Decagonal Double Cover Dodecahedron II

21 February, 2023

10-Colouring

1090109050109010905

Loop Cycles

$(t_1-,t_4+)\;$$2\mathord*(t_3)\;$

Gyre Multipliers

$-$

Growth

Single Tile Boundary Vertices

Topology Type

Toroidal

Skeleton Type

Tetrahedron (tet)

Files

graphML Download

$20230221-225400_1090109050109010905_(t1-,t4+)_3(t3)double_cover_dodecahedron_480_auto-{10,3}_10-flat.png$

Embedding

Notes

This peculiar decagonal tilegraph covers a regular dodecahedron twice as does Decagonal Double Cover Dodecahedral Net, but based on a different tile and a different looping rule. And thus the dual pictured above is also different, it forms a dodecahedral structure, but with doubled edges. The tilegraph is vertex-, edge- and face-transitive.

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_4}^{-3},\:$${t_5}^{-3},\:$${t_1}{t_3}{t_2},\:$${t_4}^{-1}{t_5}^{-1}{t_3},\:$${t_3}^2,\:$${t_4}^{-1}{t_1}$
$\|G_d\|=$	$12$

Vertex Group

$G_v=$	$\left\langle \: S_v \:\middle\|\:R_v\:\right\rangle$
$S_v=$	$a,\:$$b,\:$$c$
$R_v=$	$a^2,\:$$b^3,\:$$(bc^{-1})^2,\:$$abab^{-1},\:$$(cb^{-1}ac)^2,\:$$(c^{-1}aca)^2,\:$$(b^{-1}c^{-4})^2,\:$$(c^{-1}abc^{-1}*a)^2$
$\|G_v\|=$	$480$

\(G_d=\)	\(\left\langle \: S_d \:\middle\|\:R_d\:\right\rangle\)
\(S_d=\)	\({t_1},\:\)\({t_2},\:\)\({t_3},\:\)\({t_4},\:\)\({t_5}\)
\(R_d=\)	\({t_1}^3,\:\)\({t_2}^3,\:\)\({t_4}^{-3},\:\)\({t_5}^{-3},\:\)\({t_1}{t_3}{t_2},\:\)\({t_4}^{-1}{t_5}^{-1}{t_3},\:\)\({t_3}^2,\:\)\({t_4}^{-1}{t_1}\)
\(\|G_d\|=\)	\(12\)

\(G_v=\)	\(\left\langle \: S_v \:\middle\|\:R_v\:\right\rangle\)
\(S_v=\)	\(a,\:\)\(b,\:\)\(c\)
\(R_v=\)	\(a^2,\:\)\(b^3,\:\)\((bc^{-1})^2,\:\)\(abab^{-1},\:\)\((cb^{-1}ac)^2,\:\)\((c^{-1}aca)^2,\:\)\((b^{-1}c^{-4})^2,\:\)\((c^{-1}abc^{-1}*a)^2\)
\(\|G_v\|=\)	\(480\)