# Pentagonal pyramidal prism

Pentagonal pyramidal prism
Rank4
TypeSegmentotope
Notation
Bowers style acronymPeppyp
Coxeter diagramxx ox5oo&#x
Elements
Cells2 pentagonal pyramids, 5 triangular prisms, 1 pentagonal prism
Faces10 triangles, 10+10 squares, 2 pentagons
Edges1+5+10+10
Vertices2+10
Vertex figures2 pentagonal pyramids, edge lengths 1 (base) and 2 (legs)
10 sphenoids, edge lengths 1 (2), 2 (3), and (1+5)/2 (1)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {\frac {7+{\sqrt {5}}}{8}}}\approx 1.07448}$
Hypervolume${\displaystyle {\frac {5+{\sqrt {5}}}{24}}\approx 0.30150}$
Dichoral anglestrip–4–trip: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
Peppy–3–trip: 90°
Peppy–5–pip: 90°
trip–4–pip: ${\displaystyle \arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }}$
HeightsPeppy atop peppy: 1
Dyad atop pip: ${\displaystyle {\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573}$
Central density1
Related polytopes
ArmyPeppyp
RegimentPeppyp
DualPentagonal pyramidal tegum
ConjugatePentagrammic pyramidal prism
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryH2×A1×I, order 20
ConvexYes
NatureTame

The pentagonal pyramidal prism, or peppyp, is a CRF segmentochoron (designated K-4.38 on Richard Klitzing's list). It consists of 2 pentagonal pyramids, 1 pentagonal prism, and 5 triangular prisms.

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

Two pentagonal pyramidal prisms can be attached to the pentagonal prismatic cells of the pentagonal antiprismatic prism to produce the uniform icosahedral prism.

## Vertex coordinates

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

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

## Representations

A pentagonal pyramidal prism has the following Coxeter diagrams:

• xx o54oo&#x (full symmetry)
• oxxo5oooo&#xr (H2 axial only, pentagonal pyramid atop pentagonal pyramid)