Difference between revisions of "Antiprism"

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en>Tamfang
(→‎Volume and surface area: I didn't notice before: this long derivation is valid only (if at all) for *hexagonal* antiprism)
 
en>Tomruen
 
(4 intermediate revisions by 4 users not shown)
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{{no footnotes|date=January 2013}}
 
{| class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="280"
 
{| class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="280"
 
!bgcolor=#e7dcc3 colspan=2|Set of uniform antiprisms
 
!bgcolor=#e7dcc3 colspan=2|Set of uniform antiprisms
 
|-
 
|-
|align=center colspan=2|[[image:Hexagonal antiprism.png|280px|Hexagonal antiprism]]<br>
+
|align=center colspan=2|[[Image:Hexagonal antiprism.png|280px|Hexagonal antiprism]]<br>
 
|-
 
|-
 
|bgcolor=#e7dcc3|Type||[[uniform polyhedron]]
 
|bgcolor=#e7dcc3|Type||[[uniform polyhedron]]
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|bgcolor=#e7dcc3|[[Vertex configuration]]||3.3.3.''n''
 
|bgcolor=#e7dcc3|[[Vertex configuration]]||3.3.3.''n''
 
|-
 
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>0,1</sub>{2,2''n''}<br>s{2,''n''}
+
|bgcolor=#e7dcc3|[[Schläfli symbol]]||s{2,2''n''}<br>sr{2,''n''}<BR>{ } {{Unicode|&#x2297;}} {n}
 
|-
 
|-
 
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram]]s||{{CDD|node_h|2x|node_h|2x|n|node}}<br>{{CDD|node_h|2x|node_h|n|node_h}}
 
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram]]s||{{CDD|node_h|2x|node_h|2x|n|node}}<br>{{CDD|node_h|2x|node_h|n|node_h}}
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==Uniform antiprism==
 
==Uniform antiprism==
A '''[[Prismatic uniform polyhedron|uniform]] antiprism''' has, apart from the base faces, 2''n'' equilateral triangles as faces. As a class, the uniform antiprisms form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For {{nowrap|''n'' {{=}} 2}} we have as degenerate case the regular [[tetrahedron]], and for {{nowrap|''n'' {{=}} 3}} the non-degenerate regular [[octahedron]].
+
A '''[[Prismatic uniform polyhedron|uniform]] antiprism''' has, apart from the base faces, 2''n'' equilateral triangles as faces. As a class, the uniform antiprisms form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For {{nowrap|''n'' {{=}} 2}} we have as degenerate case the regular [[tetrahedron]] as a ''digonal antiprism'', and for {{nowrap|''n'' {{=}} 3}} the non-degenerate regular [[octahedron]] as a ''triangular antiprism''.
  
 
The [[dual polyhedra]] of the antiprisms are the [[trapezohedra]]. Their existence was first discussed and their name was coined by [[Johannes Kepler]].
 
The [[dual polyhedra]] of the antiprisms are the [[trapezohedra]]. Their existence was first discussed and their name was coined by [[Johannes Kepler]].
 +
 +
{{UniformAntiprisms}}
 +
 +
=== Schlegel diagrams===
 +
{| class=wikitable
 +
|- align=center
 +
|[[File:3-cube t2.svg|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==
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and the surface area is
 
and the surface area is
 
:<math>A = \frac{n}{2} ( \cot{\frac{\pi}{n}} + \sqrt{3}) a^2.</math>
 
:<math>A = \frac{n}{2} ( \cot{\frac{\pi}{n}} + \sqrt{3}) a^2.</math>
 +
 +
== Related polyhedra ==
 +
 +
There are an infinite set of [[Truncation (geometry)|truncated]] antiprisms, including a lower-symmetry form of the [[truncated octahedron]] (truncated triangular antiprism). These can be [[Alternation (geometry)|alternated]] to create [[snub antiprism]]s, two of which are [[Johnson solid]]s, and the ''snub triangular antiprism'' is a lower symmetry form of the [[Regular icosahedron|icosahedron]].
 +
 +
{| class=wikitable
 +
|+ Truncated antiprisms
 +
!
 +
|[[File:Truncated_octahedron_prismatic_symmetry.png|80px]]
 +
|[[File:Truncated_square_antiprism.png|80px]]
 +
|[[File:Truncated_pentagonal_antiprism.png|80px]]
 +
|...
 +
|- align=center
 +
|valign=bottom|ts{2,4}
 +
|[[truncated octahedron|ts{2,6}]]
 +
|ts{2,8}
 +
|ts{2,10}
 +
|ts{2,2n}
 +
|-
 +
!colspan=5|Snub antiprisms
 +
|-
 +
!J84
 +
!Icosahedron
 +
!J85
 +
!colspan=2|Irregular...
 +
|- align=center
 +
|[[File:Snub_digonal_antiprism.png|50px]]
 +
|[[File:snub_triangular_antiprism.png|80px]]
 +
|[[File:Snub_square_antiprism_colored.png|80px]]
 +
|[[File:Snub_pentagonal_antiprism.png|80px]]
 +
|...
 +
|- align=center
 +
|[[Snub disphenoid|ss{2,4}]]
 +
|[[Regular icosahedron|ss{2,6}]]
 +
|[[snub square antiprism|ss{2,8}]]
 +
|ss{2,10}
 +
|ss{2,2n}
 +
|}
  
