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.

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), there occurs the compound of n p/q-gonal antiprisms, with symmetry group:
• D_npd if nq is odd
• D_nph if nq is even

When p/q = 2, or equivalently p = 2, q = 1, the component is the tetrahedron (or dyadic antiprism). In this case, if n = 2 then the compound is the stella octangula, with higher symmetry (O_h).

Compounds of two antiprisms

Compounds of two n-antiprisms share their vertices with a 2n-prism, and exist as two alternated set of vertices.

Cartesian coordinates for the vertices of an antiprism with n-gonal bases and isosceles triangles are

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

Compound of two trapezohedra (duals)

The duals of the prismatic compound of antiprisms are compounds of trapezohedra:

Compound of three antiprisms

For compounds of three digonal antiprisms, they are rotated 60 degrees, while three triangular antiprisms are rotated 40 degrees.

References

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