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Set of uniform n-gonal antiprisms | |
---|---|

Type | uniform in the sense of semiregular polyhedron |

Faces | 2 regular n-gons 2 n equilateral triangles |

Edges | 4n |

Vertices | 2n |

Vertex configuration | 3.3.3.n |

Schläfli symbol | { }⊗{n} ^{ [1] }s{2,2 n}sr{2, n} |

Conway notation | An |

Coxeter diagram | |

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, 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 2*n* triangles. They are represented by the Conway notation A*n*.

- History
- Special cases
- Right antiprism
- Uniform antiprism
- Cartesian coordinates
- Volume and surface area
- Symmetry
- Generalizations
- In higher dimensions
- Self-crossing polyhedra
- See also
- References
- Further reading
- External links

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 2*n* 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 *regular*n-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 2*n* side faces are *isosceles* triangles.

A ** uniform n-antiprism** has two congruent * regular* n-gons as base faces, and 2

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 2*n* isosceles triangle side faces) are:

where 0 ≤ *k* ≤ 2*n* – 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 D_{nd} = D_{nv} of order 4*n*, except in the cases of:

*n*= 2: the regular tetrahedron, which has the larger symmetry group T_{d}of order 24 = 3 × (4 × 2), which has three versions of D_{2d}as subgroups;

*n*= 3: the regular octahedron, which has the larger symmetry group O_{h}of order 48 = 4 × (4 × 3), 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 2*n*, except in the cases of:

*n*= 2: the regular tetrahedron, which has the larger rotation group T of order 12 = 3 × (2 × 2), which has three versions of D_{2}as subgroups;

*n*= 3: the regular octahedron, which has the larger rotation group O of order 24 = 4 × (2 × 3), which has four versions of D_{3}as subgroups.

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

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:

- Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, and so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron.

- Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, and so cannot be uniform. Example: a retrograde star antiprism with regular star 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.

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 | |||
---|---|---|---|---|---|

D_{4d}[2 ^{+},8](2*4) | 3.3/2.3.4 Crossed square antiprism | ||||

D_{5h}[2,5] (*225) | 3.3.3.5/2 Pentagrammic antiprism | 3.3/2.3.5 crossed pentagonal antiprism | |||

D_{5d}[2 ^{+},10](2*5) | 3.3.3.5/3 Pentagrammic crossed-antiprism | ||||

D_{6d}[2 ^{+},12](2*6) | 3.3/2.3.6 crossed hexagonal antiprism | ||||

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 Octagrammic antiprism | 3.3.3.8/5 Octagrammic crossed-antiprism | |||

D_{9h}[2,9] (*229) | 3.3.3.9/2 Enneagrammic antiprism (9/2) | 3.3.3.9/4 Enneagrammic antiprism (9/4) | |||

D_{9d}[2 ^{+},18](2*9) | 3.3.3.9/5 Enneagrammic crossed-antiprism | ||||

D_{10d}[2 ^{+},20](2*10) | 3.3.3.10/3 Decagrammic antiprism | ||||

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 | |||

... | ... |

- Grand antiprism, a four-dimensional polytope
- Skew polygon, a three-dimensional polygon whose convex hull is an antiprism

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.

- ↑ 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 - ↑ Kepler, Johannes (1619). "Book II, Definition X".
*Harmonices Mundi*(in Latin). p. 49. See also illustration A, of a heptagonal antiprism. - ↑ 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. - ↑ Heinze, Karl (1886). Lucke, Franz (ed.).
*Genetische Stereometrie*(in German). B. G. Teubner. p. 14. - ↑ 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. - ↑ 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. - ↑ Grünbaum, Branko (2005). "Are prisms and antiprisms really boring? (Part 3)" (PDF).
*Geombinatorics*.**15**(2): 69–78. MR 2298896. - ↑ 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.

- Anthony Pugh (1976).
*Polyhedra: A visual approach*. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2: Archimedean polyhedra, prisms and antiprisms

- Media related to Antiprisms at Wikimedia Commons
- Weisstein, Eric W. "Antiprism".
*MathWorld*. - Nonconvex Prisms and Antiprisms
- Paper models of prisms and antiprisms

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