Prismatic compound of antiprisms - Alchetron, the free social encyclopedia

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

Where p/q=2, 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 a antiprism with n-gonal bases and isosceles triangles are

( cos ⁡ k π n , sin ⁡ k π n , ( − 1 ) k h )

( cos ⁡ k π n , sin ⁡ k π n , ( − 1 ) k + 1 h )

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

2 h 2 = cos ⁡ π n − cos ⁡ 2 π n .

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

Prismatic compound of antiprisms Wikipedia

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Contents

Infinite family

Compounds of two antiprisms

Compound of two trapezohedra (duals)

Compound of three antiprisms

References