Augmented truncated dodecahedron

Augmented truncated dodecahedron
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Autid
Elements
Faces	5+5+5+5+5 triangles, 5 squares, 1 pentagon, 1+5+5 decagons
Edges	5+5+5+5+5+5+5+5+5+10+10+10+10+10+10
Vertices	5+5+5+5+5+10+10+10+10
Vertex figures	5 isosceles trapezoids, edge length 1, √2, (1+√5)/2, √2
	10 irregular tetragons, edge length 1, √2, 1, √(5+√5)/2
	50 isosceles triangles, edge lengths 1, √2+√2, √2+√2
Measures (edge length 1)
Volume	$\frac{505+243\sqrt5}{12} ≈ 87.37361$
Dihedral angles	3–4 join: $\arccos\left(-\sqrt{\frac{23+3\sqrt5}{30}}\right) ≈ 174.34011°$
	3–4 cupolaic: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$
	3–10 join: $\arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242°$
	4–5: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$
	3–10 tid: $\arccos\left(-\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 142.62263°$
	10–10: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
Central density	1
Related polytopes
Army	Autid
Regiment	Autid
Dual	Rhombirhombistellated triakis icosahedron
Conjugate	Augmented quasitruncated great stellated dodecahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The augmented truncated dodecahedron is one of the 92 Johnson solids (J₆₈). It consists of 5+5+5+5+5 triangles, 5 squares, 1 pentagon, and 1+5+5 decagons. It can be constructed by attaching a pentagonal cupola to one of the decagonal faces of the truncated dodecahedron.

Vertex coordinates

An augmented truncated dodecahedron of edge length 1 has vertices given by all even permutations of:

• $\left(0,\,±\frac12,\,±\frac{5+3\sqrt5}{4}\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt5}{4},\,±\frac{3+\sqrt5}{2}\right),$
• $\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{2},\,±\frac{2+\sqrt5}{2}\right),$

plus the following additional vertices:

• $\left(±\frac12,\,\frac{15+13\sqrt5}{20},\,3\frac{5+\sqrt5}{10}\right),$
• $\left(±\frac{1+\sqrt5}{4},\,\frac{25+13\sqrt5}{20},\,\frac{25+\sqrt5}{20}\right),$
• $\left(0,\,\frac{10+9\sqrt5}{10},\,\frac{15+\sqrt5}{20}\right).$

External links

• Klitzing, Richard. "autid".

• Quickfur. "The Augmented Truncated Dodecahedron".

• Wikipedia Contributors. "Augmented truncated dodecahedron".
• McCooey, David. "Augmented Truncated Dodecahedron"