Decagonal 120-Cell Wrap

Decagonal 120-Cell Wrap

04 March, 2023

10-Colouring

1090109050301090705

n-colouring 1090109050301090705

Loop Cycles

\((t_1-,t_5-)\;\)\(2\mathord*(t_2-,t_5+)\;\)\(2\mathord*(t_3)\;\)\(5\mathord*(t_4)\;\)

Gyre Multipliers

\(-\)

Growth

Single Tile Boundary Vertices

Topology Type

Toroidal

Skeleton Type

120-Cell

Files

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Embedding

Notes

Beautiful decagonal tilegraph with immense Petrie Polygons spanning 180 edges. One of the inner three derived skeletons is the 120-cell. In the images above the three skeleton graphs are followed by images of the three skeleton modules. The skeleton modules are the decagonal subgraphs found at the module nodes of the skeleton graphs.

After a representation of the glueing rules you will also find the color sequences of all the Petrie Polygons arranged around the base 10-colouring.

Dual Group

\(G_d=\)\(\left\langle \: S_d \:\middle|\:R_d\:\right\rangle\)
\(S_d=\)\({t_1},\:\)\({t_2},\:\)\({t_3},\:\)\({t_4},\:\)\({t_5}\)
\(R_d=\)\({t_1}^3,\:\)\({t_2}^3,\:\)\({t_5}^3,\:\)\({t_1}{t_3}{t_2},\:\)\(({t_4}^{-1}{t_3})^2,\:\)\(({t_4}^{-1}{t_5})^2,\:\)\({t_3}^2,\:\)\({t_4}^5,\:\)\(({t_5}{t_2}^{-1})^2,\:\)\({t_1}^{-1}{t_5}^{-1}\)
\(|G_d|=\)\(7200\)

Vertex Group

\(G_v=\)\(\left\langle \: S_v \:\middle|\:R_v\:\right\rangle\)
\(S_v=\)\(a,\:\)\(b,\:\)\(c\)
\(R_v=\)\(b^3,\:\)\(c^2,\:\)\((a^2ba)^3,\:\)\((ab^{-1})^2,\:\)\((a^{-1}c)^2,\:\)\((a^2bcb)^2,\:\)\(a^2cb(acb^{-1}a)^2\)
\(|G_v|=\)\(7200\)