Triangular retroprism

The triangular retroprism or trirp, also called the triangular crossed antiprism, is a prismatic isogonal polyhedron. It consists of 2 base triangles and 6 side triangles. The side triangles are isosceles triangles. Each vertex joins one base triangle and three side triangles. It is a crossed antiprism based on a triangle, seen as a 3/2-gon rather than 3/1.

Triangular retroprism
Rank	3
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Trirp
Elements
Faces	6 isosceles triangles, 2 triangles
Edges	6+6
Vertices	6
Vertex figure	Bowtie
Measures (edge lengths 1 (base), a (sides))
Circumradius	$\frac{\sqrt{a^2+\frac13}}2$
Volume	$0$
Height	$\sqrt{a^2-1}$
Related polytopes
Army	Trip
Regiment	Trirp
Dual	Triangular concave antitegum
Abstract & topological properties
Euler characteristic	2
Orientable	Yes
Genus	0
Properties
Symmetry	A2×A1, order 12
Convex	No
Nature	Tame

It cannot be made uniform, because if all the edges are of the same length, the height becomes zero and all of the triangles coincide. It can be thought of as a degenerate uniform polyhedron.

It is isomorphic to the octahedron.

In vertex figures

Triangular retroprisms occur as vertex figures of seven nonconvex uniform polychora: the faceted rectified pentachoron, faceted rectified tesseract, faceted rectified icositetrachoron, faceted rectified hecatonicosachoron, faceted rectified small stellated hecatonicosachoron, faceted rectified great grand hecatonicosachoron, and faceted rectified great grand stellated hecatonicosachoron.

External links

• Klitzing, Richard. "trirp".