Pentagonal antiprism


In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of ten triangles for a total of twelve faces. Hence, it is a non-regular dodecahedron.

Geometry

If the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron. It can also be considered as a parabidiminished icosahedron, a shape formed by removing two pentagonal pyramids from a regular icosahedron leaving two nonadjacent pentagonal faces; a related shape, the metabidiminished icosahedron, is likewise form from the icosahedron by removing two pyramids, but its pentagonal faces are adjacent to each other. The two pentagonal faces of either shape can be augmented with pyramids to form the icosahedron.
The semiregular pentagonal antiprism is inscribed in a cylinder whose bases are the disks in which the pentagonal faces are inscribed. If this polygon is projected radially onto a sphere and spherical trigonometry used to solve for the angular measures of the edges, the result is the arctangent of 2, matching the regular icosahedron; this implies that the radius of the cylinder equals its height.

Relation to polytopes

The pentagonal antiprism occurs as a constituent element in some higher-dimensional polytopes. Two rings of ten pentagonal antiprisms each bound the hypersurface of the four-dimensional grand antiprism. If these antiprisms are augmented with pentagonal prism pyramids and linked with rings of five tetrahedra each, the 600-cell is obtained.

Crossed antiprism

A crossed pentagonal antiprism is topologically identical to the pentagonal antiprism, although it can't be made uniform. The sides are isosceles triangles. It has D5h symmetry group of order 20. Its vertex configuration is 3.3/2.3.5, with one triangle retrograde and its vertex arrangement is the same as a pentagonal prism.