Pentagonal cupolic prism

Pentagonal cupolic prism
Rank4
TypeSegmentotope
Notation
Bowers style acronymPecupe
Coxeter diagramxx ox5xx&#x
Elements
Cells5 triangular prisms, 5 cubes, 1 pentagonal prism, 2 pentagonal cupolas, 1 decagonal prism
Faces10 triangles, 5+5+5+10+10 squares, 2 pentagons, 2 decagons
Edges5+10+10+10+10+20
Vertices10+20
Vertex figures10 isosceles trapezoidal pyramids, base edge lengths 1, 2, (1+5)/2, 2, side edge length 2
20 irregular tetrahedra, edge lengths 1 (1), 2 (4), and (5+5)/2 (1)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 2.28825}$
Hypervolume${\displaystyle {\frac {5+4{\sqrt {5}}}{6}}\approx 2.32404}$
Dichoral anglesTrip–4–cube: ${\displaystyle \arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }}$
Cube–4–pip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Pecu–3–trip: 90°
Pecu–4–cube: 90°
Pecu–5–pip: 90°
Pecu–10–dip: 90°
Trip–4–dip: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
Cube–4–dip: ${\displaystyle \arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }}$
HeightsPecu atop pecu: 1
Pip atop dip: ${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573}$
Central density1
Related polytopes
ArmyPecupe
RegimentPecupe
DualSemibisected pentagonal trapezohedral tegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2×A1×I, order 20
ConvexYes
NatureTame

The pentagonal cupolic prism, or pecupe, is a CRF segmentochoron (designated K-4.117 on Richard Klitzing's list). It consiss of 2 pentagonal cupolas, 5 triangular prisms, 5 cubes, 1 pentagonal prism, and 1 decagonal prism.

As the name suggests, it is a prism based on the pentagonal cupola. As such, it is a segmentochoron between two pentagonal cupolas. It can also be viewed as a segmentochoron between a decagonal prism and a pentagonal prism.

It can be obtained as a segment of the small rhombicosidodecahedral prism.

Vertex coordinates

Coordinates of the vertices of a pentagonal cupolic prism of edge length 1 centered at the origin are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\sqrt {\frac {5+2{\sqrt {5}}}{20}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,{\sqrt {\frac {5+{\sqrt {5}}}{40}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(0,\,{\sqrt {\frac {5+{\sqrt {5}}}{10}}},\,{\sqrt {\frac {5-{\sqrt {5}}}{10}}},\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {5+2{\sqrt {5}}}}{2}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\sqrt {\frac {5+{\sqrt {5}}}{8}}},\,0,\,\pm {\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,0,\,0,\,\pm {\frac {1}{2}}\right).}$