# Triangular-rectified pentachoric duoprism

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Triangular-rectified pentachoric duoprism
Rank6
TypeUniform
Notation
Bowers style acronymTrarap
Coxeter diagramx3o o3x3o3o ()
Elements
Peta5 triangular-tetrahedral duoprisms, 5 triangular-octahedral duoprisms, 3 rectified pentachoric prisms
Tera15 tetrahedral prisms, 10+20 triangular duoprisms, 15 octahedral prisms, 3 rectified pentachora
Cells15 tetrahedra, 30+30+60 triangular prisms, 15 octahedra
Faces10+30+60 triangles, 90 squares
Edges30+90
Vertices30
Vertex figureTriangular prismatic scalene, edge lengths 1 (base tetrahedron and top edge) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {210}}{15}}\approx 0.96609}$
Hypervolume${\displaystyle {\frac {11{\sqrt {15}}}{384}}\approx 0.11095}$
Dipetal anglesTratet–triddip–troct: ${\displaystyle \arccos {\left(-{\frac {1}{4}}\right)}\approx 104.47751^{\circ }}$
Rappip–ope–troct: 90°
Rappip–tepe–tratet: 90°
Troct–triddip–troct: ${\displaystyle \arccos {\left({\frac {1}{4}}\right)}\approx 75.52249^{\circ }}$
Rappip–rap–rappip: 60°
HeightsRap atop rappip: ${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Tratet atop troctt: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Central density1
Number of external pieces13
Level of complexity45
Related polytopes
ArmyTrarap
RegimentTrarap
DualTriangular-joined pentachoric duotegum
ConjugateNone
Abstract & topological properties
Flag count32400
Euler characteristic0
OrientableYes
Properties
SymmetryA4×A2, order 720
ConvexYes
NatureTame

The triangular-rectified pentachoric duoprism or trarap is a convex uniform duoprism that consists of 3 rectified pentachoric prisms, 5 triangular-octahedral duoprisms, and 5 triangular-tetrahedral duoprisms. Each vertex joins 2 rectified pentachoric prisms, 2 triangular-tetrahedral duoprisms, and 3 triangular-octahedral duoprisms. It is a duoprism based on a triangle and a rectified pentachoron, and is thus also a convex segmentopeton, as a rectified pentachoron atop rectified pentachoric prism.

## Vertex coordinates

The vertices of a triangular-rectified pentachoric duoprism of edge length 1 are given by:

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

## Representations

A triangular-rectified pentachoric duoprism has the following Coxeter diagrams:

• x3o o3x3o3o () (full symmetry)
• xx3oo xo3ox3oo&#x (A3×A2 symmetry, triangular-tetrahedral duoprism atop triangular-octahedral duoprism)
• ox oo3xx3oo3oo&#x (A4×A1 symmetry, rectified pentachoron atop rectified pentachoric prism)