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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*.

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, circumradius of the bases equal to 1) 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:

The circumradius of the horizontal circumcircle of the regular -gon at the base is

The vertices at the base are at

the vertices at the top are at

Via linear interpolation, points on the outer triangular edges of the antiprism that connect vertices at the bottom with vertices at the top are at

and at

By building the sums of the squares of the and coordinates in one of the previous two vectors, the squared circumradius of this section at altitude is

The horizontal section at altitude above the base is a -gon (truncated -gon) with sides of length alternating with sides of length . (These are derived from the length of the difference of the previous two vectors.) It can be dissected into isoceless triangles of edges and (semiperimeter ) plus isoceless triangles of edges and (semiperimeter ). According to Heron's formula the areas of these triangles are

and

The area of the section is , and the volume is

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 polychora that cannot be combined with their duals to form five-dimensional antiprisms.^{[8]}

3/2-antiprism nonuniform |
5/4-antiprism nonuniform |
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

**^**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

Convex polyhedra | |||||
---|---|---|---|---|---|

Platonic solids (regular) | |||||

- triakis tetrahedron
- rhombic dodecahedron
- triakis octahedron
- tetrakis hexahedron
- deltoidal icositetrahedron
- disdyakis dodecahedron
- pentagonal icositetrahedron
- rhombic triacontahedron
- triakis icosahedron
- pentakis dodecahedron
- deltoidal hexecontahedron
- disdyakis triacontahedron
- pentagonal hexecontahedron
| |||||

Dihedral regular | |||||

Dihedral uniform |
| ||||

Dihedral others | |||||

Degenerate polyhedra are in italics. |