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Set of uniform n-gonal antiprisms
Hexagonal antiprism.png
Uniform hexagonal antiprism (n = 6)
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]
Conway notation An
Coxeter diagram CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel n.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel n.pngCDel node h.png
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
Generalized antiprisim net.svg
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  [ de ]. [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]

Special cases

Right antiprism

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.

Uniform antiprism

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).

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 Digonal antiprism.png Trigonal antiprism.png Square antiprism.png Pentagonal antiprism.png Hexagonal antiprism.png Antiprism 7.png ...
Spherical tiling image Spherical digonal antiprism with digonal face.svg Spherical trigonal antiprism.svg Spherical square antiprism.svg Spherical pentagonal antiprism.svg Spherical hexagonal antiprism.svg Spherical heptagonal antiprism.svg Plane tiling image Infinite antiprism.svg
Vertex config.∞.3.3.3

The Schlegel diagrams of these semiregular antiprisms are as follows:

Triangular antiprismatic graph.png
Square antiprismatic graph.png
Pentagonal antiprismatic graph.png
Hexagonal antiprismatic graph.png
Heptagonal antiprism graph.png
Octagonal antiprismatic graph.png

Cartesian coordinates

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:

Volume and surface area

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.


In higher dimensions

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]

Self-crossing polyhedra

Pentagrammic antiprism.png
Pentagrammic crossed antiprism.png
Antiprism 9-2.png
Antiprism 9-4.png
Antiprism 9-5.png
This shows all the non-star and star antiprisms up to 15 sides, together with those of a 29-agon. Antiprisms.pdf
This shows all the non-star and star antiprisms up to 15 sides, together with those of a 29-agon.

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/(pq) 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.

See also

Related Research Articles

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:

<span class="mw-page-title-main">Prism (geometry)</span> Solid with 2 parallel n-gonal bases connected by n parallelograms

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.

<span class="mw-page-title-main">Octagon</span> Polygon shape with eight sides

In geometry, an octagon is an eight-sided polygon or 8-gon.

<span class="mw-page-title-main">Decagon</span> Shape with ten sides

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.

<span class="mw-page-title-main">Dodecagon</span> Polygon with 12 edges

In geometry, a dodecagon, or 12-gon, is any twelve-sided polygon.

<span class="mw-page-title-main">Cupola (geometry)</span> Solid made by joining an n- and 2n-gon with triangles and squares

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.

<span class="mw-page-title-main">Gyrobifastigium</span> 26th Johnson solid (8 faces)

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.

<span class="mw-page-title-main">Uniform 4-polytope</span> Class of 4-dimensional polytopes

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.

<span class="mw-page-title-main">Duoprism</span> Cartesian product of two polytopes

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.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

<span class="mw-page-title-main">Triangular prism</span> Prism with a 3-sided base

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.

<span class="mw-page-title-main">Pentagonal antiprism</span> Antiprism with a five-sided base

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.

<span class="mw-page-title-main">Hexagonal antiprism</span> Antiprism with 6-sided caps

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.

<span class="mw-page-title-main">Prismatic compound of antiprisms</span> Polyhedral compound

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.


  1. N.W. Johnson: Geometries and Transformations, (2018) ISBN   978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3c
  2. Kepler, Johannes (1619). "Book II, Definition X". Harmonices Mundi (in Latin). p. 49. See also illustration A, of a heptagonal antiprism.
  3. Schreiber, Peter; Fischer, Gisela; Sternath, Maria Luise (July 2008). "New light on the rediscovery of the Archimedean solids during the Renaissance". Archive for History of Exact Sciences. 62 (4): 457–467. JSTOR   41134285.
  4. Heinze, Karl (1886). Lucke, Franz (ed.). Genetische Stereometrie (in German). B. G. Teubner. p. 14.
  5. Smyth, Piazzi (1881). "XVII. On the Constitution of the Lines forming the Low-Temperature Spectrum of Oxygen". Transactions of the Royal Society of Edinburgh. 30 (1): 419–425. doi:10.1017/s0080456800029112.
  6. Coxeter, H. S. M. (January 1928). "The pure Archimedean polytopes in six and seven dimensions". Mathematical Proceedings of the Cambridge Philosophical Society. 24 (1): 1–9. doi:10.1017/s0305004100011786.
  7. Grünbaum, Branko (2005). "Are prisms and antiprisms really boring? (Part 3)" (PDF). Geombinatorics. 15 (2): 69–78. MR   2298896.
  8. Dobbins, Michael Gene (2017). "Antiprismlessness, or: reducing combinatorial equivalence to projective equivalence in realizability problems for polytopes". Discrete & Computational Geometry . 57 (4): 966–984. doi:10.1007/s00454-017-9874-y. MR   3639611.

Further reading