Pentagonal cupola

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Pentagonal cupola
Bowers style acronymPecu
Coxeter diagramox5xx&#x
Faces5 triangles, 5 squares, 1 pentagon, 1 decagon
Vertex figures5 isosceles trapezoids, edge lengths 1, 2, (1+5)/2, 2
 10 scalene triangles, edge lengths 1, 2, (5+5)/2
Measures (edge length 1)
Dihedral angles3–4:
Central density1
Number of external pieces12
Level of complexity10
Related polytopes
DualSemibisected pentagonal trapezohedron
ConjugateRetrograde pentagrammic cupola
Abstract & topological properties
Flag count100
Euler characteristic2
SymmetryH2×I, order 10

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[edit | edit source]

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

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:

Representations[edit | edit source]

A pentagonal cupola has the following Coxeter diagrams:

  • ox5xx&#x
  • so10ox&#x

Related polyhedra[edit | edit source]

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[edit | edit source]