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 ...

, 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 transi ...

. Each member of this infinite family of

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.

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 mathematics, 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 equival ...

), 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 Where ''p''/''q''=2, the component is the

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 the ...

(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 depi ...

, with higher symmetry (O_h).

Compounds of two antiprisms

Compounds of two ''n''-antiprisms share their vertices with a 2''n''-

prism Prism usually refers to: * Prism (optics), a transparent optical component with flat surfaces that refract light * Prism (geometry), a kind of polyhedron Prism may also refer to: Science and mathematics * Prism (geology), a type of sedimentary ...

, and exist as two alternated set of vertices.

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...

for the vertices of an antiprism with ''n''-gonal bases and isosceles triangles are *

\left( \cos\frac, \sin\frac, (-1)^k h \right)

\left( \cos\frac, \sin\frac, (-1)^ h \right)

with ''k'' ranging from 0 to 2''n''−1; if the triangles are equilateral, :

2h^2=\cos\frac-\cos\frac.

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 is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hig ...

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