Prismatic compound of antiprisms
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| Compound of n p/q-gonal antiprisms | |||
|---|---|---|---|
n=2
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| Type | Uniform compound | ||
| Index |
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| Polyhedra | n p/q-gonal antiprisms | ||
| Schläfli symbols (n=2) |
ß{2,2p/q} ßr{2,p/q} | ||
| Coxeter diagrams (n=2) |
File:CDel node h3.pngFile:CDel 2x.pngFile:CDel node h3.pngFile:CDel 2x.pngFile:CDel p.pngFile:CDel rat.pngFile:CDel q.pngFile:CDel node.png File:CDel node h3.pngFile:CDel 2x.pngFile:CDel node h3.pngFile:CDel p.pngFile:CDel rat.pngFile:CDel q.pngFile:CDel node h3.png | ||
| Faces | 2n {p/q} (unless p/q=2), 2np triangles | ||
| Edges | 4np | ||
| Vertices | 2np | ||
| Symmetry group |
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| Subgroup restricting to one constituent |
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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 | edit source]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
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 (Oh).
Compounds of two antiprisms
[edit | edit source]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)
[edit | edit source]The duals of the prismatic compound of antiprisms are compounds of trapezohedra:
| File:Compound two cubes.png Two cubes (trigonal trapezohedra) |
Compound of three antiprisms
[edit | edit source]For compounds of three digonal antiprisms, they are rotated 60 degrees, while three triangular antiprisms are rotated 40 degrees.
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| Three tetrahedra | Three octahedra |
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References
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