# Crossed pentagrammic cupola

Crossed pentagrammic cupola
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

Family of star-cupolae and star-cuploids
n / d 3 4 5 6 7 8
2
Crossed triangular cuploid

Pentagrammic cuploid

Heptagrammic cuploid
3
Crossed square cupola

Crossed pentagrammic cupola

Heptagrammic cupola

Octagrammic cupola
4
Crossed pentagonal cuploid

Crossed heptagrammic cuploid
5 Crossed hexagonal cupola
(Flat)

Crossed heptagrammic cupola

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 Nonconvex great rhombicosidodecahedron Great dodecicosidodecahedron 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

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