Pentagonal antiprism


Uniform pentagonal antiprism
Type Prismatic uniform polyhedron
Elements F = 12, E = 20
V = 10 (χ = 2)
Faces by sides 10{3}+2{5}
Schläfli symbol s{2,10}
Wythoff symbol | 2 2 5
Coxeter diagram
Symmetry group D5d, [2+,10], (2*5), order 20
Rotation group D5, [5,2]+, (522), order 10
References U77(c)
Dual Pentagonal trapezohedron
Properties convex

Vertex figure

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.

Three Dimension model of a (uniform) pentagonal antiprism



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 (one of the Johnson solids), 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.

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.

See also


The pentagonal antiprism can be truncated and alternated to form a snub antiprism:

Snub antiprisms
s{2,10} ts{2,10} ss{2,10}
v:10; e:20; f:12 v:40; e:60; f:22 v:20; e:50; f:32
Family of uniform n-gonal antiprisms
Antiprism name Digonal antiprism (Trigonal)
Triangular antiprism
Square antiprism
Pentagonal antiprism Hexagonal antiprism Heptagonal antiprism ... Apeirogonal antiprism
Polyhedron image             ...
Spherical tiling image             Plane tiling image  
Vertex config. ... ∞.3.3.3

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 d5d symmetry, order 10. Its vertex configuration is 3.3/2.3.5, with one triangle retrograde and its vertex arrangement is the same as a pentagonal prism.

  • Weisstein, Eric W. "Antiprism". MathWorld.
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  • Pentagonal Antiprism: Interactive Polyhedron Model
  • Virtual Reality Polyhedra The Encyclopedia of Polyhedra
  • polyhedronisme A5