# Comparing nearest edges between Split Real Even (SRE) E8

I have been intrigued with Richter’s arXiv 0704.3091 “Triacontagonal Coordinates for the E8 root system” paper.

Comparing nearest edges between Split Real Even (SRE) E8 and Richter’s Complex 4D Golden Ratio Petrie Projection Model:

Please note, the vertex numbers are reordered from Richter’s paper in order to be consistent with ArXiv quant-ph 1502.04350 “Parity Proofs of Kochen-Specker theorem based on the Lie Algebra E8” by Aravind and Waegell. This paper uses the beautiful symmetries of E8 as a basis for proof sets related to the Bell inequalities of quantum mechanics. Below is a graphic showing that vertex number ordering:

Richter’s model with 74 nearest edges per vertex (Complex 4D Norm’d length Sqrt[2], with 8880 total):

Richter’s model with only 26 nearest edges per vertex (Complex 4D Norm’d length 1, 3120 total):

The above unity length edge pattern has is more consistent with the SRE E8 below.

The SRE E8 Petrie Projection (with 56 Norm’d edge length Sqrt[2] for each vertex, 6720 total). This is equivalent to folding E8 to H4 with an 8×8 rotation matrix which creates a 4D-left H4+H4*Phi and a 4D-right H4+H4*Phi.

Please note, the vertex number is that of the E8 vertices rotated (or “folded”) to H4 in 2D Petrie projection. The vertex numbers are in canonical binary order from E8’s 1:1 correspondence with the 9th row of the Pascal Triangle (eliminating the 16 generator/anti-generator vertices of E8 found in the 2nd and 8th column of the Pascal Triangle). The graphic below shows the vertex number ordering:

Here we take the 4D-left half and project to two concentric rings of H4 and H4 Phi (Golden Ratio) with 56 nearest edges per vertex (Norm’d length of unity or Phi for each vertex):

The animation frames are sorted by the ArcTan[y/x] of the vertex position in each ring (sorted H4*Phi outer to inner, then H4 outer to inner).

BTW – if you find this information useful, or provide any portion of it to others, PLEASE make sure you cite this post. If you feel a blog post citation would not be an acceptable form for academic research papers, I would be glad to clean it up and put it into LaTex format in order to provide it to arXiv (with your academic sponsorship) or Vixra. Just send me a note at:  jgmoxness@theoryofeverthing.org.

# E8 now available as a 3D printed shadow of H4

The 3D vertex shape and size represent theoretically assigned extended SM particle types based on a modified A.G. Lisi model.

# Updated My ToE Demonstrations to Wolfram Language (aka. Mathematica) 11

Please see the latest in .nb, .cdf demonstrations files and web interactive pages.

 ToE_Demonstration-Lite.cdf Latest: 08/15/2016 (10 Mb). This is a lite version of the full Mathematica version 11 demonstration in .CDF below (or as an interactive-Lite web page) (4 Mb). It only loads the first 8 panes and the last UI pane which doesn’t require the larger file and load times. It requires the free Mathematica CDF plugin. This version of the ToE_Demonstration-Lite.nb (13 Mb) is the same as CDF except it includes file I/O capability not available in the free CDF player. This requires a full Mathematica license. ToE_Demonstration.cdf Latest: 08/15/2016 (110 Mb). This is a Mathematica version 11 demonstration in .CDF (or as an interactive web page) (130 Mb) takes you on an integrated visual journey from the abstract elements of hyper-dimensional geometry, algebra, particle and nuclear physics, Computational Fluid Dynamics (CFD) in Chaos Theory and Fractals, quantum relativistic cosmological N-Body simulations, and on to the atomic elements of chemistry (visualized as a 4D periodic table arranged by quantum numbers). It requires the free Mathematica CDF plugin. This version of the ToE_Demonstration.nb (140 Mb) is the same as CDF except it includes file I/O capability not available in the free CDF player. This requires a full Mathematica license.

(The CDF player from Wolfram.com is still at v. 10.4.1, so still exhibits the bug I discovered related to clipping planes/slicing of 3D models).

