Editing Antiprism (section)

== Generalizations ==
===In higher dimensions===
Four-dimensional antiprisms can be defined as having two [[dual polyhedra]] as parallel opposite faces, so that each [[Cell (geometry)|three-dimensional face]] between them comes from two dual parts of the polyhedra: a vertex and a dual polygon, or two dual edges. Every three-dimensional convex polyhedron is combinatorially equivalent to one of the two opposite faces of a four-dimensional antiprism, constructed from its [[canonical polyhedron]] and its polar dual.<ref>{{cite journal | last = Grünbaum | first = Branko | author-link = Branko Grünbaum | issue = 2 | journal = Geombinatorics | mr = 2298896 | pages = 69–78 | title = Are prisms and antiprisms really boring? (Part 3) | url = https://faculty.washington.edu/moishe/branko/BG256.Prisms%20and%20antiprisms.%20Part%203.pdf | volume = 15 | year = 2005}}</ref> However, there exist four-dimensional polychora that cannot be combined with their duals to form five-dimensional antiprisms.<ref>{{cite journal | last = Dobbins | first = Michael Gene | doi = 10.1007/s00454-017-9874-y | issue = 4 | journal = [[Discrete & Computational Geometry]] | mr = 3639611 | pages = 966–984 | title = Antiprismlessness, or: reducing combinatorial equivalence to projective equivalence in realizability problems for polytopes | volume = 57 | year = 2017}}</ref>

===Self-crossing polyhedra===
{{unreferenced section|date=April 2025}}
{{Further|Prismatic uniform polyhedron}}
{| class="wikitable floatright" width=320
|- align=center
| [[File:Crossed-triangular_prism.png|80px]]<BR>[[crossed triangular antiprism|3/2-antiprism]]<br>nonuniform

| [[File:Crossed pentagonal antiprism.png|80px]]<BR>5/4-antiprism<br>nonuniform

|[[File:Pentagrammic antiprism.png|100px]]<BR>5/2-antiprism
|[[File:Pentagrammic crossed antiprism.png|100px]]<BR>5/3-antiprism
|- align=center
|[[Image:Antiprism 9-2.png|100px]]<BR>9/2-antiprism
|[[Image:Antiprism 9-4.png|100px]]<BR>9/4-antiprism
|[[Image:Antiprism 9-5.png|100px]]<BR>9/5-antiprism
|}

[[File:Antiprisms.pdf|400px|thumb|All the non-star and star uniform antiprisms up to 15 sides, together with those of a 29-gon (or [[List of polygons#List of n-gons by Greek numerical prefixes|icosaenneagon]]). For example, the icosaenneagrammic crossed antiprism ({{math|29/''q''}}) with the greatest {{math|''q''}}, such that it can be uniform, has {{math|''q'' {{=}} 19}} and is depicted at the bottom right corner of the image. For {{math|''q'' ≥ 20}} up to {{math|28}} the crossed antiprism cannot be uniform.<br />Note: Octagrammic crossed antiprism (8/5) is missing.]]
Uniform star antiprisms are named by their [[star polygon]] bases, {{math|{''p''/''q''},}} and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting [[vertex figure]]s, and are denoted by "inverted" fractions: {{math|''p''/(''p'' – ''q'')}} instead of {{math|''p''/''q''}}; example: (5/3) instead of (5/2).

A '''right star {{math|''n''}}-antiprism''' has two [[Congruence (geometry)|congruent]] [[#Right antiprism|coaxial]] [[Regular polygon|regular]] [[Convex polytope|'''''convex''''']] or [[Star polygon|'''''star''''']] polygon base faces, and {{math|2''n''}} [[isosceles triangle]] side faces.

Any star antiprism with ''regular'' convex or star polygon bases can be made a ''right'' star antiprism (by translating and/or twisting one of its bases, if necessary).

