View source for Antiprism

{{Short description|Polyhedron with parallel bases connected by triangles}}
[[File:Octagonal antiprism.png|thumb|Octagonal antiprism]]
{{more footnotes|date=January 2013}}
In [[geometry]], an '''{{nowrap|{{mvar|n}}-gonal}} antiprism''' or {{nowrap|'''{{mvar|n}}-antiprism'''}} is a [[polyhedron]] composed of two [[Parallel (geometry)|parallel]] [[Euclidean group|direct]] copies (not mirror images) of an {{nowrap|{{mvar|n}}-sided}} [[polygon]], connected by an alternating band of {{math|2''n''}} [[triangle]]s. They are represented by the [[Conway polyhedron notation|Conway notation]] {{math|A''n''}}.

Antiprisms are a subclass of [[prismatoid]]s, and are a (degenerate) type of [[snub polyhedron]]. 

Antiprisms are similar to [[Prism (geometry)|prisms]], except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are {{math|2''n''}} triangles, rather than {{mvar|n}} [[quadrilateral]]s.

The [[dual polyhedron]] of an {{mvar|n}}-gonal antiprism is an {{mvar|n}}-gonal [[trapezohedron]].

== History ==
In his 1619 book ''[[Harmonices Mundi]]'', [[Johannes Kepler]] observed the existence of the infinite family of antiprisms.<ref>{{cite book|title=Harmonices Mundi|first=Johannes|last=Kepler|author-link=Johannes Kepler|title-link=Harmonices Mundi|year=1619|contribution=Book II, Definition X|language=la|page=49|contribution-url=https://archive.org/details/ioanniskepplerih00kepl/page/n65}} See also [https://archive.org/details/ioanniskepplerih00kepl/page/n75 illustration A], of a heptagonal antiprism.</ref> This has conventionally been thought of as the first discovery of these shapes, but they may have been known earlier: an unsigned printing block for the [[net (geometry)|net]] of a [[hexagonal antiprism]] has been attributed to [[Hieronymus Andreae]], who died in 1556.<ref>{{cite journal
 | last1 = Schreiber | first1 = Peter
 | last2 = Fischer | first2 = Gisela | author2-link = Gisela Fischer
 | last3 = Sternath | first3 = Maria Luise
 | date = July 2008
 | issue = 4
 | journal = Archive for History of Exact Sciences
 | jstor = 41134285
 | pages = 457–467
 | title = New light on the rediscovery of the Archimedean solids during the Renaissance
 | volume = 62| doi = 10.1007/s00407-008-0024-z
 }}</ref>

The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to {{ill|Theodor Wittstein|de}}.<ref>{{cite book|title=Genetische Stereometrie|first=Karl|last=Heinze|editor-first=Franz|editor-last=Lucke|publisher=B. G. Teubner|year=1886|language=de|page=14|url=https://books.google.com/books?id=rZALAAAAYAAJ&pg=PA14}}</ref> Although the English "anti-prism" had been used earlier for an [[Prism (optics)|optical prism]] used to cancel the effects of a primary optical element,<ref>{{cite journal
 | last = Smyth | first = Piazzi
 | doi = 10.1017/s0080456800029112
 | issue = 1
 | journal = Transactions of the Royal Society of Edinburgh
 | pages = 419–425
 | title = XVII. On the Constitution of the Lines forming the Low-Temperature Spectrum of Oxygen
 | volume = 30
 | year = 1881}}</ref> the first use of "antiprism" in English in its geometric sense appears to be in the early 20th century in the works of [[H. S. M. Coxeter]].<ref>{{cite journal
 | last = Coxeter | first = H. S. M.
 | date = January 1928
 | doi = 10.1017/s0305004100011786
 | issue = 1
 | journal = Mathematical Proceedings of the Cambridge Philosophical Society
 | page = 1–9
 | title = The pure Archimedean polytopes in six and seven dimensions
 | volume = 24| bibcode = 1928PCPS...24....1C
 }}</ref>

==Special cases==
=== Right antiprism ===
For an antiprism with [[Regular polygon|regular {{mvar|n}}-gon]] bases, one usually considers the case where these two copies are twisted by an angle of {{math|{{sfrac|180|''n''}}}} degrees.

