File:7-cubePascal.svg
Summary
| Description |
English: 7-Cube in Orthogonal column graph "Pascal Triangle" projection. |
| Source | Own work |
| Author | Jgmoxness |
Detail Description:
The 8 axes are shown with locators labeled A-H. The 128 vertices are a subset of the E8 group of even (half-integer) permutations. The number of vertices in each column represents rows in Pascal's triangle, being 1:7:21:35:35:21:7:1 in the case of a 7-cube. The 448 edges of unit length have colors that can be varied based on their projected edge positions. The vertex color is varied by overlap counts. Specifically:
vertices={color=overlap,count},...,Total
{{purple=1,32},{black=2,24},{red=3,30},{green=4,32},{blue=5,10}},128
vertices InView={{color=overlap,count},...,Total
{{purple=1,32},{black=2,12},{red=3,10},{green=4,8},{blue=5,2}},64
This is constructed from VisibLie_E8 found on TheoryOfEverything.org archive copy at the Wayback Machine with the following settings:
(* This is an auto generated list from e8Flyer.nb *)
new := {
scale=0.06;
cylR=0.021;
range=2.;
limitToRange=False;
favorite=29;
eGrad=0;
showEdges=True;
edgeVal={1, 448};
edgeDimTrim=7;
vOverlapColor=True;
pListName="C7=Even=Int/2";
new;
The 2 projection vectors are:
H={1/Sqrt[7], 1/Sqrt[7], 1/Sqrt[7], 1/Sqrt[7], 1/Sqrt[7], 1/Sqrt[7], 1/Sqrt[7], 0}
V={-(3/(2 Sqrt[7])), -(1/Sqrt[7]), -(1/(2 Sqrt[7])), 0, 1/(2 Sqrt[7]), 1/Sqrt[7], 3/(2 Sqrt[7]), 0}
Credit: User:Tomruen for sharing basis vectors resulting in my code for n-cube Pascal projections:
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]];)
Licensing
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License. |
