦ, 6 (G2^S) (g) luons}

f4=Sort@so8⋃fS4
Length@%
pLbl@%%

(pE8/@so8)[[All,;;4]]==(pE8/@fS4)[[All,5;;]]

dynkinF4={{-1,0,0,1},{0,-1,0,1},{0,0,-1,1},{0,0,0,1},{0,0,1,1},{-(1/2),-(1/2),-(1/2),(1/2)},{-(1/2),-(1/2),(1/2),(1/2)},{0,1,0,1},{1,0,0,1},{-(1/2),(1/2),-(1/2),(1/2)},{-1,-1,0,0},{(1/2),-(1/2),-(1/2),(1/2)},{-(1/2),(1/2),(1/2),(1/2)},{-1,0,-1,0},{(1/2),-(1/2),(1/2),(1/2)},{0,-1,-1,0},{-1,0,0,0},{(1/2),(1/2),-(1/2),(1/2)},{0,-1,0,0},{-1,0,1,0},{(1/2),(1/2),(1/2),(1/2)},{0,0,-1,0},{0,-1,1,0},{-1,1,0,0},{1,-1,0,0},{0,1,-1,0},{0,0,1,0},{-(1/2),-(1/2),-(1/2),-(1/2)},{1,0,-1,0},{0,1,0,0},{-(1/2),-(1/2),(1/2),-(1/2)},{1,0,0,0},{0,1,1,0},{-(1/2),(1/2),-(1/2),-(1/2)},{1,0,1,0},{(1/2),-(1/2),-(1/2),-(1/2)},{-(1/2),(1/2),(1/2),-(1/2)},{1,1,0,0},{(1/2),-(1/2),(1/2),-(1/2)},{-1,0,0,-1},{0,-1,0,-1},{(1/2),(1/2),-(1/2),-(1/2)},{(1/2),(1/2),(1/2),-(1/2)},{0,0,-1,-1},{0,0,0,-1},{0,0,1,-1},{0,1,0,-1},{1,0,0,-1}};
Sort@{215,208,202,5,51,28,23,46,40,18,93,12,129,92,109,86,87,99,81,214,94,76,207,213,170,176,4,1,169,3,24,2,45,19,39,13,17,38,11,91,85,10,95,80,72,181,175,168}

dynkinF4+1/2

newf4=Sort@Select[position[e8Orig,#]&/@((Join[#,Array[0&,4]]&/@dynkinF4)⋃(Join[#,Array[-1/2&,4]]&/@dynkinF4)),NumericQ]
pLbl@%
Length@%
pE8/@%%%
%+1/2

Complement[{1,2,3,4,5,10,11,12,13,17,18,19,23,24,28,38,39,40,45,46,51,72,76,80,81,85,86,87,91,92,93,94,95,99,109,129,168,169,170,175,176,181,202,207,208,213,214,215},{1,2,3,4,5,10,11,12,17,18,23,38,39,40,45,46,51,80,85,86,91,92,93,94,168,169,170,175,176,181,202,207,208,213,214,215,252,253,254,255}]
pLbl@%
Length@%
pE8/@%%%
%+1/2

Complement[{1,2,3,4,5,10,11,12,17,18,23,38,39,40,45,46,51,80,85,86,91,92,93,94,168,169,170,175,176,181,202,207,208,213,214,215,252,253,254,255},{1,2,3,4,5,10,11,12,13,17,18,19,23,24,28,38,39,40,45,46,51,72,76,80,81,85,86,87,91,92,93,94,95,99,109,129,168,169,170,175,176,181,202,207,208,213,214,215}]
pLbl@%
Length@%
pE8/@%%%

notInNewF4={95,99,109,129}
pLbl@%
Length@%
pE8/@%%%
%+1/2
newf4=Join[newf4,notInNewF4]

(*)
f4=newf4;

cube4
pLbl@%
Length@%
pE8/@%%%+1/2

Complement[cube4,newf4]
pLbl@%
pE8/@%%+1/2

e8cell24dual=d4

f4=e8cell24⋃e8cell24dual
Length@%

f4==f4⋃newf4
pLbl@Complement[f4,newf4]

Another way to link e8Orig to DynkinF4

inDynkinF4NotInF4=Flatten[{(Subscript[(Underoverscript[(Subscript[ν, μ]), y, (_)]), spList[[#]]]),(Subscript[(Underoverscript[e, y, (_)]), spList[[#]]])}&/@Range@4];
pLbl@%

f4New=Sort@Join[Select[position[e8Orig,#]&/@(
(Join[#,Array[0&,4]]&/@dynkinF4)⋃
(Join[#,Array[-1/2&,4]]&/@dynkinF4)),NumericQ],
inDynkinF4NotInF4]
pLbl@%

f4==f4New

(*??)
f4=b3+a4;

MyToE:

Fundamental constants

New dimensional relationships

α=FineStructureConstant;

LTC=LengthUnit(1/(TimeUnit^2))/.slRep;

MTC=MassUnit(TimeUnit/(LengthUnit^3))(LTC^3)/.slRep;

CTC=ChargeUnit(TimeUnit/MassUnit)MTC/.slRep;

doStdErr[newSym_,stdName_,unitSI_,unitCNV_]:=(
stdErr=ToExpression[stdName<>"Error"];
newErr=Abs[((Convert[ToExpression@newSym,unitSI/.slRep])/(ToExpression@stdName))-1];
Print[newSym<>"=",Convert[ToExpression@newSym,unitCNV/.slRep],"(",Convert[ToExpression@stdName,unitCNV/.slRep]stdErr/unitCNV/.slRep,")"];
Print["Std. Error = ",stdErr];
Print["New/Old Std. Error = ",newErr];
Print["Std. Error Factor (<1 is inside Experimental Error) = ", newErr/stdErr])

Light

c:=(α^(-8))LengthUnit/TimeUnit/.slRep
Convert[c,(Meter/Second)/.slRep]

doStdErr["c","SpeedOfLight",(Meter/Second),(Meter/Second)]

(Subscript[μ, 0]):=1/(Sqrt[c])

(Subscript[ϵ, 0]):=(Subscript[μ, 0])

(Subscript[Ω, 0]):=(Sqrt[(Subscript[μ, 0])/(Subscript[ϵ, 0])])

(Subscript[Ω, 0])==1

Planck

ℏ:=c MassUnit LengthUnit

h:=2π ℏ

doStdErr["h","PlanckConstant",((Kilogram (Meter^2))/Second),((Kilogram (Meter^2))/Second)]

Hubble

(Subscript[H, 0]):=(LTC/(4π c))

doStdErr["(Subscript[H, 0])","HubbleConstant",(1/Second),((Kilo Meter)/Second)(1/(Mega Parsec))]

Newton

gc2:=((Sqrt[1-2(((α π/2))^2)])/(Sqrt[1-xw]))/.xw->((α π/2)^(1/3))

(Subscript[G, N]):=(α^8)((((LengthUnit/TimeUnit)))^2)(LengthUnit/MassUnit)

doStdErr["(Subscript[G, N])","GravitationalConstant",((Meter^3)/(Second^2))(1/Kilogram),((Meter^3)/(Second^2))(1/Kilogram)]

Together

(ℏ/(MassUnit LengthUnit))(1/LTC)==c(1/LTC)==(1/(Subscript[G, N]))((LTC^3)/MTC)==(1/(4π (Subscript[H, 0])))==(α^(-8))TimeUnit/.slRep

Mass and charge

Higgs mass

mH:=(Sqrt[(Sqrt[2])ℏ LengthUnit MTC/(LTC^3)])(MassUnit/(Sqrt[(MassUnit^2)]))/.slRep

doStdErr["mH","HiggsMass",Kilogram,Giga eVperC2]

Electron charge

el:=mH(Sqrt[((4π α)/(2LengthUnit))LTC(((CTC/MTC))^2)])

doStdErr["el","ElementaryCharge",Coulomb,Coulomb]

(el^2)==2α h(((CTC/LTC))^2)/MTC/.slRep

Weak charge

Clear@α

(Subscript[θ, W]):=ArcSin[(Sqrt[(Subscript[x, W])])]

(Subscript[g, w]):=(Sqrt[(((el/(Sqrt[(Subscript[x, W])])/CTC))^2)])

(Subscript[g, w]^(')):=(Sqrt[(((el/(Sqrt[1-(Subscript[x, W])])/CTC))^2)])

second@Solve[(Subscript[g, w])==(Subscript[x, W])(Sqrt[8(ℏ/(Subscript[Ω, 0]))(1/((LTC^2)MTC))])/.slRep,(Subscript[x, W])]

(Subscript[x, W]):=((α π/2)^(1/3));

α=FineStructureConstant;

(Subscript[α, s])=(Subscript[x, W])/2

Fermi constant and weak boson masses

(Subscript[G, F]):=(((2mH))^(-2))

doStdErr["(Subscript[G, F])","FermiConstant",(1/(Kilogram^2)),(1/(((Giga eVperC2))^2))]

(Subscript[g, W])=(Subscript[x, W])(Sqrt[(Sqrt[2])]);

mW=((Subscript[g, W])/(Sqrt[(Subscript[G, F])(MassUnit^2)]))MassUnit/.slRep;

doStdErr["mW","WeakBosonMass",Kilogram,Giga eVperC2]

mZ=(mW/(Cos[(Subscript[θ, W])]));

doStdErr["mZ","ZBosonMass",Kilogram,Giga eVperC2]

