# 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
Rank3
TypeIsogonal
SpaceSpherical
Notation
Bowers style acronymTrirp
Elements
Faces6 isosceles triangles, 2 triangles
Edges6+6
Vertices6
Vertex figureBowtie
Measures (edge lengths 1 (base), a (sides))
Circumradius${\displaystyle \frac{\sqrt{a^2+\frac13}}2}$
Volume${\displaystyle 0}$
Height${\displaystyle \sqrt{a^2-1}}$
Related polytopes
ArmyTrip
RegimentTrirp
DualTriangular concave antitegum
Abstract & topological properties
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryA2×A1, order 12
ConvexNo
NatureTame

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.