 
==Symmetry==
 
==Symmetry==
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The [[rotation group SO(3)|rotation group]] is D<sub>''n''</sub> of order 2''n'', except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D<sub>2</sub> as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D<sub>3</sub> as subgroups.
 
The [[rotation group SO(3)|rotation group]] is D<sub>''n''</sub> of order 2''n'', except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D<sub>2</sub> as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D<sub>3</sub> as subgroups.
  
==See also==
+
== Star antiprism ==
{{UniformAntiprisms}}
+
{| class=wikitable align=right width=320
 +
|- align=center
 +
|colspan=3|[[File:Pentagrammic antiprism.png|160px]]<BR>5/2-antiprism
 +
|colspan=3|[[File:Pentagrammic crossed antiprism.png|160px]]<BR>5/3-antiprism
 +
|- align=center
 +
|colspan=2|[[Image:Antiprism 9-2.png|120px]]<BR>9/2-antiprism
 +
|colspan=2|[[Image:Antiprism 9-4.png|120px]]<BR>9/4-antiprism
 +
|colspan=2|[[Image:Antiprism 9-5.png|120px]]<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, {p/q}, and exist in prograde and retrograde (crossed) solutions. Crossed forms have intersecting [[vertex figure]]s, and are denoted by inverted fractions, p/(p-q) instead of p/q, like 5/3 versus 5/2.
 +
 
 +
In the retrograde forms but not in the prograde forms, the triangles joining the star bases intersect the axis of rotational symmetry.
 +
 
 +
Some retrograde star antiprisms with regular star polygon bases cannot be constructed with equal edge lengths, so are not uniform polyhedra. Star antiprism compounds also can be constructed where p and q have common factors; thus a 10/4 antiprism is the compound of two 5/2 star antiprisms.
 +
 
 +
{| class="wikitable"
 +
|+ Uniform star antiprisms by symmetry, up to 12
 +
![[List of spherical symmetry groups|Symmetry group]]
 +
!colspan=4|Star forms
 +
|-
 +
!align=center| d<sub>5h</sub><BR>[2,5]<BR>(*225)
 +
| [[Image:Pentagrammic antiprism.png|64px]]<BR>[[Pentagrammic antiprism|3.3.3.5/2]]
 +
|-
 +
!align=center| d<sub>5d</sub><BR>[2<sup>+</sup>,5]<BR>(2*5)
 +
| [[Image:Pentagrammic crossed antiprism.png|64px]]<BR>[[Pentagrammic crossed-antiprism|3.3.3.5/3]]
 +
|-
 +
!align=center| d<sub>7h</sub><BR>[2,7]<BR>(*227)
 +
| [[Image:Antiprism 7-2.png|64px]]<BR>[[Heptagrammic antiprism (7/2)|3.3.3.7/2]]
 +
| [[Image:Antiprism 7-4.png|64px]]<BR>[[Heptagrammic crossed-antiprism|3.3.3.7/4]]
 +
|-
 +
!align=center| d<sub>7d</sub><BR>[2<sup>+</sup>,7]<BR>(2*7)
 +
| [[Image:Antiprism 7-3.png|64px]]<BR>[[Heptagrammic antiprism (7/3)|3.3.3.7/3]]
 +
|-
 +
!align=center| d<sub>8d</sub><BR>[2<sup>+</sup>,8]<BR>(2*8)
 +
| [[Image:Antiprism 8-3.png|64px]]<BR>[[Octagrammic antiprism|3.3.3.8/3]]
 +
| [[Image:Antiprism 8-5.png|64px]]<BR>[[Octagrammic crossed-antiprism|3.3.3.8/5]]
 +
|-
 +
!align=center| d<sub>9h</sub><BR>[2,9]<BR>(*229)
 +
| [[Image:Antiprism 9-2.png|64px]]<BR>[[Enneagrammic antiprism (9/2)|3.3.3.9/2]]
 +
| [[Image:Antiprism 9-4.png|64px]]<BR>[[Enneagrammic antiprism (9/4)|3.3.3.9/4]]
 +
|-
 +
!align=center| d<sub>9d</sub><BR>[2<sup>+</sup>,9]<BR>(2*9)
 +
| [[Image:Antiprism 9-5.png|64px]]<BR>[[Enneagrammic crossed-antiprism|3.3.3.9/5]]
 +
|-
 +
!align=center| d<sub>10d</sub><BR>[2<sup>+</sup>,10]<BR>(2*10)
 +
| [[Image:Antiprism 10-3.png|64px]]<BR>[[Decagrammic antiprism|3.3.3.10/3]]
 +
|-
 +
!align=center| 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
 +
|-
 +
!align=center| d<sub>11d</sub><BR>[2<sup>+</sup>,11]<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
 +
|-
 +
!align=center| d<sub>12d</sub><BR>[2<sup>+</sup>,12]<BR>(2*12)
 +
| [[Image:Antiprism 12-5.png|64px]]<BR>[[Dodecagrammic antiprism|3.3.3.12/5]]
 +
| [[Image:Antiprism 12-7.png|64px]]<BR>[[Dodecagrammic crossed-antiprism|3.3.3.12/7]]
 +
|-
 +
| ...
 +
|}
  
