# Rectified pentachoric prism

Rectified pentachoric prism
File:Rectified pentachoric prism.png
Rank5
TypeUniform
Notation
Bowers style acronymRappip
Coxeter diagramx o3x3o3o ()
Elements
Tera5 tetrahedral prisms, 2 rectified pentachora, 5 octahedral prisms
Cells10 tetrahedra, 10+20 triangular prisms, 10 octahedra
Faces20+40 triangles, 30 squares
Edges10+60
Vertices20
Vertex figureTriangular prismatic pyramid, edge lengths 1 (base), 2 (legs)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {85}}{10}}\approx 0.92195}$
Hypervolume${\displaystyle {\frac {11{\sqrt {5}}}{96}}\approx 0.25622}$
Diteral anglesTepe–trip–ope: ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }}$
Rap–oct–ope: 90°
Rap–tet–tepe: 90°
Ope–trip–ope: ${\displaystyle \arccos \left({\frac {1}{4}}\right)\approx 75.52249^{\circ }}$
HeightsRap atop rap: 1
Tepe atop ope: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Central density1
Number of external pieces12
Level of complexity15
Related polytopes
ArmyRappip
RegimentRappip
DualJoined pentachoric tegum
ConjugateNone
Abstract & topological properties
Flag count3600
Euler characteristic2
OrientableYes
Properties
SymmetryA4×A1, order 240
ConvexYes
NatureTame

The rectified pentachoric prism or rappip is a prismatic uniform polyteron that consists of 2 rectified pentachora, 5 octahedral prisms and 5 tetrahedral prisms. 1 rectified pentachoron, 2 tetrahedral prisms, and 3 octahedral prisms join at each vertex. As the name suggests, it is a prism based on the rectified pentachoron. As such it is also a convex segmentoteron.

## Vertex coordinates

The vertices of a rectified pentachoric prism of edge length 1 are given by:

• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{20}},\,-{\frac {\sqrt {6}}{4}},\,0,\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(-{\frac {3{\sqrt {10}}}{20}},\,{\frac {\sqrt {6}}{12}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{3}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,{\frac {\sqrt {6}}{6}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {10}}{10}},\,-{\frac {\sqrt {6}}{6}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,\pm {\frac {1}{2}}\right).}$

## Representations

A rectified pentachoric prism has the following Coxeter diagrams:

• x o3x3o3o (full symmetry)
• oo3xx3oo3oo&#x (A4 symmetry, as segmentoteron)
• xx xo3ox3oo&#x (A3×A1 axial, tetrahedral prism atop octahedral prism)
• xxx oxo oxo3oox&#xt (A2×A1×A1 symmetry, edge-first)