Triangular-pentachoric duoprism

Triangular-pentachoric duoprism
Rank6
TypeUniform
Notation
Bowers style acronymTrapen
Coxeter diagramx3o x3o3o3o ()
Tapertopic notation1311
Elements
Peta3 pentachoric prisms, 5 triangular-tetrahedral duoprisms
Tera3 pentachora, 15 tetrahedral prisms, 10 triangular duoprisms
Cells15 tetrahedra, 10+30 triangular prisms
Faces5+30 triangles, 30 squares
Edges15+30
Vertices15
Vertex figureTetrahedral scalene, edge lengths 1 (base tetrahedron and top edge) and 2 (sides)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {165}}{15}}\approx 0.85635}$
Hypervolume${\displaystyle {\frac {\sqrt {15}}{384}}\approx 0.010086}$
Dipetal anglesPenp–tepe–tratet: 90°
Tratet–triddip–tratet: ${\displaystyle \arccos {\left({\frac {1}{4}}\right)}\approx 75.52249^{\circ }}$
Penp–pen–penp: 60°
HeightsPen atop penp: ${\displaystyle {\frac {\sqrt {3}}{2}}\approx 0.86603}$
Trig atop tratet: ${\displaystyle {\frac {\sqrt {10}}{4}}\approx 0.79057}$
Trip atop perp triddip: ${\displaystyle {\frac {\sqrt {15}}{6}}\approx 0.64550}$
Central density1
Number of external pieces8
Level of complexity15
Related polytopes
ArmyTrapen
RegimentTrapen
DualTriangular-pentachoric duotegum
ConjugateNone
Abstract & topological properties
Flag count10800
Euler characteristic0
OrientableYes
Properties
SymmetryA4×A2, order 720
ConvexYes
NatureTame

The triangular-pentachoric duoprism or trapen is a convex uniform duoprism that consists of 3 pentachoric prisms and 5 triangular-tetrahedral duoprisms. Each vertex joins 2 pentachoric prisms and 4 triangular-tetrahedral duoprisms. It is a duoprism based on a triangle and a pentachoron, and is thus also a convex segmentopeton, as a pentachoron atop pentachoric prism.

Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right),}$
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right),}$
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}}\right),}$
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,0,\,0,\,0,\,{\frac {\sqrt {10}}{5}}\right),}$
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,0,\,0,\,0,\,{\frac {\sqrt {10}}{5}}\right).}$

Representations

A triangular-pentachoric duoprism has the following Coxeter diagrams:

• x3o x3o3o3o (full symmetry)
• xx3oo ox3oo3oo&#x (A3×A2 symmetry, triangle atop triangular-tetrahedral duoprism)
• ox xx3oo3oo3oo&#x (A4×A1 symmetry, pentachoron atop pentachoric prism)
• ox xo3oo xx3oo&#x (A2×A2×A1 symmetry, triangular prism atop orthogonal triangular duoprism)
• xxx3ooo3ooo3ooo&#x (A4 symmetry, 3 pentachoric layers)