 +
*[[Prism (geometry)|Prism]]
 
*[[Apeirogonal antiprism]]
 
*[[Apeirogonal antiprism]]
*[[Grand antiprism]] – a four dimensional polytope
+
*[[Grand antiprism]] – a four-dimensional polytope
 +
*[[One World Trade Center]], a building consisting primarily of an elongated square antiprism<ref name=Kabai2013>{{cite web|last=Kabai|first=Sándor|title=One World Trade Center Antiprism|url=http://demonstrations.wolfram.com/OneWorldTradeCenterAntiprism/|publisher=Wolfram Demonstrations Project|accessdate=8 October 2013}}</ref>
 +
 
 +
== References==
 +
* {{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, prisma and antiprisms
 +
{{reflist}}
  
 
==External links==
 
==External links==
 +
{{Commonscat|Antiprisms}}
 
*{{MathWorld |urlname=Antiprism |title=Antiprism}}
 
*{{MathWorld |urlname=Antiprism |title=Antiprism}}
 
*{{GlossaryForHyperspace |anchor=Antiprism |title=Antiprism}}
 
*{{GlossaryForHyperspace |anchor=Antiprism |title=Antiprism}}
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[[Category:Uniform polyhedra]]
 
[[Category:Uniform polyhedra]]
 
[[Category:Prismatoid polyhedra]]
 
[[Category:Prismatoid polyhedra]]
 
[[ca:Antiprisma]]
 
[[el:Αντιπρίσμα]]
 
[[es:Antiprisma]]
 
[[eo:Kontraŭprismo]]
 
[[fr:Antiprisme]]
 
[[ko:엇각기둥]]
 
[[it:Antiprisma]]
 
[[nl:Antiprisma]]
 
[[ja:反角柱]]
 
[[no:Antiprisme]]
 
[[pl:Antygraniastosłup]]
 
[[pt:Antiprisma]]
 
[[ru:Антипризма]]
 
[[sl:Antiprizma]]
 
[[th:แอนติปริซึม]]
 
[[zh:反棱柱]]
 

Latest revision as of 08:05, 11 January 2015

Template:No footnotes

Set of uniform antiprisms
Hexagonal antiprism
Type uniform polyhedron
Faces 2 n-gons, 2n triangles
Edges 4n
Vertices 2n
Vertex configuration 3.3.3.n
Schläfli symbol s{2,2n}
sr{2,n}
{ } {n}
Coxeter–Dynkin diagrams Template:CDD
Template:CDD
Symmetry group Dnd, [2+,2n], (2*n), order 4n
Rotation group Dn, [2,n]+, (22n), order 2n
Dual polyhedron trapezohedron
Properties convex, semi-regular vertex-transitive
Net Generalized antiprisim net.svg

In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids.

Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.

In the case of a regular n-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.

