This article includes a list of general references, but it lacks sufficient corresponding inline citations .(January 2013) |

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
- Right antiprism
- Uniform antiprism
- Schlegel diagrams
- Cartesian coordinates
- Volume and surface area
- Related polyhedra
- Symmetry
- Star antiprism
- See also
- References
- Bibliography
- 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.

At the intersection of modern-day graph theory and coding theory, the triangulation of a set of points have interested mathematicians since Isaac Newton, who fruitlessly sought a mathematical proof of the kissing number problem in 1694.^{ [2] } The existence of antiprisms 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 faces and on vertices as the Archimedean solids.^{[ citation needed ]} According to Ericson and Zinoviev, Harold Scott MacDonald Coxeter wrote at length on the topic,^{ [2] } and was among the first to apply the mathematics of Victor Schlegel to this field.

Knowledge in this field is "quite incomplete" and "was obtained fairly recently", i.e. in the 20th century^{[ citation needed ]}. For example, as of 2001 it had been proven for only a limited number of non-trivial cases that the n-gonal antiprism is the mathematically optimal arrangement of 2*n* points in the sense of maximizing the minimum Euclidean distance between any two points on the set: in 1943 by László Fejes Tóth for 4 and 6 points (digonal and trigonal antiprisms, which are Platonic solids); in 1951 by Kurt Schütte and Bartel Leendert van der Waerden for 8 points (tetragonal antiprism, which is not a cube).^{ [2] }

The chemical structure of binary compounds has been remarked to be in the family of antiprisms;^{ [3] } especially those of the family of boron hydrides (in 1975) and carboranes because they are isoelectronic. This is a mathematically real conclusion reached by studies of X-ray diffraction patterns,^{ [4] } and stems from the 1971 work of Kenneth Wade,^{ [5] } the nominative source for Wade's rules of polyhedral skeletal electron pair theory.

Rare-earth metals such as the lanthanides form antiprismatic compounds with some of the halides or some of the iodides. The study of crystallography is useful here.^{ [6] } Some lanthanides, when arranged in peculiar antiprismatic structures with chlorine and water, can form molecule-based magnets.^{ [7] }

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 regular tetrahedron as a *digonal antiprism* (degenerate antiprism); 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 | Octagonal antiprism | Enneagonal antiprism | Decagonal antiprism | Hendecagonal antiprism | Dodecagonal 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 | 8.3.3.3 | 9.3.3.3 | 10.3.3.3 | 11.3.3.3 | 12.3.3.3 | ... | ∞.3.3.3 |

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:

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 regular 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-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.

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, 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, 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 | ||||

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 ^{+},20](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 | |||

... | ... |

- Apeirogonal antiprism
- Grand antiprism – a four-dimensional polytope
- One World Trade Center, a building consisting primarily of an elongated square antiprism
- Skew polygon

A (symmetric) n-gonal **bipyramid** or **dipyramid** is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2*n* triangle faces, 3*n* edges, and 2 + *n* vertices.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular** octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

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, 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 **heptagon** or **septagon** is a seven-sided polygon or 7-gon.

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

In geometry, the **triangular bipyramid** is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces.

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, an **n-gonal****trapezohedron**, **n-trapezohedron**, **n-antidipyramid**, **n-antibipyramid**, or **n-deltohedron** is the dual polyhedron of an n-gonal antiprism. The **2***n* faces of an n-trapezohedron are congruent and symmetrically staggered; they are called *twisted kites*. With a higher symmetry, its 2*n* faces are *kites*.

The **triaugmented triangular prism**, in geometry, is a convex polyhedron with 14 equilateral triangles as its faces. It can be constructed from a triangular prism by attaching equilateral square pyramids to each of its three square faces. The same shape is also called the **tetrakis triangular prism**, **tricapped trigonal prism**, **tetracaidecadeltahedron**, or **tetrakaidecadeltahedron**; these last names mean a polyhedron with 14 triangular faces. It is an example of a deltahedron and of a Johnson solid.

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

In geometry, the **square antiprism** is the second in an infinite family of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an *anticube*.

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 polygonal base. A pyramid with an n-sided base has *n* + 1 vertices, *n* + 1 faces, and 2*n* edges. 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, an **icosahedron** is a polyhedron with 20 faces. The name comes from Ancient Greek εἴκοσι* (eíkosi)* 'twenty' and from Ancient Greek ἕδρα* (hédra)* ' seat'. The plural can be either "icosahedra" or "icosahedrons".

- ↑ 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 - 1 2 3 Ericson, Thomas; Zinoviev, Victor (2001). "Codes in dimension n = 3".
*Codes on Euclidean Spheres*. North-Holland Mathematical Library. Vol. 63. pp. 67–106. doi:10.1016/S0924-6509(01)80048-9. ISBN 9780444503299. - ↑ Beall, Herbert; Gaines, Donald F. (2003). "Boron Hydrides".
*Encyclopedia of Physical Science and Technology*. pp. 301–316. doi:10.1016/B0-12-227410-5/00073-9. ISBN 9780122274107. - ↑ “Boron Hydride Chemistry” (E. L. Muetterties, ed.), Academic Press, New York
- ↑ Wade, K. (1971). "The structural significance of the number of skeletal bonding electron-pairs in carboranes, the higher boranes and borane anions, and various transition-metal carbonyl cluster compounds".
*J. Chem. Soc. D*.**1971**(15): 792–793. doi:10.1039/C29710000792. - ↑ Meyer, Gerd (2014). [10.1016/B978-0-444-63256-2.00264-3 "Symbiosis of Intermetallic and Salt"].
*Including Actinides*. Handbook on the Physics and Chemistry of Rare Earths. Vol. 45. pp. 111–178. doi:10.1016/B978-0-444-63256-2.00264-3. ISBN 9780444632562.`{{cite book}}`

: Check`|chapter-url=`

value (help) - ↑ Bartolomé, Elena; Arauzo, Ana; Luzón, Javier; Bartolomé, Juan; Bartolomé, Fernando (2017).
*Magnetic Relaxation of Lanthanide-Based Molecular Magnets*. Handbook of Magnetic Materials. Vol. 26. pp. 1–289. doi:10.1016/bs.hmm.2017.09.002. ISBN 9780444639271.

- 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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