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In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An. Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.
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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.
In chemistry, the capped square antiprismatic molecular geometry describes the shape of compounds where nine atoms, groups of atoms, or ligands are arranged around a central atom, defining the vertices of a gyroelongated square pyramid. The symmetry group of the resulting object is C 4v
Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property.
Notes: Two of these polyhedra may be constructed from the first two snub polyhedra in the list starting with the icosahedron : the pentagonal antiprism is a parabidiminished icosahedron and a pentagrammic crossed-antiprism is a parabidiminished great icosahedron, also known as a parabireplenished great icosahedron .
If p/q is an integer, i.e. if q = 1, the prism or antiprism is convex. (The fraction is always assumed to be stated in lowest terms.) An antiprism with p/q < 2 is crossed or retrograde; its vertex figure resembles a bowtie. If p/q < 3/2 no uniform antiprism can exist, as its vertex figure would have to violate the triangle inequality.
If the faces of the pentagonal antiprism are all regular, it is a semiregular polyhedron.It can also be considered as a parabidiminished icosahedron, a shape formed by removing two pentagonal pyramids from a regular icosahedron leaving two nonadjacent pentagonal faces; a related shape, the metabidiminished icosahedron (one of the Johnson solids), is likewise form from the icosahedron by ...