Pentagonal cupola

Pentagonal cupola
(3D model) (OFF file)
Rank	3
Type	CRF
Notation
Bowers style acronym	Pecu
Coxeter diagram	ox5xx&#x
Elements
Faces	5 triangles, 5 squares, 1 pentagon, 1 decagon
Edges	5+5+5+10
Vertices	5+10
Vertex figures	5 isosceles trapezoids, edge lengths 1, √2, (1+√5)/2, √2
	10 scalene triangles, edge lengths 1, √2, √(5+√5)/2
Measures (edge length 1)
Circumradius	${\frac {\sqrt {11+4{\sqrt {5}}}}{2}}\approx 2.23295$
Volume	${\frac {5+4{\sqrt {5}}}{6}}\approx 2.32404$
Dihedral angles	3–4: $\arccos \left(-{\frac {{\sqrt {3}}+{\sqrt {15}}}{6}}\right)\approx 159.09484^{\circ }$
	4–5: $\arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }$
	3–10: $\arccos \left({\sqrt {\frac {5+2{\sqrt {5}}}{15}}}\right)\approx 37.37737^{\circ }$
	4–10: $\arccos \left({\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 31.71747^{\circ }$
Height	${\sqrt {\frac {5-{\sqrt {5}}}{10}}}\approx 0.52573$
Central density	1
Number of external pieces	12
Level of complexity	10
Related polytopes
Army	Pecu
Regiment	Pecu
Dual	Semibisected pentagonal trapezohedron
Conjugate	Retrograde pentagrammic cupola
Abstract & topological properties
Flag count	100
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	H2×I, order 10
Convex	Yes
Nature	Tame

The pentagonal cupola is one of the 92 Johnson solids (J₅). It consists of 5 triangles, 5 squares, 1 pentagon, and 1 decagon. It is a cupola based on the pentagon, and is one of three Johnson solid cupolas, the other two being the triangular cupola and the square cupola.

It can be obtained as a segment of the small rhombicosidodecahedron.

Vertex coordinates

A pentagonal cupola of edge length 1 has vertices given by the following coordinates:

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

These can be obtained from placing a pentagon and decagon in parallel planes.

Alternatively, coordinates can be obtained as a subset of vertices of the small rhombicosidodecahedron:

• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,{\frac {2+{\sqrt {5}}}{2}}\right),$
• $\left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {5+{\sqrt {5}}}{4}}\right),$
• $\left({\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,{\frac {1+{\sqrt {5}}}{2}}\right),$
• $\left({\frac {5+{\sqrt {5}}}{4}},\,0,\,{\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left({\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,{\frac {3+{\sqrt {5}}}{4}}\right),$
• $\left({\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,{\frac {1+{\sqrt {5}}}{4}}\right),$
• $\left({\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,{\frac {1}{2}}\right).$

Representations

A pentagonal cupola has the following Coxeter diagrams:

• ox5xx&#x
• so10ox&#x

Related polyhedra

Two pentagonal cupolas can be attached at their decagonal bases in the same orientation to form a pentagonal orthobicupola. If the second cupola is rotated by 36º the result is the pentagonal gyrobicupola. If a pentagonal rotunda is attached, the result is either a pentagonal orthocupolarotunda (if the base pentagons are in the same orientation) or a pentagonal gyrocupolarotunda (if the base pentagons are rotated 36º).

A decagonal prism can be attached to the pentagonal cupola's decagonal base to form the elongated pentagonal cupola. If a decagonal antiprism is attached instead, the result is the gyroelongated pentagonal cupola.

The pentagonal cupola is the pentagon-first cap of the small rhombicosidodecahedron. Gyrating or removing such caps results in Johnson Solids 72 to 83. The pentagonal cupola can also be augmented onto a truncated dodecahedron, producing the augmented, parabiaugmented, metabiaugmented and triaugmented truncated dodecahedra.

External links

• Klitzing, Richard. "pecu".
• Quickfur. "The Pentagonal Cupola".

• Weisstein, Eric W. "Pentagonal Cupola" ("Johnson solid") at MathWorld.

• Wikipedia contributors. "Pentagonal cupola".
• McCooey, David. "Pentagonal Cupola"

• Hi.gher.Space Wiki Contributors. "Pentagonal cupola".