In the retrograde forms, but not in the prograde forms, the triangles joining the convex or star bases intersect the axis of rotational symmetry. Thus:

*Retrograde star antiprisms with regular convex polygon bases cannot have all equal edge lengths, and so cannot be uniform. "Exception": a retrograde star antiprism with equilateral triangle bases (vertex configuration: 3.3/2.3.3) can be uniform; but then, it has the appearance of an equilateral triangle: it is a degenerate star polyhedron.
*Similarly, some retrograde star antiprisms with regular star polygon bases cannot have all equal edge lengths, and so cannot be uniform. Example: a retrograde star antiprism with regular star {{mset|7/5}}-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.

Also, star antiprism compounds with regular star {{math|{{mset|''p''/''q''}}}}-gon bases can be constructed if {{mvar|p}} and {{mvar|q}} have common factors. Example: a star (10/4)-antiprism is the compound of two star (5/2)-antiprisms.

====Number of uniform crossed antiprisms====
If the notation {{math|(''p''/''q'')}} is used for an antiprism, then for {{math|''q'' > ''p''/2}} the antiprism is crossed (by definition) and for {{math|''q'' < ''p''/2}} is not. In this section all antiprisms are assumed to be non-degenerate, i.e. {{math|''p'' ≥ 3}}, {{math|''q'' ≠ ''p''/2}}. Also, the condition {{math|(''p'',''q'') {{=}} 1}} ({{mvar|p}} and {{mvar|q}} are relatively prime) holds, as compounds are excluded from counting. The number of uniform crossed antiprisms for fixed {{mvar|p}} can be determined using simple inequalities. The condition on possible {{mvar|q}} is

: {{math|{{sfrac|''p''|2}} < ''q'' < {{sfrac|2|3}} ''p''}} and {{math|1=(''p'',''q'') = 1.}}

Examples:
* {{mvar|''p''}} = 3: 2 ≤ {{mvar|q}} ≤  1 – a uniform triangular crossed antiprism does not exist.
* {{mvar|''p''}} = 5: 3 ≤ {{mvar|q}} ≤ 3 – one antiprism of the type (5/3) can be uniform.
* {{mvar|''p''}} = 29: 15 ≤ {{mvar|q}} ≤ 19 – there are five possibilities (15 thru 19) shown in the rightmost column, below the (29/1) convex antiprism, on the image above.
* {{mvar|''p''}} = 15: 8 ≤ {{mvar|q}} ≤ 9 – antiprism with {{mvar|q}} = 8 is a solution, but {{mvar|q}} = 9 must be rejected, as (15,9) = 3 and {{sfrac|15|9}} = {{sfrac|5|3}}. The antiprism (15/9) is a compound of three antiprisms (5/3). Since 9 satisfies the inequalities, the compound can be uniform, and if it is, then its parts must be. Indeed, the antiprism (5/3) can be uniform by example 2.

