Hexagonal antiprism
In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
| Uniform hexagonal antiprism | |
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| Type | Prismatic uniform polyhedron |
| Elements | F = 14, E = 24 V = 12 (χ = 2) |
| Faces by sides | 12{3}+2{6} |
| Schläfli symbol | s{2,12} sr{2,6} |
| Wythoff symbol | | 2 2 6 |
| Coxeter diagram |      |
| Symmetry group | D6d, [2+,12], (2*6), order 24 |
| Rotation group | D6, [6,2]+, (622), order 12 |
| References | U77(d) |
| Dual | Hexagonal trapezohedron |
| Properties | convex |
 Vertex figure 3.3.3.6 | |
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
In the case of a regular 6-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.
If faces are all regular, it is a semiregular polyhedron.
Crossed antiprism
A crossed hexagonal antiprism is a star polyhedron, topologically identical to the convex hexagonal antiprism with the same vertex arrangement, but it can't be made uniform; the sides are isosceles triangles. Its vertex configuration is 3.3/2.3.6, with one triangle retrograde. It has D6d symmetry, order 24.
Related polyhedra
The hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles.
| Uniform hexagonal dihedral spherical polyhedra | ||||||||||||||
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| Symmetry: [6,2], (*622) | [6,2]+, (622) | [6,2+], (2*3) | ||||||||||||
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| {6,2} | t{6,2} | r{6,2} | t{2,6} | {2,6} | rr{6,2} | tr{6,2} | sr{6,2} | s{2,6} | ||||||
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| V62 | V122 | V62 | V4.4.6 | V26 | V4.4.6 | V4.4.12 | V3.3.3.6 | V3.3.3.3 | ||||||
| Antiprism name | Digonal antiprism | (Trigonal) Triangular antiprism |
(Tetragonal) Square antiprism |
Pentagonal antiprism | Hexagonal antiprism | Heptagonal antiprism | Octagonal antiprism | Enneagonal antiprism | Decagonal antiprism | Hendecagonal antiprism | Dodecagonal antiprism | ... | Apeirogonal antiprism |
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| Spherical tiling image |  |
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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 | 8.3.3.3 | 9.3.3.3 | 10.3.3.3 | 11.3.3.3 | 12.3.3.3 | ... | ∞.3.3.3 |
External links
- Weisstein, Eric W. "Antiprism". MathWorld.
- Hexagonal Antiprism: Interactive Polyhedron model
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- polyhedronisme A6