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)
Schläfli symbol {5/3} || t{5/3}
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 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.

Family of star-cupolae
n / d 4 5 7 8
3
{4/3}
{5/3}
{7/3}
{8/3}
5 — —
{7/5}
{8/5}

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:

References

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.