Uniform antiprism

A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For n = 2 we have as degenerate case the regular tetrahedron as a digonal antiprism, and for n = 3 the non-degenerate regular octahedron as a triangular antiprism.

The dual polyhedra of the antiprisms are the trapezohedra. Their existence was first discussed and their name was coined by Johannes Kepler.

Template:UniformAntiprisms

Schlegel diagrams

3-cube t2.svg
A3
Square antiprismatic graph.png
A4
Pentagonal antiprismatic graph.png
A5
Hexagonal antiprismatic graph.png
A6
Heptagonal antiprism graph.png
A7
Octagonal antiprismatic graph.png
A8

Cartesian coordinates

Cartesian coordinates for the vertices of a right antiprism with n-gonal bases and isosceles triangles are

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

Volume and surface area

Let a be the edge-length of a uniform antiprism. Then the volume is

and the surface area is

Related polyhedra

There are an infinite set of truncated antiprisms, including a lower-symmetry form of the truncated octahedron (truncated triangular antiprism). These can be alternated to create snub antiprisms, two of which are Johnson solids, and the snub triangular antiprism is a lower symmetry form of the icosahedron.

Truncated antiprisms
Truncated octahedron prismatic symmetry.png Truncated square antiprism.png Truncated pentagonal antiprism.png ...
ts{2,4} ts{2,6} ts{2,8} ts{2,10} ts{2,2n}
Snub antiprisms
J84 Icosahedron J85 Irregular...
Snub digonal antiprism.png Snub triangular antiprism.png Snub square antiprism colored.png Snub pentagonal antiprism.png ...
ss{2,4} ss{2,6} ss{2,8} ss{2,10} ss{2,2n}

Symmetry

The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is Dnd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group Td of order 24, which has three versions of D2d as subgroups, and the octahedron, which has the larger symmetry group Oh of order 48, which has four versions of D3d as subgroups.

The symmetry group contains inversion if and only if n is odd.

The rotation group is Dn of order 2n, except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D2 as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.

Star antiprism

Pentagrammic antiprism.png
5/2-antiprism
Pentagrammic crossed antiprism.png
5/3-antiprism
Antiprism 9-2.png
9/2-antiprism
Antiprism 9-4.png
9/4-antiprism
Antiprism 9-5.png
9/5-antiprism
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, {p/q}, and exist in prograde and retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by inverted fractions, p/(p-q) instead of p/q, like 5/3 versus 5/2.

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

Some retrograde star antiprisms with regular star polygon bases cannot be constructed with equal edge lengths, so are not uniform polyhedra. Star antiprism compounds also can be constructed where p and q have common factors; thus a 10/4 antiprism is the compound of two 5/2 star antiprisms.

Uniform star antiprisms by symmetry, up to 12
Symmetry group Star forms
d5h
[2,5]
(*225)
Pentagrammic antiprism.png
3.3.3.5/2
d5d
[2+,5]
(2*5)
Pentagrammic crossed antiprism.png
3.3.3.5/3
d7h
[2,7]
(*227)
Antiprism 7-2.png
3.3.3.7/2
Antiprism 7-4.png
3.3.3.7/4
d7d
[2+,7]
(2*7)
Antiprism 7-3.png
3.3.3.7/3
d8d
[2+,8]
(2*8)
Antiprism 8-3.png
3.3.3.8/3
Antiprism 8-5.png
3.3.3.8/5
d9h
[2,9]
(*229)
Antiprism 9-2.png
3.3.3.9/2
Antiprism 9-4.png
3.3.3.9/4
d9d
[2+,9]
(2*9)
Antiprism 9-5.png
3.3.3.9/5
d10d
[2+,10]
(2*10)
Antiprism 10-3.png
3.3.3.10/3
d11h
[2,11]
(*2.2.11)
Antiprism 11-2.png
3.3.3.11/2
Antiprism 11-4.png
3.3.3.11/4
Antiprism 11-6.png
3.3.3.11/6
d11d
[2+,11]
(2*11)
Antiprism 11-3.png
3.3.3.11/3
Antiprism 11-5.png
3.3.3.11/5
Antiprism 11-7.png
3.3.3.11/7
d12d
[2+,12]
(2*12)
Antiprism 12-5.png
3.3.3.12/5
Antiprism 12-7.png
3.3.3.12/7
...

References

  • {{#invoke:citation/CS1|citation

|CitationClass=book }} Chapter 2: Archimedean polyhedra, prisma and antiprisms

External links

Template:Commonscat

Template:Polyhedron navigator