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Set of uniform n-gonal antiprisms | |
---|---|
Type | uniform in the sense of semiregular polyhedron |
Faces | 2 regular n-gons 2n equilateral triangles |
Edges | 4n |
Vertices | 2n |
Vertex configuration | 3.3.3.n |
Schläfli symbol | { }⊗{n} [1] s{2,2n} sr{2,n} |
Conway notation | An |
Coxeter diagram | |
Symmetry group | Dnd, [2+,2n], (2*n), order 4n |
Rotation group | Dn, [2,n]+, (22n), order 2n |
Dual polyhedron | convex dual-uniform n-gonal trapezohedron |
Properties | convex, vertex-transitive, regular polygon faces, congruent & coaxial bases |
Net | |
Net of uniform enneagonal antiprism (n = 9) |
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.
Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.
Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are 2n triangles, rather than n quadrilaterals.
The dual polyhedron of an n-gonal antiprism is an n-gonal trapezohedron.
In his 1619 book Harmonices Mundi , Johannes Kepler observed the existence of the infinite family of antiprisms. [2] This has conventionally been thought of as the first discovery of these shapes, but they may have been known earlier: an unsigned printing block for the net of a hexagonal antiprism has been attributed to Hieronymus Andreae, who died in 1556. [3]
The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to Theodor Wittstein . [4] Although the English "anti-prism" had been used earlier for an optical prism used to cancel the effects of a primary optimal element, [5] the first use of "antiprism" in English in its geometric sense appears to be in the early 20th century in the works of H. S. M. Coxeter. [6]
For an antiprism with regular n-gon bases, one usually considers the case where these two copies are twisted by an angle of 180/n degrees.
The axis of a regular polygon is the line perpendicular to the polygon plane and lying in the polygon centre.
For an antiprism with congruent regularn-gon bases, twisted by an angle of 180/n degrees, more regularity is obtained if the bases have the same axis: are coaxial ; i.e. (for non-coplanar bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a right antiprism, and its 2n side faces are isosceles triangles.
A uniform n-antiprism has two congruent regular n-gons as base faces, and 2n equilateral triangles as side faces.
Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For n = 2, we have the digonal antiprism (degenerate antiprism), which is visually identical to the regular tetrahedron; for n = 3, the regular octahedron as a triangular antiprism (non-degenerate antiprism).
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 |
The Schlegel diagrams of these semiregular antiprisms are as follows:
A3 | A4 | A5 | A6 | A7 | A8 |
Cartesian coordinates for the vertices of a right n-antiprism (i.e. with regular n-gon bases and 2n isosceles triangle side faces) are:
where 0 ≤ k ≤ 2n – 1;
if the n-antiprism is uniform (i.e. if the triangles are equilateral), then:
Let a be the edge-length of a uniform n-gonal antiprism; then the volume is:
and the surface area is:
Furthermore, the volume of a regular right n-gonal antiprism with side length of its bases l and height h is given by:
Note that the volume of a right n-gonal prism with the same l and h is:
which is smaller than that of an antiprism.
The symmetry group of a right n-antiprism (i.e. with regular bases and isosceles side faces) is Dnd = Dnv of order 4n, except in the cases of:
The symmetry group contains inversion if and only if n is odd.
The rotation group is Dn of order 2n, except in the cases of:
Note: The right n-antiprisms have congruent regular n-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform n-antiprism, for n ≥ 4.
Four-dimensional antiprisms can be defined as having two dual polyhedra as parallel opposite faces, so that each three-dimensional face between them comes from two dual parts of the polyhedra: a vertex and a dual polygon, or two dual edges. Every three-dimensional convex polyhedron is combinatorially equivalent to one of the two opposite faces of a four-dimensional antiprism, constructed from its canonical polyhedron and its polar dual. [7] However, there exist four-dimensional polyhedra that cannot be combined with their duals to form five-dimensional antiprisms. [8]
5/2-antiprism | 5/3-antiprism | ||||
9/2-antiprism | 9/4-antiprism | 9/5-antiprism |
Uniform star antiprisms are named by their star polygon bases, {p/q}, and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: p/(p – q) instead of p/q; example: 5/3 instead of 5/2.
