Triangular antiprism

The triangular antiprism, or trap, is a triangle-based antiprism. The version with all equal edges is the regular octahedron, one of the Platonic solids, but other versions exist with isosceles triangles as their sides. In the latter case, their Coxeter diagram could be given as xo3ox&#y. They are formed by alternating a general hexagonal prism. Given a triangular antiprism with base edges of length b and side edges of length l, the corresponding hexagonal prism has base edges of length ${\frac {b{\sqrt {3}}}{3}}$ and side edges of length ${\sqrt {l^{2}-{\frac {b^{2}}{3}}}}$.

Triangular antiprism
(3D model)
Rank	3
Type	Isogonal
Notation
Bowers style acronym	Trap
Coxeter diagram	s2s6o ()
Elements
Faces	6 isosceles triangles, 2 triangles
Edges	6+6
Vertices	6
Vertex figure	Rectangle
Measures (edge lengths b  [base edge length], l  [other edge length])
Circumradius	${\frac {\sqrt {b^{2}+l^{2}}}{2}}$
Height	${\sqrt {\frac {3l^{2}-b^{2}}{3}}}$
Central density	1
Related polytopes
Army	Trap
Regiment	Trap
Dual	Triangular antitegum
Conjugate	Triangular antiprism
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	(G2×A1)/2, order 12
Convex	Yes
Nature	Tame

Any such triangular antiprism has an equatorial rectangle section with edges of the same lengths as the antiprism.

The bases of a triangular antiprism are rotated by 60° with respect to each other. If the rotation angle is different the resulting polyhedron has scalene triangles as lateral faces and is called a triangular gyroprism.

A notable case occurs as the alternation of the uniform hexagonal prism. This specific case has base edges of length ${\sqrt {3}}$ and side edges of length ${\sqrt {2}}$.

Vertex coordinates

Cartesian coordinates for a triangular antiprism created from two triangles of edge length b laced by edges of length ℓ, centered at the origin, are given by:

• $\pm \left(0,\,{\frac {{\sqrt {3}}b}{3}},\,{\sqrt {\frac {3l^{2}-b^{2}}{12}}}\right),$
• $\pm \left(\pm {\frac {b}{2}},\,-{\frac {{\sqrt {3}}b}{6}},\,{\sqrt {\frac {3l^{2}-b^{2}}{12}}}\right).$

In vertex figures

A triangular antiprism with base edges of length 1 and side edges of length √2 occurs as the vertex figure of the small prismatodecachoron.