Hofstadter’s Quantum-Mechanical Butterfly relates to the fractional Quantum Hall Effect, which is (IMHO) at the heart of understanding (interpreting) how QM really works!

I modified Wolfram Demonstration code by Enrique Zeleny to produce a short video of the emergence of the Hofstadter’s Quantum-Mechanical Butterfly

I found an interesting pattern. By modifying the integer used in the solution i=12, the Golden Ratio Ф=(1+Sqrt[5])/2=1.618 … emerges within the butterfly! Can you find it (or 1/Ф=.618…)?
Hint: Use the interactive version and mouse-over the red or green dots in the white space of the wings of butterfly.

A snapshot of the code and last frame@n=50:

If you have the free Mathematica CDF plugin on a non-Chrome browser, you can interactively analyze this with n<11 being presented with symbolic solutions in a mouse-over Tooltip (below).

# Interactive Reimann Zeta Function Zeros Demonstration

This web enabled demonstration shows a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points) along the critical line Zeta(1/2+it) for real values of t running from 0 to 50. The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to show the relative distance between their values of t. Gram’s law states that the curve usually crosses the real axis once between zeros.

Note: The interactive CDF plug-in as required below does not currently work on Chrome browsers.

A Snapshot picture for those w/o Wolfram CDF interactivity:

Selectable example code snippet:
[wlcode]Show[ListPlot[Style[pts2, Red], PlotRange -> {{-2, 4}, {-3, 3}},
AspectRatio -> 1, ImageSize -> imageSize,
AxesStyle ->
Directive[Thick, If[artPrint && ! localize, Large, Medium]],
Graphics[{PointSize@.01,
tttxt := If[artPrint && ! localize, tttxt1, ttxt0];
If[ttxt0 = ToString[# – 1];
Abs@zeroY[[#, 1]] < 10 chop, (* Magenta Critical Line Zeta Zeros *) tttxt1 = Column[{ToString[# - 1], "t=" <> ToString@zeroY[[#, 4]]},
Center];
ttLoc =
zeroY[[#, 4]] If[artPrint && ! localize, 1, 2] imagesize/40000;
(* Flip Point Labels above/below the X axis *)
ttLoc1 = {-1, (-1)^Round[#/2]} ttLoc/Sqrt[2];
{Magenta, Point@ttLoc1,
Circle[zeroY[[#, ;; 2]], ttLoc, {1, 3} \[Pi]/2],
Black,
Tooltip[Text[
Style[tttxt, If[artPrint && ! localize, Large, Medium, Bold]],
(* Shift the Labels off the Point *)
ttLoc1 (1 – (-1)^Round[#/2] .05)], tttxt1]},
(* Orange Critical Line Imaginary zeros w/Real>0 *)
tttxt1 =
Column[{ToString[# – 1], “x=” <> ToString@zeroY[[#, 1]],
“t=” <> ToString@zeroY[[#, 4]]}, Center];
{Tooltip[{Orange, Point@zeroY[[#, ;; 2]], Text[Style[

Column[If[EvenQ[Round[(# – 1)/2]], Prepend,
Append][{“\[UpDownArrow]”}, tttxt], Center,
Frame -> True],
Black, If[artPrint && ! localize, Large, Medium],
Background -> White],
(* Flip Point Labels above/below the X axis *)

zeroY[[#, ;; 2]] + {0, (-1)^Round[(# – 1)/2]} If[
artPrint && ! localize, 1,
If[artPrint, 4/1, 3]] imagesize/5000]},
Column[{ToString[# – 1], “x=” <> ToString@zeroY[[#, 1]],
“t=” <> ToString@zeroY[[#, 4]]}, Center]]}] & /@
Range@Length@zeroY,
Magenta, Disk[{0, 0}, .03]}]][/wlcode]

More plots with various scaling functions and multi-color coding along with Tooltip on mouse-over. Bear in mind the last Smith Chart with a division by Abs@Zeta indicates where the increments go exponential near the 0.