The '''axis''' of a regular polygon is the line [[perpendicular]] to the polygon plane and lying in the polygon centre.

For an antiprism with [[Congruence (geometry)|congruent]] ''regular'' {{mvar|n}}-gon bases, twisted by an angle of {{math|{{sfrac|180|''n''}}}} degrees, more regularity is obtained if the bases have the same axis: are ''[[coaxial]]''; i.e. (for non-[[Coplanarity|coplanar]] bases): if the line connecting the base centers is perpendicular to the base planes. Then the antiprism is called a '''right antiprism''', and its {{math|2''n''}} side faces are [[Isosceles triangle|''isosceles'' triangles]].

=== Uniform antiprism ===
A '''[[Prismatic uniform polyhedron|uniform]] {{mvar|n}}-antiprism''' has two [[Congruence (geometry)|congruent]] [[Regular polygon|'''''regular''''']] {{mvar|n}}-gons as base faces, and {{math|2''n''}} [[Equilateral triangle|'''''equilateral''''']] triangles as side faces.

Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For {{math|''n'' {{=}} 2}}, we have the ''digonal antiprism'' (degenerate antiprism), which is visually identical to the regular [[tetrahedron]]; for {{math|''n'' {{=}} 3}}, the regular [[octahedron]] as a ''triangular antiprism'' (non-degenerate antiprism).

{{UniformAntiprisms}}

The [[Schlegel diagram]]s of these semiregular antiprisms are as follows:

{| class=wikitable
|- align=center
|[[File:Triangular antiprismatic graph.png|100px]]<BR>A3
|[[File:Square antiprismatic graph.png|100px]]<BR>A4
|[[File:Pentagonal antiprismatic graph.png|100px]]<BR>A5
|[[File:Hexagonal antiprismatic graph.png|100px]]<BR>A6
|[[File:Heptagonal antiprism graph.png|100px]]<BR>A7
|[[File:Octagonal antiprismatic graph.png|100px]]<BR>A8
|}

===Cartesian coordinates===
[[Cartesian coordinates]] for the vertices of a [[#Right antiprism|right]] {{mvar|n}}-antiprism (i.e. with regular {{mvar|n}}-gon bases and {{math|2''n''}} isosceles triangle side faces, circumradius of the bases equal to 1) are:
:<math>\left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^k h \right)</math>

where {{math|0 ≤ ''k'' ≤ 2''n'' – 1}};

if the {{mvar|n}}-antiprism is uniform (i.e. if the triangles are equilateral), then:
<math display=block>2h^2 = \cos\frac{\pi}{n} - \cos\frac{2\pi}{n}.</math>

===Volume and surface area===
Let {{mvar|a}} be the edge-length of a [[Uniform polyhedron|uniform]] {{mvar|n}}-gonal antiprism; then the volume is:
<math display=block>V = \frac{n\sqrt{4\cos^2\frac{\pi}{2n}-1}\sin \frac{3\pi}{2n} }{12\sin^2\frac{\pi}{n}}~a^3,</math>

and the surface area is:
<math display=block>A = \frac{n}{2} \left( \cot\frac{\pi}{n} + \sqrt{3} \right) a^2.</math>

Furthermore, the volume of a regular [[Antiprism#Right_antiprism|right {{mvar|n}}-gonal antiprism]] with side length of its bases {{mvar|l}} and height {{mvar|h}} is given by:
<math display=block>V = \frac{nhl^2}{12} \left( \csc\frac{\pi}{n} + 2\cot\frac{\pi}{n}\right).</math>

