Difference between revisions of "Antiprism"
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+ | {{no footnotes|date=January 2013}} | ||
+ | {| class="wikitable" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="280" | ||
+ | !bgcolor=#e7dcc3 colspan=2|Set of uniform antiprisms | ||
+ | |- | ||
+ | |align=center colspan=2|[[Image:Hexagonal antiprism.png|280px|Hexagonal antiprism]]<br> | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|Type||[[uniform polyhedron]] | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|Faces||2 [[polygon|''n''-gon]]s, 2''n'' [[triangle]]s | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|Edges||4''n'' | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|Vertices||2''n'' | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|[[Vertex configuration]]||3.3.3.''n'' | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|[[Schläfli symbol]]||s{2,2''n''}<br>sr{2,''n''}<BR>{ } ⨂ {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|[[List of spherical symmetry groups|Symmetry group]]||D<sub>''n''d</sub>, [2<sup>+</sup>,2''n''], (2*''n''), order 4''n'' | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|[[Point_groups_in_three_dimensions#Rotation_groups|Rotation group]]||D<sub>''n''</sub>, [2,''n'']<sup>+</sup>, (22''n''), order 2''n'' | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|[[Dual polyhedron]]||[[trapezohedron]] | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|Properties||convex, semi-regular [[vertex-transitive]] | ||
+ | |- | ||
+ | |bgcolor=#e7dcc3|[[Net (polyhedron)|Net]]||[[Image:Generalized antiprisim net.svg|150px]] | ||
+ | |} | ||
+ | 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 [[triangle]]s. Antiprisms are a subclass of the [[prismatoid]]s. | ||
+ | Antiprisms are similar to [[prism (geometry)|prism]]s 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-gon|''n''-gonal]] bases and, connecting those bases, 2''n'' isosceles triangles. | ||
+ | |||
+ | ==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]] 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]]. | ||
+ | |||
+ | {{UniformAntiprisms}} | ||
+ | |||
+ | ==Cartesian coordinates== | ||
+ | [[Cartesian coordinates]] for the vertices of a right 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> | ||
+ | |||
+ | 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> | ||
+ | |||
+ | ==Volume and surface area== | ||
+ | Let ''a'' be the edge-length of a [[uniform polyhedron|uniform]] antiprism. Then the volume is | ||
+ | :<math>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>A = \frac{n}{2} ( \cot{\frac{\pi}{n}} + \sqrt{3}) a^2.</math> | ||
+ | |||
+ | ==Symmetry== | ||
+ | The [[symmetry group]] of a right ''n''-sided antiprism with regular base and isosceles side faces is D<sub>''n''d</sub> of order 4''n'', except in the case of a tetrahedron, which has the larger symmetry group T<sub>d</sub> of order 24, which has three versions of D<sub>2d</sub> as subgroups, and the octahedron, which has the larger symmetry group O<sub>h</sub> of order 48, which has four versions of D<sub>3d</sub> as subgroups. | ||
+ | |||
+ | The symmetry group contains [[inversion in a point|inversion]] [[if and only if]] ''n'' is odd. | ||
+ | |||
+ | 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. | ||
+ | |||
+ | == Star antiprism == | ||
+ | |||
+ | {| class="wikitable" | ||
+ | ![[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]] | ||
+ | *[[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== | ||
+ | {{Commonscat|Antiprisms}} | ||
+ | *{{MathWorld |urlname=Antiprism |title=Antiprism}} | ||
+ | *{{GlossaryForHyperspace |anchor=Antiprism |title=Antiprism}} | ||
+ | **{{GlossaryForHyperspace |anchor=Prismatic |title=Prismatic polytopes}} | ||
+ | *[http://home.comcast.net/~tpgettys/nonconvexprisms.html Nonconvex Prisms and Antiprisms] | ||
+ | *[http://www.software3d.com/Prisms.php Paper models of prisms and antiprisms] | ||
+ | |||
+ | {{Polyhedron navigator}} | ||
+ | |||
+ | [[Category:Uniform polyhedra]] | ||
+ | [[Category:Prismatoid polyhedra]] |
Revision as of 13:40, 31 July 2014
Set of uniform antiprisms | |
---|---|
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 | D_{nd}, [2^{+},2n], (2*n), order 4n |
Rotation group | D_{n}, [2,n]^{+}, (22n), order 2n |
Dual polyhedron | trapezohedron |
Properties | convex, semi-regular vertex-transitive |
Net |
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.
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
Symmetry
The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is D_{nd} of order 4n, except in the case of a tetrahedron, which has the larger symmetry group T_{d} of order 24, which has three versions of D_{2d} as subgroups, and the octahedron, which has the larger symmetry group O_{h} of order 48, which has four versions of D_{3d} as subgroups.
The symmetry group contains inversion if and only if n is odd.
The rotation group is D_{n} 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 D_{2} as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D_{3} as subgroups.
Star antiprism
Symmetry group | Star forms | |||
---|---|---|---|---|
d_{5h} [2,5] (*225) |
3.3.3.5/2 | |||
d_{5d} [2^{+},5] (2*5) |
3.3.3.5/3 | |||
d_{7h} [2,7] (*227) |
3.3.3.7/2 |
3.3.3.7/4 | ||
d_{7d} [2^{+},7] (2*7) |
3.3.3.7/3 | |||
d_{8d} [2^{+},8] (2*8) |
3.3.3.8/3 |
3.3.3.8/5 | ||
d_{9h} [2,9] (*229) |
3.3.3.9/2 |
3.3.3.9/4 | ||
d_{9d} [2^{+},9] (2*9) |
3.3.3.9/5 | |||
d_{10d} [2^{+},10] (2*10) |
3.3.3.10/3 | |||
d_{11h} [2,11] (*2.2.11) |
3.3.3.11/2 |
3.3.3.11/4 |
3.3.3.11/6 | |
d_{11d} [2^{+},11] (2*11) |
3.3.3.11/3 |
3.3.3.11/5 |
3.3.3.11/7 | |
d_{12d} [2^{+},12] (2*12) |
3.3.3.12/5 |
3.3.3.12/7 | ||
... |
- Prism
- Apeirogonal antiprism
- Grand antiprism – a four-dimensional polytope
- One World Trade Center, a building consisting primarily of an elongated square antiprism^{[1]}
References
- {{#invoke:citation/CS1|citation
|CitationClass=book }} Chapter 2: Archimedean polyhedra, prisma and antiprisms