Decagonal Ugly Duckling

Decagonal Ugly Duckling

01 March, 2023

10-Colouring

1090109050301090705

n-colouring 1090109050301090705

Loop Cycles

\((t_1-,t_5+)\;\)\(2\mathord*(t_2-,t_4-)\;\)\(2\mathord*(t_3)\;\)\(4\mathord*(t_4)\;\)

Gyre Multipliers

\(-\)

Growth

Single Tile Boundary Vertices

Topology Type

Toroidal

Skeleton Type

Unknown

Files

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Embedding

Notes

Symmetry and asymmetry in one. The n-colored tile shows only mirror symmetry, while all petrie polygons are of size 30. The dual graph forms two interconnected icosahedra.

Dual Group

\(G_d=\)\(\left\langle \: S_d \:\middle|\:R_d\:\right\rangle\)
\(S_d=\)\({t_1},\:\)\({t_2},\:\)\({t_3},\:\)\({t_4},\:\)\({t_5}\)
\(R_d=\)\({t_1}^3,\:\)\({t_2}^3,\:\)\({t_5}^3,\:\)\({t_1}{t_3}{t_2},\:\)\(({t_4}^{-1}{t_3})^2,\:\)\(({t_4}^{-1}{t_5})^2,\:\)\({t_3}^2,\:\)\({t_4}^4,\:\)\(({t_2}{t_4})^2,\:\)\({t_5}^{-1}{t_1}\)
\(|G_d|=\)\(24\)

Vertex Group

\(G_v=\)\(\left\langle \: S_v \:\middle|\:R_v\:\right\rangle\)
\(S_v=\)\(a,\:\)\(b\)
\(R_v=\)\(a^3,\:\)\(b^4,\:\)\((ba)^2,\:\)\((ba^{-1})^4\)
\(|G_v|=\)\(24\)