Elongated pentagonal pyramid

Elongated pentagonal pyramid
Rank3
TypeCRF
Notation
Bowers style acronymEpeppy
Coxeter diagramoxx5ooo&#xt
Elements
Faces5 triangles, 5 squares, 1 pentagon
Edges5+5+5+5
Vertices1+5+5
Vertex figures1 pentagon, edge length 1
5 kites, edge lengths 1 and 2
5 isosceles triangles, edge lengths (1+5)/2, 2, 2
Measures (edge length 1)
Volume${\displaystyle {\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\approx 2.02198}$
Dihedral angles3–3: ${\displaystyle \arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }}$
3–4: ${\displaystyle \arccos \left(-{\sqrt {\frac {10-2{\sqrt {5}}}{15}}}\right)\approx 127.37737^{\circ }}$
4–4: 108°
4–5: 90°
Central density1
Number of external pieces11
Level of complexity8
Related polytopes
ArmyEpeppy
RegimentEpeppy
DualElongated pentagonal pyramid
ConjugateElongated pentagrammic pyramid
Abstract & topological properties
Flag count80
Euler characteristic2
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryH2×I, order 10
ConvexYes
NatureTame

The elongated pentagonal pyramid is one of the 92 Johnson solids (J9). It consists of 5 triangles, 5 squares, and 1 pentagon. It can be constructed by attaching a pentagonal prism to the base of the pentagonal pyramid..

If a second pyramid is attached to the other base of the pentagonal prism, the result is the elongated pentagonal bipyramid.

Vertex coordinates

An elongated pentagonal pyramid of edge length 1 has the following vertices:

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