Tag Archives: Fano

Introducing the Sedenion Fano Tesseract Mnemonic

The Sedenion “Fano Tesseract” Mnemonic is an extension of the “Fano Cube” idea introduced by John Baez in his much cited blog post. The Fano Cube identifies each valid each triad by a hyper-plane which intersects the e_0 node.

Notice in this VisibLie_E8 output for the pane #3 “Fano Visualization Demonstration”, there are 35 sedenion triads, 7 of which are from the octonion used as an upper left quadrant base for a Cayley-Dickson doubling (highlighted in red).

The 16 vertices of the tesseract are sorted by the same “triad flattening” process used to construct a consistent Fano Plane Mnemonic for all 480 unique octonion multiplication tables.

As in the Fano Cube, the edges are highlighted in Cyan if they are selected in the n1-n3 buttons. Unlike the Fano Plane and Cube, the edges represented by the split octonion multiplication table columns/rows are not highlighted in red.

While there are 32 edges in the formal tesseract, each valid sedenion triad is identified by a hyper-plane which intersects the e_0 node, which are not necessarily those of the formal tesseract.

Here is the computation of the same sedenion table given in the Sedenion Wikipedia article as well as from this website: http://www.derivativesinvesting.net/article/307057068/a-few-hypercomplex-numbers/, which uses the harder to read IJKL style notation.

Incorporating Sedenions into the VisibLie_E8 visualizer

Here is a snapshot of the UI.

It is taken from an example given on this website: http://www.derivativesinvesting.net/article/307057068/a-few-hypercomplex-numbers/, which uses the harder to read IJKL style notation.

Compare the upper left quadrant of the multiplication table. It is the same as the octonion of E8 #164 (non-Flipped) used to generate the sedenion using the Cayley-Dickson doubling. It can be read off using the Fano Plane (or Fano Cube) mnemonic. My next post will show the Sedenion forms of these, namely the Fano Tetrahedron and Tesseract!

Notice the highlighting of the interesting aspect of sedenions, where some non-zero p=a+b and q=c+d give a product pq=0. These are given in a list of the {a,b,c,d} where p={a+b} and q={c+d} and pq=0 (along with the deltas between the indices), always 42 excluding trivial juxtapositions. In this case, it is the row entry with {3,12,5,10}.

Here is the computation of the same table in the format given in the Sedenion Wikipedia article.

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






quat2oct@in_ := Block[{a3},
a3 = FromQuaternion@in;
a3[[2 ;;]]],




[wlcode](* Commutator *)

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

(* Derivation matrix and vector *)



outG2Null := MatrixForm[parallelMap[Grid[{

[wlcode]formG2Null := MatrixForm@{
g2NullTbl = Table[{
setFM[#, flip, split];
{split, 0, 7}];
(* Color the common Der in each column *)
Length@# == 1 &];
cmnDclr = Style[cmnD, Magenta];

getG2Null := ColumnForm[{
Center] &;[/wlcode]

Emergent patterns:



Octonion triality testing


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″]

QuasiCrystals and Geometry


12-gon Rectified E8

18-gon Rectified E8

30-gon Rectified E8




Mathematica MyToE on the Wolfram Cloud



Here are a few results…


ByzPbCqIQAAzUJq.png large


Octonions: The Fano Plane & Cubic


Dynkin Diagrams


E8 and Subgroup Projections


Particle Selector


Hadron Builder




Solar System (from NBody Universe Simulator)

4D Periodic Table


Another look at integrating the Pascal Triangle to Clifford Algebra, E8 Lie Algebra/Groups, Octonions and Particle Physics Standard Model


Modified Lisi split real even E8 particle assignment quantum bit patterns:


Assigning a specific mass, length, time, and charge metrics based on new dimensional relationships and the Planck constant (which defines Higgs mass).


The split real even E8 group used has been determined from this simple root matrix (which gives the Cartan matrix upon dot product with a transpose of itself):


This Dynkin diagram builds the Cartan matrix and determines the root/weight/height with corresponding Hasse diagrams.



The Comprehensive Split Octonions and their Fano Planes

I am pleased to announce the availability of splitFano.pdf, a 321 page pdf file with the 3840=480*8 split octonion permutations (with Fano planes and multiplication tables). These are organized into “flipped” and “non-flipped” pairs associated with the 240 assigned particles to E8 vertices (sorted by Fano plane index or fPi).

There are 7 sets of split octonions for each of the 480 “parent” octonions (each of which is defined by 30 sets of 7 triads and 16 7 bit “sign masks” which reverse the direction of the triad multiplication). The 7 split octonions are identified by selecting a triad. The complement of {1,2,3,4,5,6,7} and the triad list leaves 4 elements which are the rows/colums 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. The selected triad nodes are yellow, and the other 4 are cyan (25MB).


I believe this is the only comprehensive presentation of all 3840 Split Fano Planes with their multiplication tables available.

Below is the first page of the comprehensive split octonion list.


Connecting the Octonion Fano Plane to the Atomic Elements

There are 28 octonion Fano plane triangles that correspond directly to the 28 Trott quartic curve bitangents. These bitangents are directly related to the Legendre functions used in the Shroedinger spherical harmonic electron orbital probability densities. Shown below is a graphic of these overlaid onto the n=5, l=2, m=1 element, which is assigned to gold (Au). When using an algorithm based on the E8 positive algebra root assignments, the “flipped” Fano plane has E8 algebra root number 79 (the atomic number of Au) and split real even group number of 228 (in Clifford/Pascal triangle order). For more information, please see the complete list of integrated Fano indices here with full explanation of the layout here.