Uniform square antiprism
Square antiprism.png
Type Prismatic uniform polyhedron
Elements F = 10, E = 16
V = 8 (χ = 2)
Faces by sides 8{3}+2{4}
Schläfli symbol s{2,8}
sr{2,4}
Wythoff symbol | 2 2 4
Coxeter diagram CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png
Symmetry group D4d, [2+,8], (2*4), order 16
Rotation group D4, [4,2]+, (442), order 8
References U77(b)
Dual Tetragonal trapezohedron
Properties convex
Square antiprism vertfig.png

Vertex figure
3.3.3.4
3D model of a (uniform) square antiprism

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.[1]

If all its faces are regular, it is a semiregular polyhedron or uniform polyhedron.

A nonuniform D4-symmetric variant is the cell of the noble square antiprismatic 72-cell.

Points on a sphere

When eight points are distributed on the surface of a sphere with the aim of maximising the distance between them in some sense, the resulting shape corresponds to a square antiprism rather than a cube. Specific methods of distributing the points include, for example, the Thomson problem (minimizing the sum of all the reciprocals of distances between points), maximising the distance of each point to the nearest point, or minimising the sum of all reciprocals of squares of distances between points.

Molecules with square antiprismatic geometry

Main article: Square antiprismatic molecular geometry

According to the VSEPR theory of molecular geometry in chemistry, which is based on the general principle of maximizing the distances between points, a square antiprism is the favoured geometry when eight pairs of electrons surround a central atom. One molecule with this geometry is the octafluoroxenate(VI) ion (XeF2−
8
) in the salt nitrosonium octafluoroxenate(VI); however, the molecule is distorted away from the idealized square antiprism.[2] Very few ions are cubical because such a shape would cause large repulsion between ligands; PaF3−
8
is one of the few examples.[3]

In addition, the element sulfur forms octatomic S8 molecules as its most stable allotrope. The S8 molecule has a structure based on the square antiprism, in which the eight atoms occupy the eight vertices of the antiprism, and the eight triangle-triangle edges of the antiprism correspond to single covalent bonds between sulfur atoms.

In architecture

The main building block of the One World Trade Center (at the site of the old World Trade Center destroyed on September 11, 2001) has the shape of an extremely tall tapering square antiprism. It is not a true antiprism because of its taper: the top square has half the area of the bottom one.

Topologically identical polyhedra

Twisted prism

A twisted prism can be made (clockwise or counterclockwise) with the same vertex arrangement. It can be seen as the convex form with 4 tetrahedrons excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices. It has half of the symmetry of the uniform solution: D4 order 4.[4][5]

Twisted square antiprism.png

Crossed antiprism

A crossed square antiprism is a star polyhedron, topologically identical to the square antiprism with the same vertex arrangement, but it can't be made uniform; the sides are isosceles triangles. Its vertex configuration is 3.3/2.3.4, with one triangle retrograde. It has d4d symmetry, order 8.

Crossed square antiprism.png

Related polyhedra

Derived polyhedra

The gyroelongated square pyramid is a Johnson solid (specifically, J10) constructed by augmenting one a square pyramid. Similarly, the gyroelongated square bipyramid (J17) is a deltahedron (a polyhedron whose faces are all equilateral triangles) constructed by replacing both squares of a square antiprism with a square pyramid.

The snub disphenoid (J84) is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles. The snub square antiprism (J85) can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle. The sphenocorona (J86) and the sphenomegacorona (J88) are other Johnson solids that, like the square antiprism, consist of two squares and an even number of equilateral triangles.

The square antiprism can be truncated and alternated to form a snub antiprism:

Snub antiprisms
Antiprism Truncated
t
Alternated
ht
Square antiprism.png

s{2,8}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node.png
Truncated square antiprism.png

ts{2,8}
Snub square antiprism colored.png

ss{2,8}

Symmetry mutation

As an antiprism, the square antiprism belongs to a family of polyhedra that includes the octahedron (which can be seen as a triangle-capped antiprism), the pentagonal antiprism, the hexagonal antiprism, and the octagonal antiprism.

Family of uniform n-gonal antiprisms
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
Digonal antiprism.png
Trigonal antiprism.png
Square antiprism.png
Pentagonal antiprism.png
Hexagonal antiprism.png
Antiprism 7.png
Octagonal antiprism.png
Enneagonal antiprism.png
Decagonal antiprism.png
Hendecagonal antiprism.png
Dodecagonal antiprism.png
...
Spherical tiling image
Spherical digonal antiprism.png
Spherical trigonal antiprism.png
Spherical square antiprism.png
Spherical pentagonal antiprism.png
Spherical hexagonal antiprism.png
Spherical heptagonal antiprism.png
Spherical octagonal antiprism.png
Plane tiling image
Infinite antiprism.svg
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

The square antiprism is first in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
Spherical square antiprism.png
Spherical snub cube.png
Uniform tiling 44-snub.png
H2-5-4-snub.svg
Uniform tiling 64-snub.png
Uniform tiling 74-snub.png
Uniform tiling 84-snub.png
Uniform tiling i42-snub.png
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.∞
Gyro
figures
Spherical tetragonal trapezohedron.png
Spherical pentagonal icositetrahedron.png
Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
H2-5-4-floret.svg
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.∞

Examples

See also

Notes

  1. ^ Holleman-Wiberg. Inorganic Chemistry, Academic Press, Italy, p. 299. ISBN 0-12-352651-5.
  2. ^ Peterson, W.; Holloway, H.; Coyle, A.; Williams, M. (Sep 1971). "Antiprismatic Coordination about Xenon: the Structure of Nitrosonium Octafluoroxenate(VI)". Science. 173 (4003): 1238–1239. Bibcode:1971Sci...173.1238P. doi:10.1126/science.173.4003.1238. ISSN 0036-8075. PMID 17775218. S2CID 22384146.
  3. ^ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 1275. ISBN 978-0-08-037941-8.
  4. ^ The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172
  5. ^ "Pictures of Twisted Prisms".