# Triangular retroprism

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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

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.

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.