Crossed pentagrammic cupola

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Crossed pentagrammic cupola
Crossed pentagrammic cupola.png
Type Johnson isomorph
Cupola
Faces 5 triangles
5 squares
1 pentagram
1 decagram
Edges 25
Vertices 15
Vertex configuration 5+5(3.4.10/3)
5(3.4.5/3.4)
Symmetry group C5v, [5], (*55)
Rotation group C5, [5]+, (55)
Dual polyhedron -

In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex pentagonal cupola. It can be obtained as a slice of the nonconvex great rhombicosidodecahedron or quasirhombicosidodecahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is a decagram.

It may be seen as a cupola with a retrograde pentagrammic base, so that the squares and triangles connect across the bases in the opposite way to the pentagrammic cuploid, hence intersecting each other more deeply.

Related polyhedra[edit]

Family of star-cupolae and star-cuploids
n / d 3 4 5 6 7 8
2 Tetrahemihexahedron.png
Crossed triangular cuploid
Pentagrammic cuploid.png
Pentagrammic cuploid
Heptagrammic cuploid.png
Heptagrammic cuploid
3 Crossed square cupola.png
Crossed square cupola
Crossed pentagrammic cupola.png
Crossed pentagrammic cupola
Heptagrammic cupola.png
Heptagrammic cupola
Octagrammic cupola.png
Octagrammic cupola
4 Crossed pentagonal cuploid.png
Crossed pentagonal cuploid
Crossed heptagrammic cuploid.png
Crossed heptagrammic cuploid
5 Crossed hexagonal cupola
(Flat)
Crossed heptagrammic cupola.png
Crossed heptagrammic cupola
Crossed octagrammic cupola.png
Crossed octagrammic cupola

The crossed pentagonal cupola may be seen as a part of the uniform polyhedra known as the nonconvex great rhombicosidodecahedron, great dodecicosidodecahedron, and great rhombidodecahedron.

Crossed pentagrammic cupola.png
Crossed pentagrammic cupola
Uniform great rhombicosidodecahedron.png
Nonconvex great rhombicosidodecahedron
Great dodecicosidodecahedron.png
Great dodecicosidodecahedron
Great rhombidodecahedron.png
Great rhombidodecahedron

Various crossed pentagrammic cupolae on the nonconvex great rhombicosidodecahedron may be diminished or gyrated (rotated) to produce a set of 12 polyhedra isomorphic to the Johnson solids J72 to J83. They are the gyrate, metabigyrate, parabigyrate, trigyrate, diminished, metabidiminished, parabidiminished, tridiminished, metagyrate diminished, paragyrate diminished, bigyrate diminished, and gyrate bidiminished quasirhombicosidodecahedra.

Dual polyhedron[edit]

The dual of the crossed pentagrammic cupola has 10 triangular and 5 kite faces:

Dual crossed pentagrammic cupola.png

References[edit]

External links[edit]