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| 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} sr{2,5} |
| 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 3.3.3.5 | |
Ingeometry, thepentagonal antiprism is the third in an infinite set ofantiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of twopentagons joined to each other by a ring of tentriangles for a total of twelve faces. Hence, it is a non-regulardodecahedron.
If the faces of the pentagonal antiprism are all regular, it is asemiregular polyhedron. It can also be considered as aparabidiminishedicosahedron, a shape formed by removing twopentagonal pyramids from aregular icosahedron leaving two nonadjacent pentagonal faces; a related shape, themetabidiminished icosahedron (one of theJohnson 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.
The pentagonal antiprism occurs as a constituent element in some higher-dimensionalpolytopes. Two rings of ten pentagonal antiprisms each bound the hypersurface of the four-dimensionalgrand antiprism. If these antiprisms are augmented with pentagonal prism pyramids and linked with rings of five tetrahedra each, the600-cell is obtained.
Thepentagonal antiprism can be truncated and alternated to form asnub antiprism:
| Antiprism A5 | Truncated tA5 | Alternated htA5 |
|---|---|---|
|  |  |
| 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 |
| Antiprism name | Digonal antiprism | (Trigonal) Triangular antiprism | (Tetragonal) Square antiprism | Pentagonal antiprism | Hexagonal antiprism | Heptagonal antiprism | ... | Apeirogonal antiprism |
|---|---|---|---|---|---|---|---|---|
| Polyhedron image |  |  |  | |  |  | ... | |
| Spherical tiling image |  |  |  |  |  |  | Plane tiling image |  |
| Vertex config. | 2.3.3.3 | 3.3.3.3 | 4.3.3.3 | 5.3.3.3 | 6.3.3.3 | 7.3.3.3 | ... | ∞.3.3.3 |
A crossed pentagonal antiprism is topologically identical to thepentagonal antiprism, although it can't be made uniform. The sides areisosceles triangles. It has D5h symmetry group of order 20. Itsvertex configuration is 3.3/2.3.5, with one triangle retrograde and itsvertex arrangement is the same as apentagonal prism.
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