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Uniform n-gonal antiprisms | |
---|---|
Example hexagonal antiprism | |
Type | uniform in the sense of semiregular polyhedron |
Faces | 2 n-gons, 2n triangles |
Edges | 4n |
Vertices | 2n |
Conway polyhedron notation | An |
Vertex configuration | 3.3.3.n |
Schläfli symbol | { }?{n}^{[1]} s{2,2n} sr{2,n} |
Coxeter diagrams | |
Symmetry group | D_{nd}, [2^{+},2n], (2*n), order 4n |
Rotation group | D_{n}, [2,n]^{+}, (22n), order 2n |
Dual polyhedron | convex dual-uniform n-gonal trapezohedron |
Properties | convex, vertex-transitive, regular polygon faces |
Net |
In geometry, an n-gonal antiprism or n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of prismatoids and are a (degenerate) type of snub polyhedra.
Antiprisms are similar to prisms except that the bases are twisted relatively to each other, and that the side faces are triangles, rather than quadrilaterals.
In the case of a regular n-sided base, one usually considers the case where its copy is twisted by an angle of degrees. Extra regularity is obtained when the line connecting the base centers is 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.
A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For we have the regular tetrahedron as a digonal antiprism (degenerate antiprism), and for the regular octahedron as a triangular antiprism (non-degenerate antiprism).
Dual polyhedra of antiprisms are trapezohedra. Their existence was discussed and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the same conditions on vertices as the Archimedean solids.
Family of uniform n-gonal antiprisms | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Polyhedron image | ... | Apeirogonal antiprism | ||||||||||||
Spherical tiling image | Plane tiling image | |||||||||||||
Vertex configuration n.3.3.3 | 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 |
Cartesian coordinates for the vertices of a right antiprism with (regular) n-gonal bases and isosceles triangles are
with k ranging from 0 to 2n - 1; if the triangles are equilateral,
Let a be the edge-length of a uniform antiprism. Then the volume is
and the surface area is
There are an infinite set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron (truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower symmetry form of the icosahedron.
Antiprisms | ||||
---|---|---|---|---|
... | ||||
s{2,4} | s{2,6} | s{2,8} | s{2,10} | s{2,2n} |
Truncated antiprisms | ||||
... | ||||
ts{2,4} | ts{2,6} | ts{2,8} | ts{2,10} | ts{2,2n} |
Snub antiprisms | ||||
J_{84} | Icosahedron | J_{85} | Irregular faces... | |
... | ||||
ss{2,4} | ss{2,6} | ss{2,8} | ss{2,10} | ss{2,2n} |
The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is D_{nd} of order 4n, except in the case of a tetrahedron, which has the larger symmetry group T_{d} of order 24, which has three versions of D_{2d} as subgroups, and the octahedron, which has the larger symmetry group O_{h} of order 48, which has four versions of D_{3d} as subgroups.
The symmetry group contains inversion if and only if n is odd.
The rotation group is D_{n} of order 2n, except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D_{2} as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D_{3} as subgroups.
Uniform star antiprisms are named by their star polygon bases, {p/q}, and exist in prograde and retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, p/(p - q) instead of p/q, e.g. 5/3 instead of 5/2.
In the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry.
Some retrograde star antiprisms with regular convex polygon bases cannot be constructed with equal edge lengths, so are not uniform polyhedra.
Star antiprism compounds also can be constructed where p and q have common factors; example: a 10/4 star antiprism is the compound of two 5/2 star antiprisms.
Star antiprisms by symmetry, up to 12 | |||||
---|---|---|---|---|---|
Symmetry group | Uniform stars | Other stars | |||
D_{4d} [2^{+},8] (2*5) |
3.3/2.3.4 | ||||
D_{5h} [2,5] (*225) |
3.3.3.5/2 |
3.3/2.3.5 | |||
D_{5d} [2^{+},10] (2*5) |
3.3.3.5/3 | ||||
D_{6d} [2^{+},12] (2*6) |
3.3/2.3.6 | ||||
D_{7h} [2,7] (*227) |
3.3.3.7/2 |
3.3.3.7/4 | |||
D_{7d} [2^{+},14] (2*7) |
3.3.3.7/3 | ||||
D_{8d} [2^{+},16] (2*8) |
3.3.3.8/3 |
3.3.3.8/5 | |||
D_{9h} [2,9] (*229) |
3.3.3.9/2 |
3.3.3.9/4 | |||
D_{9d} [2^{+},18] (2*9) |
3.3.3.9/5 | ||||
D_{10d} [2^{+},12] (2*10) |
3.3.3.10/3 | ||||
D_{11h} [2,11] (*2.2.11) |
3.3.3.11/2 |
3.3.3.11/4 |
3.3.3.11/6 | ||
D_{11d} [2^{+},22] (2*11) |
3.3.3.11/3 |
3.3.3.11/5 |
3.3.3.11/7 | ||
D_{12d} [2^{+},24] (2*12) |
3.3.3.12/5 |
3.3.3.12/7 | |||
... |