In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a prismatic compound of antiprism is a category 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 ...
. Each member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of
antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s sharing a common axis of rotational symmetry.
Infinite family
This infinite family can be enumerated as follows:
*For each positive integer ''n'' ≥ 1 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 ''n'' ''p''/''q''-gonal antiprisms, with symmetry group:
**D
''np''d if ''nq'' is odd
**D
''np''h if ''nq'' is even
When ''p''/''q'' = 2, or equivalently p = 2, q = 1, the component is the
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 ...
(or dyadic antiprism). In this case, if ''n'' = 2 then the compound is the
stella octangula
The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicte ...
, with higher symmetry (O
h).
Compounds of two antiprisms
Compounds of two ''n''-antiprisms share their vertices with a 2''n''-
prism
PRISM is a code name for a program under which the United States National Security Agency (NSA) collects internet communications from various U.S. internet companies. The program is also known by the SIGAD . PRISM collects stored internet ...
, and exist as two
alternated set of vertices.
Cartesian coordinates
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
for the vertices of an antiprism with ''n''-gonal bases and isosceles triangles are
*
*
with ''k'' ranging from 0 to 2''n''−1; if the triangles are equilateral,
:
Compound of two trapezohedra (duals)
The duals of the prismatic compound of antiprisms are compounds of
trapezohedra
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron Remarks: the faces of a deltohedron are deltoids; a (non-twisted) kite or deltoid can be dissected into two isosceles triangles or "deltas" (Δ), ba ...
:
Compound of three antiprisms
For compounds of three digonal antiprisms, they are rotated 60 degrees, while three triangular antiprisms are rotated 40 degrees.
References
*.
Polyhedral compounds
{{polyhedron-stub