A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and 2n isosceles triangle side faces.
Any star antiprism with regular convex or star polygon bases can be made a right star antiprism (by translating and/or twisting one of its bases, if necessary).
In the retrograde forms, but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus:
Also, star antiprism compounds with regular star p/q-gon bases can be constructed if p and q have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.
Symmetry group | Uniform stars | Right stars | |||
---|---|---|---|---|---|
D4d [2+,8] (2*4) | 3.3/2.3.4 Crossed square antiprism | ||||
D5h [2,5] (*225) | 3.3.3.5/2 Pentagrammic antiprism | 3.3/2.3.5 crossed pentagonal antiprism | |||
D5d [2+,10] (2*5) | 3.3.3.5/3 Pentagrammic crossed-antiprism | ||||
D6d [2+,12] (2*6) | 3.3/2.3.6 crossed hexagonal antiprism | ||||
D7h [2,7] (*227) | 3.3.3.7/2 | 3.3.3.7/4 | |||
D7d [2+,14] (2*7) | 3.3.3.7/3 | ||||
D8d [2+,16] (2*8) | 3.3.3.8/3 Octagrammic antiprism | 3.3.3.8/5 Octagrammic crossed-antiprism | |||
D9h [2,9] (*229) | 3.3.3.9/2 Enneagrammic antiprism (9/2) | 3.3.3.9/4 Enneagrammic antiprism (9/4) | |||
D9d [2+,18] (2*9) | 3.3.3.9/5 Enneagrammic crossed-antiprism | ||||
D10d [2+,20] (2*10) | 3.3.3.10/3 Decagrammic antiprism | ||||
D11h [2,11] (*2.2.11) | 3.3.3.11/2 | 3.3.3.11/4 | 3.3.3.11/6 | ||
D11d [2+,22] (2*11) | 3.3.3.11/3 | 3.3.3.11/5 | 3.3.3.11/7 | ||
D12d [2+,24] (2*12) | 3.3.3.12/5 | 3.3.3.12/7 | |||
... | ... |
In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two pyramids together base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise specified the base vertices are usually coplanar and a bipyramid is usually symmetric, meaning the two pyramids are mirror images across their common base plane. When each apex of the bipyramid is on a line perpendicular to the base and passing through its center, it is a right bipyramid; otherwise it is oblique. When the base is a regular polygon, the bipyramid is also called regular.
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent regular polygons, and the same number of faces meet at each vertex. There are only five such polyhedra:
In geometry, a prism is a polyhedron comprising an n-sided polygon base, a second base which is a translated copy of the first, and n other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.
In geometry, an octagon is an eight-sided polygon or 8-gon.
In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.
In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.
In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of isosceles triangles and rectangles. If the triangles are equilateral and the rectangles are squares, while the base and its opposite face are regular polygons, the triangular, square, and pentagonal cupolae all count among the Johnson solids, and can be formed by taking sections of the cuboctahedron, rhombicuboctahedron, and rhombicosidodecahedron, respectively.
In geometry, the gyrobifastigium is the 26th Johnson solid. It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space.
In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are dimensions of 2 (polygon) or higher.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is oblique. A uniform triangular prism is a right triangular prism with equilateral bases, and square sides.
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.
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.
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with a polygonal base. Many types of pyramids can be found by determining the shape of bases, or cutting off the apex. It can be generalized into higher dimension, known as hyperpyramid. All pyramids are self-dual.
In geometry, a prismatic compound of antiprism is a category of uniform polyhedron compound. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry.
In geometry of 4 dimensions or higher, a double pyramid, duopyramid, or fusil is a polytope constructed by 2 orthogonal polytopes with edges connecting all pairs of vertices between the two. The term fusil is used by Norman Johnson as a rhombic-shape. The term duopyramid was used by George Olshevsky, as the dual of a duoprism.