Elongated pentagonal pyramid

Elongated pentagonal pyramid
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Epeppy
Coxeter diagram	oxx5ooo&#xt
Elements
Faces	5 triangles, 5 squares, 1 pentagon
Edges	5+5+5+5
Vertices	1+5+5
Vertex figures	1 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	${\frac {5+{\sqrt {5}}+6{\sqrt {25+10{\sqrt {5}}}}}{24}}\approx 2.02198$
Dihedral angles	3–3: $\arccos \left(-{\frac {\sqrt {5}}{3}}\right)\approx 138.18969^{\circ }$
	3–4: $\arccos \left(-{\sqrt {\frac {10-2{\sqrt {5}}}{15}}}\right)\approx 127.37737^{\circ }$
	4–4: 108°
	4–5: 90°
Central density	1
Number of external pieces	11
Level of complexity	8
Related polytopes
Army	Epeppy
Regiment	Epeppy
Dual	Elongated pentagonal pyramid
Conjugate	Elongated pentagrammic pyramid
Abstract & topological properties
Flag count	80
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

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:

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

External links

• Klitzing, Richard. "epeppy".
• Quickfur. "The Elongated Pentagonal Pyramid".

• Wikipedia contributors. "Elongated pentagonal pyramid".
• McCooey, David. "Elongated Pentagonal Pyramid"