In the first column of the following table, the symbols are Schoenflies, Coxeter, and orbifold notation, in this order.
{| class="wikitable mw-collapsible mw-collapsed" style="text-align:center"
|+ class="nowrap"| Star ({{math|''p''/''q''}})-antiprisms by symmetry, for {{math|''p'' ≤ 12}}
! [[List of spherical symmetry groups|Symmetry group]]
! colspan=4 | Uniform stars
! Right stars
|-
! {{math|D<sub>3h</sub><br>[2,3]<br>(2*3)}}
| colspan=4 |
| [[Image:Crossed triangular antiprism.svg|64px]]<br>3.3/2.3.3<br>[[Crossed triangular antiprism]]
|-
! {{math|D<sub>4d</sub><BR>[2<sup>+</sup>,8]<BR>(2*4)}}
| colspan=4 |
| [[Image:Crossed square antiprism.png|64px]]<BR>3.3/2.3.4<br>[[Crossed square antiprism]]
|-
! {{math|D<sub>5h</sub><BR>[2,5]<BR>(*225)}}
| [[Image:Pentagrammic antiprism.png|64px]]<BR>3.3.3.5/2<br>[[Pentagrammic antiprism]]
| colspan=3 |
| [[Image:Crossed pentagonal antiprism.png|64px]]<BR>3.3/2.3.5<br>[[Crossed pentagonal antiprism]]
|-
! {{math|D<sub>5d</sub><BR>[2<sup>+</sup>,10]<BR>(2*5)}}
| [[Image:Pentagrammic crossed antiprism.png|64px]]<BR>3.3.3.5/3<br>[[Pentagrammic crossed-antiprism]]
|-
! {{math|D<sub>6d</sub><BR>[2<sup>+</sup>,12]<BR>(2*6)}}
| colspan=4 |
| [[Image:Crossed hexagonal antiprism.png|64px]]<BR>3.3/2.3.6<br>[[Crossed hexagonal antiprism]]
|-
! {{math|D<sub>7h</sub><BR>[2,7]<BR>(*227)}}
| [[Image:Antiprism 7-2.png|64px]]<BR>3.3.3.7/2<br>Heptagrammic antiprism (7/2)
| [[Image:Antiprism 7-4.png|64px]]<BR>3.3.3.7/4<br>Heptagrammic crossed antiprism (7/4)
|-
! {{math|D<sub>7d</sub><BR>[2<sup>+</sup>,14]<BR>(2*7)}}
| [[Image:Antiprism 7-3.png|64px]]<BR>3.3.3.7/3<br>Heptagrammic antiprism (7/3)
|-
! {{math|D<sub>8d</sub><BR>[2<sup>+</sup>,16]<BR>(2*8)}}
| [[Image:Antiprism 8-3.png|64px]]<BR>3.3.3.8/3<br>[[Octagrammic antiprism]]
| [[Image:Antiprism 8-5.png|64px]]<BR>3.3.3.8/5<br>[[Octagrammic crossed-antiprism]]
|-
! {{math|D<sub>9h</sub><BR>[2,9]<BR>(*229)}}
| [[Image:Antiprism 9-2.png|64px]]<BR>3.3.3.9/2<br>[[Enneagrammic antiprism (9/2)]]
| [[Image:Antiprism 9-4.png|64px]]<BR>3.3.3.9/4<br>[[Enneagrammic antiprism (9/4)]]
|-
! {{math|D<sub>9d</sub><BR>[2<sup>+</sup>,18]<BR>(2*9)}}
| [[Image:Antiprism 9-5.png|64px]]<BR>3.3.3.9/5<br>[[Enneagrammic crossed-antiprism]]
|-
! {{math|D<sub>10d</sub><BR>[2<sup>+</sup>,20]<BR>(2*10)}}
| [[Image:Antiprism 10-3.png|64px]]<BR>3.3.3.10/3<br>[[Decagrammic antiprism]]
|-
! {{math|D<sub>11h</sub><BR>[2,11]<BR>(*2.2.11)}}
| [[Image:Antiprism 11-2.png|64px]]<BR>3.3.3.11/2<br>Undecagrammic (11/2)
| [[Image:Antiprism 11-4.png|64px]]<BR>3.3.3.11/4<br>Undecagrammic (11/4)
| [[Image:Antiprism 11-6.png|64px]]<BR>3.3.3.11/6<br>Undecagrammic crossed (11/6)
|-
! {{math|D<sub>11d</sub><BR>[2<sup>+</sup>,22]<BR>(2*11)}}
| [[Image:Antiprism 11-3.png|64px]]<BR>3.3.3.11/3<br>Undecagrammic (11/3)
| [[Image:Antiprism 11-5.png|64px]]<BR>3.3.3.11/5<br>Undecagrammic (11/5)
| [[Image:Antiprism 11-7.png|64px]]<BR>3.3.3.11/7<br>Undecagrammic crossed (11/7)
|-
! {{math|D<sub>12d</sub><BR>[2<sup>+</sup>,24]<BR>(2*12)}}
| [[Image:Antiprism 12-5.png|64px]]<BR>3.3.3.12/5<br>Dodecagrammic
| [[Image:Antiprism 12-7.png|64px]]<BR>3.3.3.12/7<br>Dodecagrammic crossed
|-
! ...
| ...
|}