Square antiprism
Encyclopedia
In geometry
, the square antiprism is the second in an infinite set of antiprism
s formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube.
If all its faces are regular, it is a semiregular polyhedron
.
When eight points are distributed on the surface of a sphere with the aim of maximising the distance between them in some sense, then the resulting shape corresponds to a square anti-prism rather than a cube. Different examples include maximising the distance to the nearest point, or using electrons to maximise the sum of all reciprocals of squares of distances.
of molecular geometry
in chemistry, which is based on the general principle of maximizing the distances between points, a square antiprism is a 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.
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.
, 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 gyroelongated square pyramid
is a Johnson solid
(specifically, J10) constructed by replacing one square of a square antiprism with a square pyramid
. Similarly, the gyroelongated square bipyramid (J17) is a deltahedron
(a polyhedron
whose face
s 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.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the square antiprism is the second in an infinite set of antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
s formed by an even-numbered sequence of triangle sides closed by two polygon caps. It is also known as an anticube.
If all its faces are regular, it is a semiregular polyhedron
Semiregular polyhedron
The term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...
.
When eight points are distributed on the surface of a sphere with the aim of maximising the distance between them in some sense, then the resulting shape corresponds to a square anti-prism rather than a cube. Different examples include maximising the distance to the nearest point, or using electrons to maximise the sum of all reciprocals of squares of distances.
Molecules with square antiprism geometry
According to the VSEPR theoryVSEPR theory
Valence shell electron pair repulsion theory is a model in chemistry used to predict the shape of individual molecules based upon the extent of electron-pair electrostatic repulsion. It is also named Gillespie–Nyholm theory after its two main developers...
of molecular geometry
Molecular geometry
Molecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...
in chemistry, which is based on the general principle of maximizing the distances between points, a square antiprism is a 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)
Nitrosonium octafluoroxenate(VI)
Nitrosonium octafluoroxenate is a chemical compound of xenon with nitrogen, oxygen, and fluorine, having formula . It is an ionic compound containing well-separated nitrosonium cations and octafluoroxenate anions...
; however, the molecule is distorted away from the idealized square antiprism.
In addition, the element sulfur
Sulfur
Sulfur or sulphur is the chemical element with atomic number 16. In the periodic table it is represented by the symbol S. It is an abundant, multivalent non-metal. Under normal conditions, sulfur atoms form cyclic octatomic molecules with chemical formula S8. Elemental sulfur is a bright yellow...
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.
Related polyhedra
As an antiprismAntiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
, the square antiprism belongs to a family of polyhedra that includes the octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
(which can be seen as a triangle-capped antiprism), the pentagonal antiprism
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 a total of 12 faces...
, the hexagonal antiprism
Hexagonal antiprism
In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...
, and the octagonal antiprism
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.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...
.
The gyroelongated square pyramid
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
Johnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
(specifically, J10) constructed by replacing one square of a square antiprism with a square pyramid
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 will have C4v symmetry.- Johnson solid :...
. Similarly, the gyroelongated square bipyramid (J17) is a deltahedron
Deltahedron
A deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
(a polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
whose face
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
s are all equilateral triangles) constructed by replacing both squares of a square antiprism with a square pyramid.
The snub disphenoid
Snub disphenoid
In geometry, the snub disphenoid is one of the Johnson solids . It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron. It is not a regular polyhedron because some vertices have four faces and others have five...
(J84) is another deltahedron, constructed by replacing the two squares of a square antiprism by pairs of equilateral triangles. The snub square antiprism
Snub square antiprism
In geometry, the snub square antiprism is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the Icosahedron that has fourfold symmetry instead of...
(J85) can be seen as a square antiprism with a chain of equilateral triangles inserted around the middle. The sphenocorona
Sphenocorona
In geometry, the sphenocorona is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids....
(J86) and the sphenomegacorona
Sphenomegacorona
In geometry, the sphenomegacorona is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids....
(J88) are other Johnson solids that, like the square antiprism, consist of two squares and an even number of equilateral triangles.
External links
- Square Antiprism interactive model
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- VRMLVRMLVRML is a standard file format for representing 3-dimensional interactive vector graphics, designed particularly with the World Wide Web in mind...
model - Conway Notation for Polyhedra Try: "A4"
- VRML