Each member of this infinite family of
uniform polyhedron compound
In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts t ...
s is a symmetric arrangement of
antiprisms
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 ...
sharing a common axis of rotational symmetry. It arises from superimposing two copies of the corresponding
prismatic compound of antiprisms
In geometry, a prismatic compound of antiprism is a category of uniform polyhedron compound. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry ...
(without rotational freedom), and rotating each copy by an equal and opposite angle.
This infinite family can be enumerated as follows:
*For each positive integer ''n''>0 and for each rational number ''p''/''q''>3/2 (expressed with ''p'' and ''q''
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
), there occurs the compound of 2''n'' ''p''/''q''-gonal antiprisms (with rotational freedom), with symmetry group:
**D
''np''d if ''nq'' is odd
**D
''np''h if ''nq'' is even
Where ''p''/''q''=2 the component is a
tetrahedron
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, sometimes not considered a true antiprism.
References
*.
Polyhedral compounds
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