Augmented triangular prism

Augmented triangular prism
(3D model) (OFF file)
Rank	3
Type	CRF
Space	Spherical
Notation
Bowers style acronym	Autip
Coxeter diagram	oxx oxo&#xt
Elements
Faces	2+2+2 triangles, 2 squares
Edges	1+2+2+4+4
Vertices	1+2+4
Vertex figures	1 square, edge length 1
	4 trapezoids, edge lengths 1, 1, 1, √2
	2 isosceles triangles, edge lengths 1, √2, √2
Measures (edge length 1)
Volume	$\frac{2\sqrt2+3\sqrt3}{12} ≈ 0.66871$
Dihedral angles	3–3 join: $\arccos\left(-\frac{\sqrt6}{3}\right) ≈ 144.73561^\circ$
	3–4 join: $\arccos\left(-\frac{3\sqrt2-\sqrt3}{6}\right) ≈ 114.73561^\circ$
	3–3 pyramidal: $\arccos\left(-\frac13\right) ≈ 109.47122^\circ$
	3–4 prismatic: 90°
	4–4: 60°
Central density	1
Number of external pieces	8
Level of complexity	13
Related polytopes
Army	Autip
Regiment	Autip
Dual	Lateromonotruncated triangular tegum
Conjugate	Excavated triangular prism
Abstract & topological properties
Flag count	52
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	K2×I, order 4
Convex	Yes
Nature	Tame

The augmented triangular prism is one of the 92 Johnson solids (J₄₉). It consists of 2+2+2 triangles and 2 squares. It can be constructed by attaching a square pyramid to one of the square faces of the triangular prism.

Vertex coordinates

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

• $\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12\right),$
• $\left(0,\,\frac{\sqrt3}{3},\,±\frac12\right),$
• $\left(0,\,-\frac{3\sqrt2+\sqrt3}{6},\,0\right).$

In vertex figures

The augmented triangular prism appears as the vertex figure of the nonuniform digonal-triangular duoantiprism. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson augmented triangular prism has no circumscribed sphere.

Variants made by changing the edge opposite to the vertex also appear as the vertex figure of the nonuniform duoantiprisms made out of a dyad and a regular polygon.

External links

• Klitzing, Richard. "autip".

• Quickfur. "The Augmented Triangular Prism".

• Wikipedia Contributors. "Augmented triangular prism".
• McCooey, David. "Augmented Triangular Prism"