Decagonal Double Cover Dodecahedral Net

Decagonal Double Cover Dodecahedral Net

24 May, 2022

10-Colouring

1090304040703060607

n-colouring 1090304040703060607

Loop Cycles

\((t_1+,t_2+,t_2+)\;\)\((t_1+,t_3-,t_4-)\;\)

Gyre Multipliers

\(-\)

Growth

Single Tile Boundary Vertices

Topology Type

Toroidal

Skeleton Type

Unknown

Files

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Embedding

Notes

This 10-gon based tilegraph can be mapped onto the net of a dodecahedron as a double cover of the edges. The genus 5 graph has 480 automorphisms. This tilegraph can be found under the name R5.2 in the list of "Regular orientable maps of genus 2 to 101" generated by Marston Conder and Peter Dobesányi.

Dual Group

\(G_d=\)\(\left\langle \: S_d \:\middle|\:R_d\:\right\rangle\)
\(S_d=\)\({t_1},\:\)\({t_2},\:\)\({t_3},\:\)\({t_4},\:\)\({t_5}\)
\(R_d=\)\({t_1}^3,\:\)\({t_3}^{-1}{t_4}{t_2},\:\)\({t_4}^{-1}{t_5}{t_3},\:\)\({t_2}^{-1}{t_5}^{-1}{t_1}^{-1},\:\)\({t_2}{t_2}{t_1},\:\)\({t_3}^{-1}{t_4}^{-1}{t_1}\)
\(|G_d|=\)\(12\)

Vertex Group

\(G_v=\)\(\left\langle \: S_v \:\middle|\:R_v\:\right\rangle\)
\(S_v=\)\(a,\:\)\(b\)
\(R_v=\)\(a^2,\:\)\((abab^{-1})^2,\:\)\((b^2ab^{-2}ab)^2,\:\)\((b^{-2}ab^2ab^{-1})^2,\:\)\((ab)^2b^4(ab^{-1})^2b^{-4}\)
\(|G_v|=\)\(480\)