Augmented dodecahedron (Johnson solid)

Augmented dodecahedron (Johnson solid)
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Aud
Coxeter diagram	oxfoo ooofx&#xt
Elements
Faces	5 triangles, 1+5+5 pentagons
Edges	5+5+5+5+5+10
Vertices	1+5+5+5+5
Vertex figures	1 pentagon, edge length 1
	5 kites, edge lengths 1 and (1+√5)/2
	5+5+5 triangles, edge lengths (1+√5)/2
Measures (edge length 1)
Volume	$\frac{95+43\sqrt5}{24} ≈ 7.96462$
Dihedral angles	3–5: $\arccos\left(-\sqrt{\frac{65-2\sqrt5}{75}}\right) ≈ 153.94242°$
	3–3: $\arccos\left(-\frac{\sqrt5}{3}\right) ≈ 138.18969°$
	5–5: $\arccos\left(-\frac{\sqrt5}{5}\right) ≈ 116.56505°$
Central density	1
Related polytopes
Army	Aud
Regiment	Aud
Dual	Monotruncated icosahedron
Conjugate	Monoexcavated 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

Not to be confused with Augmented dodecahedron (isohedral deltahedron).

The augmented dodecahedron is one of the 92 Johnson solids (J₅₈). It consists of 5 triangles and 1+5+5 pentagons. It can be constructed by attaching a pentagonal pyramid to one of the faces of the regular dodecahedron.

Vertex coordinates

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

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

As well as:

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

External links

• Klitzing, Richard. "aud".

• Quickfur. "The Augmented Dodecahedron".

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