==== Derivation ====
The circumradius of the horizontal circumcircle of the regular <math>n</math>-gon at the base is
:<math>
R(0) = \frac{l}{2\sin\frac{\pi}{n}}.
</math>
The vertices at the base are at
:<math>\left(\begin{array}{c}R(0)\cos\frac{2\pi m}{n} \\ R(0)\sin\frac{2\pi m}{n} \\ 0\end{array}\right),\quad  m=0..n-1;</math>
the vertices at the top are at
:<math>\left(\begin{array}{c}R(0)\cos\frac{2\pi (m+1/2)}{n}\\R(0)\sin\frac{2\pi (m+1/2)}{n}\\h\end{array}\right), \quad m=0..n-1.</math>
Via linear interpolation, points on the outer triangular edges of the antiprism that connect vertices at the bottom with vertices at the top
are at
:<math>\left(\begin{array}{c}
\frac{R(0)}{h}[(h-z)\cos\frac{2\pi m}{n}+z\cos\frac{\pi(2m+1)}{n}]\\
\frac{R(0)}{h}[(h-z)\sin\frac{2\pi m}{n}+z\sin\frac{\pi(2m+1)}{n}]\\
\\z\end{array}\right), \quad 0\le z\le h, m=0..n-1</math>
and at
:<math>\left(\begin{array}{c}
\frac{R(0)}{h}[(h-z)\cos\frac{2\pi (m+1)}{n}+z\cos\frac{\pi(2m+1)}{n}]\\
\frac{R(0)}{h}[(h-z)\sin\frac{2\pi (m+1)}{n}+z\sin\frac{\pi(2m+1)}{n}]\\
\\z\end{array}\right), \quad 0\le z\le h, m=0..n-1.</math>
By building the sums of the squares of the <math>x</math> and <math>y</math> coordinates in one of the previous two vectors,
the squared circumradius of this section at altitude <math>z</math> is
:<math>
R(z)^2 = \frac{R(0)^2}{h^2}[h^2-2hz+2z^2+2z(h-z)\cos\frac{\pi}{n}].
</math> 
The horizontal section at altitude <math>0\le z\le h</math> above the base is a <math>2n</math>-gon (truncated <math>n</math>-gon)
with <math>n</math> sides of length <math>l_1(z)=l(1-z/h)</math> alternating with <math>n</math> sides of length <math>l_2(z)=lz/h</math>.
(These are derived from the length of the difference of the previous two vectors.)
It can be dissected into <math>n</math> isoceless triangles of edges <math>R(z),R(z)</math> and <math>l_1</math> (semiperimeter <math>R(z)+l_1(z)/2</math>)
plus <math>n</math>
isoceless triangles of edges <math>R(z),R(z)</math> and <math>l_2(z)</math> (semiperimeter <math>R(z)+l_2(z)/2</math>).
According to Heron's formula the areas of these triangles are
:<math>
Q_1(z) = \frac{R(0)^2}{h^2} (h-z)\left[(h-z)\cos\frac{\pi}{n}+z\right] \sin\frac{\pi}{n}
</math>
and
:<math>
Q_2(z) = 
\frac{R(0)^2}{h^2} z\left[z\cos\frac{\pi}{n}+h-z\right] \sin\frac{\pi}{n}
. 
</math>
The area of the section is <math>n[Q_1(z)+Q_2(z)]</math>, and the volume is
:<math>
V = n\int_0^h [Q_1(z)+Q_2(z)] dz 
= \frac{nh}{3}R(0)^2\sin\frac{\pi}{n}(1+2\cos\frac{\pi}{n})
= \frac{nh}{12}l^2\frac{1+2\cos\frac{\pi}{n}}{\sin\frac{\pi}{n}}
. 
</math>




Note that the volume of a right {{mvar|n}}-gonal [[Prism (geometry)|prism]] with the same {{mvar|l}} and {{mvar|h}} is:
<math display=block>V_{\mathrm{prism}}=\frac{nhl^2}{4} \cot\frac{\pi}{n}</math>
which is smaller than that of an antiprism.