A Snapshot picture for those w/o Wolfram CDF interactivity:

# E8 in E6 Petrie Projection

An article (interview) with John Baez used an E8 projection which I introduced to Wikipedia in Feb of 2010 here. Technically, it is E8 projected to the E6 Coxeter plane.

The projection uses X Y basis vectors of:
X = {-Sqrt[3] + 1, 0, 1, 1, 0, 0, 0, 0};
Y = {0, Sqrt[3] – 1, -1, 1, 0, 0, 0, 0};

Resulting in vertex overlaps of:
24 Yellow with 1 overlap
24 Dark Blue each with 8 overlaps (192 vertices)
1 Light Blue with 24 overlaps (24 vertices)

After doing this for a few example symmetries, Tom took my idea of projecting higher dimensional objects to the 2D (and 3D) symmetries of lower dimensional subgroups – and ran with it in 2D – producing a ton of visualizations across WP. 🙂

It was one of those that was subsequently used that article from the 4_21 E8 WP page.

Here is a representation of E6 in the E6 Coxeter plane:

Resulting in vertex overlaps of:
24 Yellow with 1 overlap
24 Orange each with 2 overlaps (48 vertices)

# G2 as the automorphism group of the (split) octonion algebra

This post describes the derivation of G2 automorphisms for each of the 480 unique octonion multiplication tables, as well as each of the 7 split octonions (created from negating the 4 row/column entries which are not members of the split #, which is an index to one of the 7 triads that make up the octonion).

The Exceptional Lie Algebra/Group G2 is identified by its Dynkin diagram and/or associated Cartan Matrix (also shown here with its Hasse diagram):

The particular octonion multiplication table selected in the example referenced above is associated in my model as a “Non Flipped E8 #164”. The multiplication table in various formats (IJKL, e_n, and Numeric) is:

The selected multiplication table (fm) is the basis for the following octonion symbolic math:
[wlcode](* Octonion math from the matrix using 8D vector representations for x and y, this has the Subscript[\[ScriptE], 0]=1 first *)
octProduct[a_, b_] := Block[{c3 = Array[0 &, 8]},
Do[c3[[If[i != j, Abs@fm[[i, j]] + 1, 1]]]+=Sign@fm[[i, j]] a[[i]] b[[j]],
{i, 8}, {j, 8}];
Chop@c3];

(* Complex, quaternion and octonian multiplication Subscript[\[ScriptE], a]\[SmallCircle]Subscript[\[ScriptE], b] *)
quat2oct@in_ := Block[{a3},
If[QuaternionQ@in,
a3 = FromQuaternion@in;
Switch[Length@a3,
0, Re@a3 + Im@a3 Subscript[\[ScriptE], 1],
2, If[Length@a[[1]] > 0, a3,
1\[SmallCircle](Re@a3[[1]] +
Im@a3[[1]] Subscript[\[ScriptE], 1]) + a3[[2]]]],
3, Re@a3[[1]] + Im@a3[[1]] Subscript[\[ScriptE], 1] +
a3[[2 ;;]]],
“not Quaternion”] /. Most@Thread[octIJKL -> oct]];

(* Octonion math with conversions, leave final as octonion *)
SmallCircle[a_List, b_List] := octonion@octProduct[a, b];
SmallCircle[a_, b_Quaternion] := a\[SmallCircle]quat2oct@b;
SmallCircle[a_Quaternion, b_] := quat2oct@a\[SmallCircle]b;
SmallCircle[a_, b_Complex] := a\[SmallCircle]ToQuaternion@b;
SmallCircle[a_Complex, b_] := ToQuaternion@a\[SmallCircle]b;
SmallCircle[a_, b_] := oct2List@a\[SmallCircle]oct2List@b;

associator[a_List, b_List,c_List] := (a\[SmallCircle]b)\[SmallCircle]c-a\[SmallCircle](b\[SmallCircle]c);
associator[a_, b_, c_] :=associator[oct2List@a, oct2List@b, oct2List@c];[/wlcode]

Each set of 21 upper triangle pairs of octonion elements (utOct) has 14 “derivations” which are null
(alternatively, the NullSpace of the derivation vectors of utOct has 7 which are NOT null…)

