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Prismatic compound of antiprisms

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(Redirected from Compound of two octahedra) Polyhedral compound
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

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

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

  • ( cos k π n , sin k π n , ( 1 ) k h ) {\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)}
  • ( cos k π n , sin k π n , ( 1 ) k + 1 h ) {\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k+1}h\right)}

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

2 h 2 = cos π n cos 2 π n . {\displaystyle 2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.}
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)

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


Two cubes
(trigonal trapezohedra)

Compound of three antiprisms

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

Three tetrahedra Three octahedra

References


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