Jump to content

Prismatic compound of antiprisms

From Wikipedia, the free encyclopedia
Compound of n p/q-gonal antiprisms
n=2

5/3-gonal

5/2-gonal
Type Uniform compound
Index
  • q odd: UC23
  • q even: UC25
Polyhedra n p/q-gonal antiprisms
Schläfli symbols
(n=2)
ß{2,2p/q}
ßr{2,p/q}
Coxeter diagrams
(n=2)

Faces 2n {p/q} (unless p/q=2), 2np triangles
Edges 4np
Vertices 2np
Symmetry group
Subgroup restricting to one constituent

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

[edit]

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:
    • Dnpd if nq is odd
    • Dnph 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 (Oh).

Compounds of two antiprisms

[edit]

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,

Compounds of 2 antiprisms





2 digonal
antiprisms

(tetrahedra)
2 triangular
antiprisms

(octahedra)
2 square
antiprisms
2 hexagonal
antiprisms
2 pentagrammic
crossed
antiprism

Compound of two trapezohedra (duals)

[edit]

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


Two cubes
(trigonal trapezohedra)

Compound of three antiprisms

[edit]

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

Three tetrahedra Three octahedra

References

[edit]
  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79 (3): 447–457, doi:10.1017/S0305004100052440, MR 0397554.