A derivation pair D_{x, y} is defined by:
[wlcode](* Commutator *)
commutator[x_,y_]:=x\[SmallCircle]y-y\[SmallCircle]x;
CircleDot[x_,y_]:=commutator[x, y];

(* Commutator matrix *)
comMat[x_, y_] := x.y – y.x;
comMat[{x_, y_}] := comMat[x, y];

(* Derivation matrix and vector *)
derivation[x_,y_][a_]:=(x\[CircleDot]y)\[CircleDot]a-3 associator[x,y,a];
Square[{x_, y_, a_}] := derivation[x, y][a];
derMatrix[x_,y_]:=Table[Coefficient[\[Square]{x,y,oct[[i]]},oct[[j]]], {j, 7}, {i,7}];
derVector[x_,y_]:=derMatrix[x, y]//Flatten;[/wlcode]
The 14 null pairs define G2 for a SPECIFIC octonion multiplication matrix.

For each of 7 non-null derivations there is a triple of D_{x1, y1}=D_{x2, y2}+D_{x3, y3} related derivations.

There is a 7×21 matrix describing the non-NullSpace for the octonion given in the Wolfram MathSource reference above along with the 7 non-null triples. This is generated by g2Null:
[wlcode]g2Null:=NullSpace[derVector[Subscript[\[ScriptE], #1], Subscript[\[ScriptE],#2]]&@@@utOct//Transpose];

(* Formatting all non-Null upper triangle pairs derivation entries *)
outG2Null := MatrixForm[parallelMap[Grid[{
{Row[
Subscript[D, Row@Flatten@##] & /@
Partition[Flatten@{utOct[[Flatten@Position[#, -1]]]}, 2]]},{“==”},
{Row[
Subscript[D, Row@Flatten@##] & /@
Partition[Flatten@{utOct[[Flatten@Position[#, 1]]]}, 2]]}(*),
{“\[LongDash]\[LongDash]\[LongDash]\[LongDash]”}**)}] &,
g2Null]];[/wlcode]

getG2Null (below) retrieves two octonion multiplication matrices for the flipped and non-flipped index on one of 240 E8 particles.
[wlcode]formG2Null := MatrixForm@{
Row@{Style[Row@{“flip=”, flip}, Blue],
Style[” split # \[DownArrow],”, Red]},
g2NullTbl = Table[{
split,
setFM[#, flip, split];
outG2Null},
{split, 0, 7}];
MatrixForm[Column[#, Center] & /@ noNull@# & /@ Table[
(* Color the common Der in each column *)
cmnD =first@Select[g2NullTbl[[All,2,1,i,1,3,1,1,All]],
Length@# == 1 &];
cmnDclr = Style[cmnD, Magenta];
If[j == 1, {“”, Row@{
(* Highlight the missing index in the triple D_{x,y} for each row *)
Style[Complement[Range@7,Union@Flatten@g2NullTbl[[All,2,1,i,1,3,1,1,All,2,1]]][[1]],Darker@Green],
g2NullTbl[[1, 2, 1, i]] /. cmnD -> cmnDclr}},
(* Display only non-null derivations from the splits if not the same as the base octonion *)
If[g2NullTbl[[1,2,1,i]] =!= g2NullTbl[[j,2,1,i]],
{Style[g2NullTbl[[j, 1]], Red],
g2NullTbl[[j,2,1,i]] /. cmnD -> cmnDclr}]],
{i,7},{j,8}]]} &;

getG2Null := ColumnForm[{
Style[Row@{“E8#=”, #}, Darker@Green],
Row@{flip = False; formG2Null@#, flip = True; formG2Null@#}},
Center] &;[/wlcode]

Here we show the output of getG2Null for the example non-Flipped E8 #164):

Emergent patterns:

1. For each of the 7 non-null derivation triples (the first column of the 7 rows), there are precisely 3 (of the 7) split octonions that don’t share the exact same non-null triple derivation pair as the parent (non-split) octonion (these deviations are shown in column 2 thru 4, along with the associated Fano plane diagrams for the parent and 7 split octonions of E8 #164).
2. Each non-null derivation triple contains 6 of 7 indices, and each row is missing a different index (highlighted in green before the column 1 triple). The sequence of missing element numbers in each row of the given example follows the row number.
3. It is only the equality relationship (signs) of the non-Null triple that change in 3 of the 7 splits, not the D_{x,y} itself.
4. There are always 2 pairs of 2 D_{x,y} which are negative (i.e. with a -1 entry in the NullSpace matrix located above the “==”) that occur across the row.
5. There is always a common positive D_{x,y} (i.e. with a +1 entry in the NullSpace matrix located below the “==”) in each entry of the row (colored magenta). The common positive entries in the 7 rows suggest a “distinguished” non-null indicator for the 14=21-7 G2 automorphism.
6. All octonion multiplication matrices have the first 3 rows of distinguished entries of {6,7},{5,7} and {5,6} in that order (i.e. the last 2 rows of utOct).
7. There are several possible choices for G2 automorphism sets of 14 elements within each of 480 octonions based on the 7 non-Null entries. Interestingly, there are the 4 sets of rows in utOct which sum to 14 elements, specifically rows {{1, 2, 4}, {1, 2, 5, 6}, {1, 2, 4, 6}, {2, 3, 4, 5}}). The distinguished entries of the flipped E8 #164 example below (as in the MathSource post referenced above) suggests a G2 created by rows {1,2,4} of utOct. Although, not all G2 sets must use complete rows as in this example.

This is octonionG2-nonNull, a 480 page (15 MB) which lists each of the 7 non-Null parent octonion derivations (and for each of those, the 3 other split octonion non-null derivations) which are NOT a member of the 14 null derivations for each of 480 octonion multiplication matrices and 7 splits for each (480*8=3840). It includes the Fano plane mnemonics for the parent and 7 split octionions. This is a smaller 240 page (1MB) version of octonionG2-nonNull.pdf without the Fano plane mnemonics.

# Octonion triality testing

I am working on validating some theoretical work on the triality automorphisms of the split octonions. This is a post with preliminary work on that… for those who are interested 🙂

BTW – It requires the here.

An addendum to the original post (below):

[WolframCDF source=”http://theoryofeverything.org/TOE/JGM/octonion triality checks.cdf” width=”900″ height=”8000″ altimage=”http://theoryofeverything.org/TOE/JGM/octonion triality checks.png” altimagewidth=”900″ altimageheight=”8000″]

# Visualizing E6 as a subgroup of E8 for a Leptoquark SUSY GUT?

MyToE was mentioned in a Luboš Motl blog post comment, so I thought I would throw out a few ideas and offer to help Luboš visualize the models/thinking around developing an E6 Leptoquark SUSY GUT.

Here are the E6 Dynkin related constructs, including the detail Hasse visualization.

For this post, I use Split Real Even (SRE) E8 vertices created by dot product of the 120 positive (and 120 negative roots) with the following Simple Roots Matrix (SRM), which generates the Cartan as well:

This is a list of E6 vertices as a subset of SRE E8 by taking off the orthogonal right-side 2 dimensions (or Dynkin nodes in red) vertices with identical entries).

This is an interesting E6 projection with basis vectors of:
X={2, -1, 1, 1/2, -1, -(1/2), 0, 0}
Y= {0, 0, Sqrt[3], Sqrt[3]/2, Sqrt[3], Sqrt[3]/2, 0, 0}

Adding a third Z={0,0,Sqrt[3],Sqrt[3]/2,Sqrt[3],Sqrt[3]/2,0,0}, we get a 3D projection:

Luboš> “the maximum subgroup we may embed to E6 is actually SO(10)×U(1).”

Since D5 (which is the same as E5 in terms of the Dynkin diagram topology) relates to SO(10), it is interesting to look at the complement of E6 vertices and the 40 D5 vertices contained in E6 (leaving 32):

Here is the D5 Dynkin and its Cartan algebra: