Antiprism

Antiprism with a hexagon as the base

octahedron

An antiprism (plural: antiprism ) is a geometric body that has a polygon as a base and a similar polygon as a top surface, which are parallel to each other but twisted. These two surfaces are connected by a circumferential band of triangular surfaces , the tips of which point alternately to the base and top surface.

Antiprisms are special polyhedra . They resemble prisms and, like them, belong to the prismatoid group . In contrast to prisms, however, they have twice as many lateral surfaces (with the same number of corners) and the lateral surfaces are triangles instead of parallelograms .

Special antiprisms

In the case of straight antiprisms , the base and top surface are regular n -gons and 2n isosceles triangles form the jacket. The centers of the base and top surfaces are exactly one above the other and the top surface is rotated by the angle with respect to the base surface. With straight antiprisms all corners are of the same type. The dual polyhedra to straight antiprisms are trapezoids . ${\displaystyle {\tfrac {180^{\circ }}{n}}}$

A uniform antiprism also has the condition that its lateral surfaces are equilateral triangles . An example of this is the octahedron , which can (also) be described as a uniform antiprism with a triangular base.

The diagonal antiprism, also called disphenoid , is a degenerate special case in which the base and top surface are reduced to a two-cornered shape . A uniform diagonal antiprism is a regular tetrahedron .

Examples of uniform antiprisms:

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  Diagonal (two-cornered) antiprism: tetrahedron

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  Trigonal (triangular) antiprism: octahedron

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  Tetragonal (square) antiprism

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  Pentagonal (pentagonal) antiprism
  s. a. Icosahedron

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  Hexagonal (hexagonal) antiprism

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  Heptagonal (heptagonal) antiprism

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  Octagonal (octagonal) antiprism

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  Nonagonal (nine-cornered) antiprism

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  Decagonal (decagonal) antiprism

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  Decagonal (eleven) antiprism

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  Dodecagonal (twelve-sided) antiprism

symmetry

If an antiprism has a regular corner as its base and the top surface is rotated by the angle with respect to the base, the antiprism is point-symmetrical if and only if it is odd. ${\displaystyle n}$ ${\displaystyle {\tfrac {180^{\circ }}{n}}}$ ${\displaystyle n}$

Examples:

• The octahedron with an odd number of corner points on the base side (triangle) is point-symmetrical.
• The tetrahedron with an even number of corner points on the base side (triangle) is not point-symmetrical.

Cartesian coordinates

Cartesian coordinates for the corners of a straight antiprism with an n -angular base and isosceles triangles are

${\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)}$

where k takes values ​​from 0 to 2 n −1; if the triangles are equilateral, is

${\displaystyle 2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.}$

Volume and surface

The edge length of a uniform n -angular antiprism is given by a . Then its volume

${\displaystyle V={\frac {n{\sqrt {4\cos ^{2}{\frac {\pi }{2n}}-1}}\sin {\frac {3\pi }{2n}}}{12\sin ^{2}{\frac {\pi }{n}}}}\;a^{3}}$

and its surface

${\displaystyle A={\frac {n}{2}}\left(\cot {\frac {\pi }{n}}+{\sqrt {3}}\right)a^{2}.}$

Umkugel

The radius of the sphere with a given height and given circumferential radius of the base area is calculated according to the Pythagorean theorem : ${\displaystyle R}$ ${\displaystyle h}$ ${\displaystyle r_{u}}$

${\displaystyle R={\sqrt {r_{u}^{2}+{\frac {h^{2}}{4}}}}}$

(Because of its four corners, every irregular tetrahedron also has a sphere. However, this pathological prism is ignored here.)

Edge ball

Only uniform antiprisms have an edge sphere . The radius of the edge sphere at a given height and a given incircle radius is calculated according to the Pythagorean theorem: ${\displaystyle R}$ ${\displaystyle h}$ ${\displaystyle r_{i}}$

${\displaystyle R={\sqrt {r_{i}^{2}+{\frac {h^{2}}{4}}}}}$

Incugel

Because of its four faces, each quadruple has an inc sphere .

Otherwise, only straight antiprisms, the height of which is twice the radius of the incugel, have an incugel. For these bodies, the radius of the incugel for a given circumferential radius and a given incircle radius of the base or top surface is: ${\displaystyle R}$ ${\displaystyle r_{u}}$ ${\displaystyle r_{i}}$

${\displaystyle R={\sqrt {r_{i}\cdot r_{u}}}}$

An antiprism with an inc sphere generally consists of isosceles triangles as lateral surfaces. Only the octahedron (next to the regular tetrahedron) is the only antiprism with equilateral triangles as lateral surfaces.

network

There are various possibilities for the network of antiprisms. One is to first define a band of triangles for the jacket and then to attach the base and top surface to this band. The following graphics show two other possibilities.

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  Network of a straight / uniform anti-prism

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  Another way of building the network

Occurrence

Antiprisms are found, among other things, as coordination polyhedra in crystal structures and complex compounds , as capsids of viruses , and in roller cubes .

Scaffolding made of rods or tubes, which form the edges of anti-prisms, are much more torsion-resistant and kink-resistant around the main axis than corresponding prisms, which is why they are often used as platforms in the circus.

The antiprism is occasionally used in architecture. The One World Trade Center in New York City mainly has the shape of an elongated square anti-prism, although the upper square is less than the lower. The Diagonal Tower in Seoul consists of three square antiprisms placed one on top of the other. Its design is intended to reduce the wind forces that arise and, at the same time, save material.

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  8-sided anti-prism dice

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  The One World Trade Center

Web links

Commons : Antiprisms  - collection of images, videos and audio files

Wiktionary: Antiprisma  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Antiprism . In: MathWorld (English).

Individual evidence

1. ↑ SOM: diagonal tower yongsan international business district
2. ^ Diagonal Tower, Yongsan International Business District