View source for Square antiprism

{{Short description|Solid with 10 faces}}
{{Prism polyhedra db|Prism polyhedron stat table|AP4}}
{{hatnote|Not to be confused with [[Square antiprismatic molecular geometry]] and [[Square antiprismatic prism]]}}
[[File:Square antiprism.stl|thumb|3D model of a (uniform) square antiprism]]
In [[geometry]], the '''square antiprism''' is the second in an infinite family of [[antiprism]]s formed by an [[even number|even-numbered]] sequence of triangle sides closed by two [[polygon]] caps. It is also known as an ''anticube''.<ref>Holleman-Wiberg. ''Inorganic Chemistry'', Academic Press, Italy, p.&nbsp;299. {{isbn|0-12-352651-5}}.</ref>

If all its faces are [[regular polygon|regular]], it is a [[semiregular polyhedron]] or [[uniform polyhedron]].

A nonuniform ''D''<sub>4</sub>-symmetric variant is the cell of the [[Noble polyhedron|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 (geometry)|cube]]. Specific methods of distributing the points include, for example, the [[Thomson problem]] (minimizing the sum of all the [[reciprocal (mathematics)|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|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]] ({{chem|XeF|8|2-}}) in the salt [[nitrosonium octafluoroxenate(VI)]]; however, the molecule is distorted away from the idealized square antiprism.<ref name="peterson">{{Cite journal | first3 = A.| first2 = H. | first4 = M.| title = Antiprismatic Coordination about Xenon: the Structure of Nitrosonium Octafluoroxenate(VI)| first1 = W.| last2 = Holloway| last3 = Coyle| volume = 173| journal = Science| issue = 4003| pages = 1238–1239| issn = 0036-8075| doi = 10.1126/science.173.4003.1238| last4 = Williams| pmid = 17775218| last1 = Peterson| date = Sep 1971|bibcode = 1971Sci...173.1238P | s2cid = 22384146 }}</ref> Very few ions are cubical because such a shape would cause large repulsion between [[ligand]]s; {{chem|Pa|F|8|3-}} is one of the few examples.<ref>{{Greenwood&Earnshaw2nd|p=1275}}</ref>

In addition, the element [[sulfur]] forms octatomic S<sub>8</sub> molecules as its most stable [[Allotropy|allotrope]]. The S<sub>8</sub> 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 (1973–2001)|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 [[tetrahedron]]s excavated around the sides. However, after this it can no longer be triangulated into tetrahedra without adding new vertices. It has half of the [[List of spherical symmetry groups#Dihedral symmetry|symmetry]] of the uniform solution: D<sub>4</sub> group of order 8.<ref>The facts on file: Geometry handbook, Catherine A. Gorini, 2003, {{isbn|0-8160-4875-4}}, p.172</ref><ref>{{Cite web|url=http://www.korthalsaltes.com/cuadros.php?type=twisted-prisms|title=Pictures of Twisted Prisms}}</ref>
: [[File:Twisted square antiprism.png|frameless]]

=== Crossed antiprism===
A '''crossed square antiprism''' is a [[star polyhedron]], [[topology|topologically]] identical to the ''square antiprism'' with the same [[vertex arrangement]], but it cannot be made uniform; the sides are [[isosceles triangle]]s. Its [[vertex configuration]] is 3.3/2.3.4, with one triangle retrograde. It has D<sub>4d</sub> symmetry group of order 16.

:[[File:Crossed square antiprism.png|frameless]]

== Related polyhedra ==

===Derived polyhedra===
The [[gyroelongated square pyramid]] is a [[Johnson solid]] (''J''<sub>10</sub>) constructed by augmenting one square face with a [[square pyramid]]. Similarly, the [[gyroelongated square bipyramid]] (''J''<sub>17</sub>) is a [[deltahedron]] (a [[polyhedron]] whose [[face (geometry)|face]]s are all [[equilateral triangle]]s) constructed by replacing both squares of a square antiprism with a square pyramid.

