geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the square antiprism is the second in an infinite family of

antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...

s formed by an even-numbered sequence of triangle sides closed by two

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...

caps. It is also known as an ''anticube''. If all its faces are regular, it is a

semiregular polyhedron In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...

uniform polyhedron In geometry, a uniform polyhedron has regular polygons as Face (geometry), faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruence (geometry), congruent. Unifor ...

. A nonuniform ''D''₄-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 A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

with the aim of maximising the distance between them in some sense, the resulting shape corresponds to a square antiprism rather than a

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...

. Specific methods of distributing the points include, for example, the

Thomson problem The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. ...

(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

According to the

VSEPR theory Valence shell electron pair repulsion (VSEPR) theory ( , ), is a model used in chemistry to predict the geometry of individual molecules from the number of electron pairs surrounding their central atoms. It is also named the Gillespie-Nyholm the ...

molecular geometry Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that dete ...

chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...

, 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 Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...

. One

molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...

with this geometry is the octafluoroxenate(VI) ion () in the salt nitrosonium octafluoroxenate(VI); however, the molecule is distorted away from the idealized square antiprism. Very few ions are cubical because such a shape would cause large repulsion between

ligand In coordination chemistry, a ligand is an ion or molecule (functional group) that binds to a central metal atom to form a coordination complex. The bonding with the metal generally involves formal donation of one or more of the ligand's elect ...

s; is one of the few examples. In addition, the element

sulfur Sulfur (or sulphur in British English) is a chemical element with the symbol S and atomic number 16. It is abundant, multivalent and nonmetallic. Under normal conditions, sulfur atoms form cyclic octatomic molecules with a chemical formul ...

forms octatomic S₈ molecules as its most stable

allotrope Allotropy or allotropism () is the property of some chemical elements to exist in two or more different forms, in the same physical State of matter, state, known as allotropes of the elements. Allotropes are different structural modifications o ...

. 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 One World Trade Center (also known as One World Trade, One WTC, and formerly Freedom Tower) is the main building of the rebuilt World Trade Center complex in Lower Manhattan, New York City. Designed by David Childs of Skidmore, Owings & Mer ...

(at the site of the old World Trade Center destroyed on

September 11, 2001 The September 11 attacks, commonly known as 9/11, were four coordinated suicide terrorist attacks carried out by al-Qaeda against the United States on Tuesday, September 11, 2001. That morning, nineteen terrorists hijacked four commerc ...

) 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 Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an op ...

of the bottom one.

Topologically identical polyhedra

Twisted prism

A twisted prism can be made (clockwise or counterclockwise) with the same

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

. It can be seen as the convex form with 4

tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all ...

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

symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

of the uniform solution: D₄ order 4. : Twisted square antiprism

Crossed antiprism

A crossed square antiprism is a

star polyhedron In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave p ...

, topologically identical to the ''square antiprism'' with the same

, but it can't be made uniform; the sides are

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

s. Its vertex configuration is 3.3/2.3.4, with one triangle retrograde. It has d_4d symmetry, order 8. : Crossed square antiprism

Related polyhedra

Derived polyhedra

The

gyroelongated square pyramid In geometry, the gyroelongated square pyramid is one of the Johnson solids (). As its name suggests, it can be constructed by taking a square pyramid and "gyroelongating" it, which in this case involves joining a square antiprism to its base. ...

is a

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johnso ...

(specifically, ''J''₁₀) constructed by augmenting one a

square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyrami ...

. Similarly, the gyroelongated square bipyramid (''J''₁₇) is a deltahedron (a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...

whose

face The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...

s are all

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each oth ...

s) constructed by replacing both squares of a square antiprism with a square pyramid. The

snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some ve ...

(''J''₈₄) is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles. The snub square antiprism (''J''₈₅) can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle. The sphenocorona (''J''₈₆) and the sphenomegacorona (''J''₈₈) 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:

Symmetry mutation

As an

, the square antiprism belongs to a family of polyhedra that includes the

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...

(which can be seen as a triangle-capped antiprism), the

pentagonal antiprism In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for ...

, the hexagonal antiprism, and the

octagonal antiprism In geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. Antiprisms are similar to prisms except the bases are twisted relative to each other ...

. The ''square antiprism'' is first in a series of snub polyhedra and tilings with

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

3.3.4.3.''n''.

Examples

File:Square-antiprismatic-3D-balls.png, File:OneWorldTradeCenter.jpg, File:Antiprisma Base Quadrada.jpg, File:Antiprisma quadrado torcido.jpg,

Notes

External links

*
Square Antiprism
interactive model

www.georgehart.com: The Encyclopedia of Polyhedra **

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graph ...

br>model

polyhedronisme
A4 {{Polyhedron navigator Prismatoid polyhedra Snub tilings