Augmented pentagonal prism

Augmented pentagonal prism
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Aupip
Coxeter diagram	oxxx oxfo&#xt
Elements
Faces	2+2 triangles, 2+2 squares, 2 pentagons
Edges	1+2+2+2+4+4+4
Vertices	1+2+2+4
Vertex figures	1 square, edge length 1
	4 irregular tetragons, edge lengths 1, 1, √2, (1+√5)/2
	2+4 isosceles triangles, edge lengths (1+√5)/2, √2, √2
Measures (edge length 1)
Volume	$\frac{2\sqrt2+3\sqrt{25+10\sqrt5}}{12} ≈ 1.95618$
Dihedral angles	3–4 join: $\arccos\left(-\sqrt{\frac{13+\sqrt5+4\sqrt{5-\sqrt5}}{24}}\right) ≈ 162.73561°$
	3–5 join: $\arccos\left(-\frac{\sqrt6}{3}\right) 144.73561°$
	3–3 pyramidal: $\arccos\left(-\frac13\right) ≈ 109.47122°$
	4–4: 108°
	4–5: 90°
Central density	1
Related polytopes
Army	Aupip
Regiment	Aupip
Dual	Lateromonotruncated pentagonal tegum
Conjugate	Augmented pentagrammic prism
Abstract properties
Euler characteristic	2
Topological properties
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	K2×I, order 4
Convex	Yes
Nature	Tame

The augmented pentagonal prism is one of the 92 Johnson solids (J₅₂). It consists of 2+2 triangles, 2 +2squares, and 2 pentagons. It can be constructed by attaching a square pyramid to one of the square faces of the pentagonal prism.

Vertex coordinates

An augmented pentagonal prism of edge length 1 has the following vertices:

• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12\right),$
• $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12\right),$
• $\left(0,\,±\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12\right),$
• $\left(0,\,-\frac{\sqrt2+\sqrt{\frac{5+2\sqrt5}{5}}}{2},\,0\right).$

External links

• Klitzing, Richard. "aupip".

• Quickfur. "The Augmented Pentagonal Prism".

• Wikipedia Contributors. "Augmented pentagonal prism".
• McCooey, David. "Augmented Pentagonal Prism"