# Pentagonal cupolic prism

Pentagonal cupolic prism
Rank4
TypeSegmentotope
SpaceSpherical
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$\frac{\sqrt2+\sqrt{10}}{2} ≈ 2.28825$ Hypervolume$\frac{5+4\sqrt5}{6} ≈ 2.32404$ Dichoral anglesTrip–4–cube: $\arccos\left(-\frac{\sqrt3+\sqrt{15}}{6}\right) ≈ 159.09484°$ Cube–4–pip: $\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253°$ Pecu–3–trip: 90°
Pecu–4–cube: 90°
Pecu–5–pip: 90°
Pecu–10–dip: 90°
Trip–4–dip: $\arccos\left(\sqrt{\frac{5+2\sqrt5}{15}}\right) ≈ 37.37737°$ Cube–4–dip: $\arccos\left(\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 31.71747°$ HeightsPecu atop pecu: 1
Pip atop dip: $\sqrt{\frac{5-\sqrt5}{10}} ≈ 0.52573$ 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:

• $\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,\sqrt{\frac{5-\sqrt5}{10}},\,±\frac12\right),$ • $\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5+\sqrt5}{40}},\,\sqrt{\frac{5-\sqrt5}{10}},\,±\frac12\right),$ • $\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,\sqrt{\frac{5-\sqrt5}{10}},\,±\frac12\right),$ • $\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0,\,±\frac12\right),$ • $\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0,\,±\frac12\right),$ • $\left(±\frac{1+\sqrt5}{2},\,0,\,0,\,±\frac12\right).$ 