Leptons

Electrons

me@oGen_:= First[{8π MassUnit/.slRep}]/;oGen==0

me@oGen_:=If[oGen==2,6,1]((me[oGen-1])/(α (2^oGen)))/;oGen>0

me@1

L@oGen_:=TimeUnit (MassUnit^(11-3oGen))(((2π))^20)(((π/2))^(2(oGen^2)))(π^(-(1+12oGen)))(3^(-(1-oGen)(4-oGen)/2))(α^(-8(3-oGen)))/.slRep

τ[pMass_,oGen_]:=(Sqrt[(((((L@oGen)/(pMass^(11-3oGen)))))^2)])(TimeUnit/(Sqrt[(TimeUnit^2)]))/.slRep

doStdErr["me[1]","ElectronMass",Kilogram,Kilo eVperC2]

doStdErr["me[2]","MuonMass",Kilogram,Mega eVperC2]

Neutrinos

mν@oGen_:=(MassUnit^2)/me[4-oGen] /.slRep

mν@oGen_:=(MassUnit^2)/me@oGen /.slRep

Quarks

qu@oGen_:=2π (2^(2oGen-3))((((2/(Sqrt[(Sqrt[2])]))(1/(3π))(1/(4π α))))^((1-oGen)(2-oGen)/2))

qd@oGen_:=qu@oGen (2^(3-2oGen))(((((Sqrt[(Sqrt[2])])/2)(1/4)))^((oGen-(((2-oGen))^2))/2))

mq[f_,oGen_]:=f@oGen(((L@oGen)/(τ[me@oGen,oGen]))^(1/(11-3oGen)))(MassUnit/((MassUnit^(11-3oGen))^(1/(11-3oGen))))/.slRep

mu@oGen_:=mq[qu,oGen]
md@oGen_:=mq[qd,oGen]

Together

M[oP_,oC_,oGen_,oS_]:=
(* Row 1 Leptons *)
If[oC==1&&oGen>0,
If[oP==If[ZLM,2,1],mν@oGen,me@oGen],
(* Row 2 Quarks *)
If[oC>1&&oGen>0,
If[oP==If[ZLM,2,1],mu@oGen,md@oGen],
(* Row 5 Excluded *)
If[oC==1&&oGen==0,zeroMass,
(* Row 3 & 4 Blue (Z & W) *)
If[oC==4&&oGen==0&&oP==If[ZLM,2,1],
If[oS==4,mW,mZ],
(* Row 3 *)
If[oGen==0&&oS==4,zeroMass,
(* Row 4 *)
mH]]]]]

mν/@Range@3

me/@Range@3

mu/@Range@3

md/@Range@3

zeroMass

mW

mZ

mH

Cosmological Constant

ρc:=(3/(8π))(Subscript[H, 0]^2)(gc2/(Subscript[G, N]))/.slRep

(Subscript[R, H]):=(c/(Subscript[H, 0]))

Convert[(Subscript[R, H]),Giga LightYear/.slRep]

ρΛ:=ΩΛ ρc/.slRep

Λ:=2(4π((Subscript[G, N])/gc2))ρΛ/.slRep

Convert[Λ,1/(Second^2)/.slRep]

ΩΛ==(Λ/(3(Subscript[H, 0]^2)))==((ρΛ)/(ρc))

cγ@t_:=(∫_0^t)\[DifferentialD]t/.slRep

(Subscript[ΩΛ, 0])=(∫_0^1)(Sqrt[cγ@t])\[DifferentialD]t/.slRep

H0@t_:=(1/(4π cγ@t))/.slRep

Λ=1/∫((4π)/(H0@t))\[DifferentialD]t

Λ==(Subscript[ΩΛ, 0])3(Subscript[H, 0])@(t^2)

(Subscript[Ω, mVis])=1/24;

dm=5;

(Subscript[Ω, γ])=(((((((3/2)))^(2/3))(((Subscript[H, 0])TimeUnit)/2)))^(1/dm))/.slRep;

1/%

(Subscript[Ω, (Subscript[mDrk, 0])])=1-(Subscript[ΩΛ, 0])

(Subscript[Ω, mDrk])=(Subscript[Ω, (Subscript[mDrk, 0])])-3(Subscript[Ω, mVis])

(Subscript[Ω, m])=(Subscript[Ω, mDrk])+(Subscript[Ω, mVis])

ΩΛ=(Subscript[ΩΛ, 0])+2(Subscript[Ω, mVis])

(Subscript[Ω, k])=-(Subscript[Ω, γ]);

Ω==((Subscript[Ω, m])+ΩΛ)+((Subscript[Ω, γ])+(Subscript[Ω, k]))

The List:

Sort by Pascal triangle/Clifford algebra E8

pascalRow@1

pascalRow@2

pascalRow@3

pascalRow@4

pascalRow@5

pascalRow@6

pascalRow@7

pascalRow@8

pascalRow@9