Square antiprism

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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

In geometry, the square antiprism is the second in an infinite set 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.

Points on a sphere[edit]

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[edit]

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 (XeF82−) 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[edit]

One World Trade Center

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[edit]

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: Dn, [4,2]+, order 8.[4][5]

SquareAntiprism.png

Related polyhedra[edit]

Derived polyhedra[edit]

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[edit]

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.

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

See also[edit]

Notes[edit]

  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. 
  3. ^ Greenwood, Norman N.; Earnshaw, Alan (1997). Chemistry of the Elements (2nd ed.). Butterworth-Heinemann. p. 1275. ISBN 0-08-037941-9. 
  4. ^ The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172
  5. ^ [1]

External links[edit]