Square anti - prism is the second polyhedron in an endless series of anti-prisms , formed by a sequence of triangular faces, closed on both sides by polygons. Square antiprism is also known as anti-cube [1] .
| Uniform square antiprism | ||
|---|---|---|
 Square antiprism | ||
| Type of | Prismatic homogeneous polyhedron | |
| Properties | convex polyhedron | |
| Combinatorics | ||
| Items |
| |
| Verge of | 8 triangles 2 squares | |
| Vertex configuration | 3.3.3.4 | |
| Dual Polyhedron | ||
Scan
| ||
| Classification | ||
| SchlΓ€fli Symbol | s {2,8} sr {2,4} | |
| | 2 2 4 | ||
| Dynkin diagram |      | |
| Symmetry group | D 4 , [4,2] + , (442), order = 8 | |
If all faces are regular polygons , the antiprism is a semi-regular polyhedron or a .
Content
Points on a sphere
If eight points are placed on a sphere in order to maximize the distances between them in a certain sense, the resulting figure corresponds to a square anti-prism rather than a cube . Specific methods for the distribution of points include, for example, (minimizing the sum of reciprocals of distances between points), maximizing the distances from the point to the nearest point, or minimizing the sum of all inverse squares of the distances between points.
Molecules with square anti-prismatic geometry
According to the theory of OEPVO in chemistry, which is based on the principle of maximizing the distances between points, square antiprism is the most preferable geometry if eight pairs of electrons surround the central atom. One of the molecules with such a geometry is the octafluoroxenate (VI) ion (XeF 8 2β ) in the salt of the . However, this molecule is far from the ideal square anti-prism [2] . Very few ions have a cubic form, since such a form would lead to a strong repulsion of the ligands . PaF 8 3β is one of the few examples [3] .
In addition, sulfur forms eight-atomic molecules S 8 as the most stable allotropic form . The S 8 molecule has a structure based on square anti-prism. In this molecule, the atoms occupy eight vertices of the anti-prism, and the eight edges between the edges correspond to the covalent bond between the sulfur atoms.
In architecture
The main building in the complex of the World Trade Center (in the place of the old World Trade Center , destroyed on September 11, 2001 ) has the form of a very high square anti-prism narrowing to the top. The building is not a true anti-prism, because it narrows towards the top - the upper square has twice the area compared to the base.
Topologically equivalent polyhedra
A twisted prism (clockwise or counterclockwise) can have the same vertex arrangement. This polyhedron can be considered as a form assembled from 4 tetrahedra with cut out parts. However, after cutting, the body cannot be divided into tetrahedra without adding new vertices. The body has half the symmetries of a homogeneous body: D n , [4,2] + [4] [5] .
Related polyhedra
Derived polyhedra
The twisted elongated quadrangular pyramid is a regular-faced polyhedron ( J 10 = Π 2 + Π 4 ), obtained by lengthening a square pyramid . Similarly, a twisted elongated quadrangular bipyramid ( J 17 = M 2 + A 4 + M 2 ) is a deltahedron (a polyhedron whose faces are regular triangles ), constructed by replacing both squares of a square anti-prism with square pyramids.
The flat-headed dopate ( J 84 = M 25 ) is another deltahedron, which is obtained by replacing two squares of square anti-prism with pairs of equilateral triangles. A flat-square square antiprism ( J 85 = M 28 ) can be considered as a square anti-prism, obtained by inserting a chain of equilateral triangles. The clino-crown ( J 86 = M 21 ) and the large clino-crown ( J 88 = M 23 ) are other regular- faced polyhedra, which, like other square antiprisms, consist of two squares and an even number of equilateral triangles.
Square antiprism can be truncated and alternated to form flat-fisted antiprism :
| Antiprism | Truncation t | ht |
|---|---|---|
| s {2,8} | ts {2,8} | ss {2,8} |
Similar polyhedra
Being anti-prism , square anti-prism belongs to the family of polyhedrons, which include an octahedron (which can be considered as a triangular anti-prism), pentagonal anti-prism , hexagonal anti-prism, and .
| Polyhedron | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Mosaic | ||||||||||||
| Configuration | V2.3.3.3 | 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 | ... β.3.3.3 |
Square antiprism is the first in a series of flat-nosed polyhedra and mosaics with a vertex figure 3.3.4.3. n .
| 4 n 2 symmetries of flat- nose mosaics: 3.3.4.3.n | ||||||||
|---|---|---|---|---|---|---|---|---|
| Symmetry 4 n 2 | Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||
| 242 | 342 | 442 | 542 | 642 | 742 | 842 | β42 | |
| Flat-footed mosaics | ||||||||
| Config | 3.3.4.3.2 | 3.3.4.3.3 | 3.3.4.3.4 | 3.3.4.3.5 | 3.3.4.3.6 | 3.3.4.3.7 | 3.3.4.3.8 | 3.3.4.3.β |
| Gyro- mosaics | ||||||||
| Config | V3.3.4.3.2 | V3.3.4.3.3 | V3.3.4.3.4 | V3.3.4.3.5 | V3.3.4.3.6 | V3.3.4.3.7 | V3.3.4.3.8 | V3.3.4.3.β |
See also
Notes
- β Holleman-Wiberg, 2001 , p. 299.
- β Peterson, Holloway, Coyle, Williams, 1971 , p. 1238β1239.
- β Norman & Earnshaw, 1997 , p. 1275.
- β Gorini, 2003 , p. 172.
- β Drawings of twisted prisms and antiprisms
Literature
- Inorganic Chemistry / AF Holleman, Nils Wiberg, Egon Wiberg. - Academic Press, 2001. - ISBN 0-12-352651-5 .
- W. Peterson, A. Holloway, H. Coyle, M. Williams. Antiprismatic Coordination about Xenon: the Structure of Nitrosonium Octafluoroxenate (VI) // Science. - 1971. - V. 173 , no. 4003 . - ISSN 0036-8075 . - DOI : 10.1126 / science.173.4003.1238 . - . - PMID 17775218 .
- Catherine A. Gorini. The Facts on File Geometry Handbook. - New York: Facts On File, Inc., 2003. - ISBN 0-8160-4875-4 .
- Norman N. Greenwood, Alan Earnshaw. Chemistry of the Elements (2nd ed.). - Butterworth-Heinemann, 1997. - ISBN 0-08-037941-9 .
Links
- Weisstein, Eric W. Antiprism (English) on the Wolfram MathWorld website.
- Square Antiprism Interactive Model
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- VRML model
- Conway Notation for Polyhedra Try: "A4"