View source for Prismatic compound of antiprisms

{{Short description|Polyhedral compound}}
{| class=wikitable align="right" width="300"
!bgcolor=#e7dcc3 colspan=2|Compound of ''n'' ''p''/''q''-gonal antiprisms
|-
|align=center colspan=2|''n''=2
{|
|- align=center
|[[Image:UC23-k n-m-gonal antiprisms.png|150px]]<BR>5/3-gonal
|[[Image:UC25-k n-m-gonal antiprisms.png|150px]]<BR>5/2-gonal
|}
|-
|bgcolor=#e7dcc3|Type||[[Uniform polyhedron compound|Uniform compound]]
|-
|bgcolor=#e7dcc3|Index||
*''q'' odd: UC<sub>23</sub>
*''q'' even: UC<sub>25</sub>
|-
|bgcolor=#e7dcc3|Polyhedra||''n'' ''p''/''q''-gonal [[prismatic uniform polyhedron|antiprisms]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s<BR>(n=2)||ß{2,2p/q}<BR>ßr{2,p/q}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s<BR>(n=2)||{{CDD|node_h3|2x|node_h3|2x|p|rat|q|node}}<BR>{{CDD|node_h3|2x|node_h3|p|rat|q|node_h3}}
|-
|bgcolor=#e7dcc3|Faces||2''n'' {''p''/''q''} (unless ''p''/''q''=2), 2''np'' [[triangles]]
|-
|bgcolor=#e7dcc3|Edges||4''np''
|-
|bgcolor=#e7dcc3|Vertices||2''np''
|-
|bgcolor=#e7dcc3|[[Symmetry group]]||
*''nq'' odd: ''np''-fold [[Dihedral symmetry in three dimensions|antiprismatic]] ('''D<sub>''np''d</sub>''')
*''nq'' even: ''np''-fold [[Dihedral symmetry in three dimensions|prismatic]] ('''D<sub>''np''h</sub>''')
|-
|bgcolor=#e7dcc3|[[Subgroup]] restricting to one constituent||
*''q'' odd: ''p''-fold [[Dihedral symmetry in three dimensions|antiprismatic]] ('''D<sub>''p''d</sub>''')
*''q'' even: ''p''-fold [[Dihedral symmetry in three dimensions|prismatic]] ('''D<sub>''p''h</sub>''')
|}
In [[geometry]], a '''prismatic compound of antiprism''' is a category of [[uniform polyhedron compound]]. Each member of this infinite family of [[uniform polyhedron compound]]s 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<sub>''np''d</sub>''' if ''nq'' is odd
**'''D<sub>''np''h</sub>''' 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<sub>h</sub>''').

== Compounds of two antiprisms ==
Compounds of two ''n''-antiprisms share their vertices with a 2''n''-[[prism (geometry)|prism]], and exist as two [[alternation (geometry)|alternated]] set of vertices.

[[Cartesian coordinates]] for the vertices of an antiprism with ''n''-gonal bases and isosceles triangles are
*<math>\left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^k h \right)</math>
*<math>\left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^{k+1} h \right)</math>

with ''k'' ranging from 0 to 2''n''−1; if the triangles are equilateral,
:<math>2h^2=\cos\frac{\pi}{n}-\cos\frac{2\pi}{n}.</math>

{| class=wikitable
|+ Compounds of 2 antiprisms
!{{CDD|node_h3|2x|node_h3|4|node}}<BR>{{CDD|node_h3|2x|node_h3|2x|node_h3}}
!{{CDD|node_h3|2x|node_h3|6|node}}<BR>{{CDD|node_h3|2x|node_h3|3|node_h3}}
!{{CDD|node_h3|2x|node_h3|8|node}}<BR>{{CDD|node_h3|2x|node_h3|4|node_h3}}
!{{CDD|node_h3|2x|node_h3|12|node}}<BR>{{CDD|node_h3|2x|node_h3|6|node_h3}}
!{{CDD|node_h3|2x|node_h3|10|rat|3x|node}}<BR>{{CDD|node_h3|2x|node_h3|5|rat|3x|node_h3}}
|-
|[[File:Compound of two tetrahedra.png|100px]]
|[[File:Compound_two_triangle_prisms.png|100px]]
|[[File:Compound_two_square_antiprisms.png|100px]]
|[[File:Compound_two_hexagonal_antiprisms.png|100px]]
|[[File:Compound_two_pentagram_crossed_antiprism.png|100px]]
|- align=center valign=top
|2 [[tetrahedron|digonal<BR>antiprisms]]<BR>(tetrahedra)
|2 [[octahedron|triangular<BR>antiprisms]]<BR>(octahedra)
|2 [[square antiprism|square<BR>antiprisms]]
|2 [[hexagonal antiprism|hexagonal<BR>antiprisms]]
|2 [[Pentagrammic crossed-antiprism|pentagrammic<BR>crossed<BR>antiprism]]
|}

=== Compound of two trapezohedra (duals) ===
The duals of the prismatic compound of antiprisms are compounds of [[trapezohedra]]:
{| class=wikitable
|- align=center
|[[File:Compound_two_cubes.png|160px]]<BR>Two cubes<BR>(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.
{| class=wikitable
|[[File:Compound_three_digonal_antiprisms.png|160px]]
|[[File:Compound_three_triangular_antiprisms.png|160px]]
|-
![[Compound of three tetrahedra|Three tetrahedra]]
!Three octahedra
|}

== References ==
*{{citation|first=John|last=Skilling|title=Uniform Compounds of Uniform Polyhedra|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|volume=79|pages=447–457|year=1976|doi=10.1017/S0305004100052440|mr=0397554|issue=3}}.

[[Category:Polyhedral compounds]]


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