===Symmetry===
The [[symmetry group]] of a [[#Right antiprism|right]] {{mvar|n}}-antiprism (i.e. with regular bases and isosceles side faces) is {{math|1=D{{sub|''n''d}} = D{{sub|''n''v}}}} of order {{math|4''n''}}, except in the cases of:
*{{math|1=''n'' = 2}}: the regular [[tetrahedron]], which has the larger symmetry group {{math|T<sub>d</sub>}} of order {{math|1=24 = 3 × (4 × 2)}}, which has three versions of {{math|D<sub>2d</sub>}} as subgroups;

*{{math|1=''n'' = 3}}: the regular [[octahedron]], which has the larger symmetry group {{math|O<sub>h</sub>}} of order {{math|1=48 = 4 × (4 × 3)}}, which has four versions of {{math|D<sub>3d</sub>}} as subgroups.

The symmetry group contains [[Inversion in a point|inversion]] [[if and only if]] {{mvar|n}} is odd.

The [[Rotation group SO(3)|rotation group]] is {{math|D<sub>''n''</sub>}} of order {{math|2''n''}}, except in the cases of:
*{{math|1=''n'' = 2}}: the regular tetrahedron, which has the larger rotation group {{math|T}} of order {{math|1=12 = 3 × (2 × 2)}}, which has three versions of {{math|D<sub>2</sub>}} as subgroups;

*{{math|1=''n'' = 3}}: the regular octahedron, which has the larger rotation group {{math|O}} of order {{math|1=24 = 4 × (2 × 3)}}, which has four versions of {{math|D<sub>3</sub>}} as subgroups.

Note: The right {{mvar|n}}-antiprisms have congruent regular {{mvar|n}}-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform {{mvar|n}}-antiprism, for {{math|''n'' ≥ 4}}.

== 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===
{{See|Prismatic uniform polyhedron}}
{| class="wikitable floatright" align=right 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|360px|thumb|This shows all the non-star and star antiprisms up to 15 sides, together with those of a 29-agon.]]
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 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 7/5-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.

Also, star antiprism compounds with regular star {{math|''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.

{| 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>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
| [[Image:Antiprism 7-4.png|64px]]<BR>3.3.3.7/4
|-
! {{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
|-
! {{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
| [[Image:Antiprism 11-4.png|64px]]<BR>3.3.3.11/4
| [[Image:Antiprism 11-6.png|64px]]<BR>3.3.3.11/6
|-
! {{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
| [[Image:Antiprism 11-5.png|64px]]<BR>3.3.3.11/5
| [[Image:Antiprism 11-7.png|64px]]<BR>3.3.3.11/7
|-
! {{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
| [[Image:Antiprism 12-7.png|64px]]<BR>3.3.3.12/7
|-
! ...
| ...
|}

== See also ==
*[[Grand antiprism]], a four-dimensional polytope
*[[Skew polygon]], a three-dimensional polygon whose convex hull is an antiprism

== References ==
{{reflist}}

==Further reading==
*{{cite book | author= Anthony Pugh | year= 1976 | title= Polyhedra: A visual approach | publisher= University of California Press Berkeley | location= California | isbn= 0-520-03056-7  }} Chapter 2: Archimedean polyhedra, prisms and antiprisms

==External links==
*{{Commonscatinline}}
*{{MathWorld|urlname=Antiprism|title=Antiprism}}
*[http://home.comcast.net/~tpgettys/nonconvexprisms.html Nonconvex Prisms and Antiprisms]
*[https://web.archive.org/web/20141019220935/http://www.software3d.com/Prisms.php Paper models of prisms and antiprisms]

{{Polyhedron navigator}}

[[Category:Uniform polyhedra]]
[[Category:Prismatoid polyhedra]]