The [[snub disphenoid]] (''J''<sub>84</sub>) is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles.

The [[snub square antiprism]] (''J''<sub>85</sub>) can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle, or as an alternated truncated square antiprism:
{| class=wikitable
|+ Snub antiprisms
!Antiprism
![[Truncation (geometry)|Truncated]]<br />t
![[Alternation (geometry)|Alternated]]<br />ht
|- align=center
|[[File:Square antiprism.png|120px]]<br />[[pentagonal antiprism|s{2,8}]]<br />{{CDD|node_h|2x|node_h|8|node}}
|[[File:Truncated_square_antiprism.png|120px]]<br />ts{2,8}
|[[File:Snub_square_antiprism_colored.png|120px]]<br />ss{2,8}
|}

The [[sphenocorona]] (''J''<sub>86</sub>) and the [[sphenomegacorona]] (''J''<sub>88</sub>) are other Johnson solids that, like the square antiprism, consist of two squares and an even number of equilateral triangles.

=== 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]].

{{UniformAntiprisms}}

The ''square antiprism'' is first in a series of [[snub polyhedron|snub polyhedra]] and tilings with [[vertex figure]] 3.3.4.3.''n''.
{{Snub4 table}}

==Euclidean geometry==
If <math>a</math> denotes the side length of each of the two squares, and <math>h</math> the (vertical) distance between the two squares, one can embed the 8 corners into Euclidean geometry by placing the 4 corners of the top square at Cartesian coordinates <math>(\pm a/2, \pm a/2, h/2)</math> and the 4 corners of the bottom square—rotated by 45 degrees around the vertical axis relative to the top square—at <math>(\pm a/\surd 2 ,0,-h/2)</math> and <math>(0,\pm a/\surd 2,-h/2)</math>.

The distance of each of these 8 corners to the center of coordinates is <math>\sqrt{2a^2+h^2}/2</math>.

If the length of the edges of the 8 triangular sides are isosceles, also equal to <math>a</math>, <math>h</math> must be set to <math>h=a/\sqrt[4]{2}</math>, and the antiprism is equilateral. The distance of each of the 8 corners to the center of coordinates then is the radius of the circumsphere, <math>r= \sqrt{8+2^{3/2}}a/4\approx 0.82266 a</math>. By building dot products between vectors from the center to pairs of corners one finds that the smallest angle <math>\phi</math> at which edges appear from the center is <math>\cos\phi = 1/(1+2^{3/2})</math>, which gives <math>\phi\approx 1.30652 \approx 74.858^\circ</math>. (This is the analog to the tetrahedral bond angle of the regular tetrahedron.)

== Examples ==
<gallery class="center" mode="nolines" widths="200" heights="200">
File:Square-antiprismatic-3D-balls.png|{{center|[[Square antiprismatic molecular geometry]]}}
File:OneWorldTradeCenter.jpg|{{center|[[One World Trade Center]] building}}
File:Antiprisma Base Quadrada.jpg|{{center|Square antiprism<br />(at [[:pt:Matemateca IME-USP|Matemateca Ime-USP]])}}
File:Antiprisma quadrado torcido.jpg|{{center|Snub square antiprism<br />(at [[:pt:Matemateca IME-USP|Matemateca IME-USP]])}}
</gallery>

== See also ==
* [[Biscornu]]

== Notes ==
{{reflist}}

== External links ==
* {{mathworld | urlname = Antiprism | title = Antiprism }}
* [https://web.archive.org/web/20070711044031/http://polyhedra.org/poly/show/27/square_antiprism Square Antiprism] interactive model
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
** [[VRML]] [https://web.archive.org/web/20061017014204/http://www.georgehart.com/virtual-polyhedra/vrml/square_antiprism.wrl model]
* [http://levskaya.github.io/polyhedronisme/?recipe=C100A4 polyHédronisme] A4

{{Polyhedron navigator}}

[[Category:Prismatoid polyhedra]]
[[Category:Snub tilings]]