Difference between revisions of "Antiprism"
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|bgcolor=#e7dcc3|[[Vertex configuration]]||3.3.3.''n'' | |bgcolor=#e7dcc3|[[Vertex configuration]]||3.3.3.''n'' | ||
|- | |- | ||
− | |bgcolor=#e7dcc3|[[Schläfli symbol]]||s{2,2''n''}<br>sr{2,''n''}<BR>{ } &# | + | |bgcolor=#e7dcc3|[[Schläfli symbol]]||s{2,2''n''}<br>sr{2,''n''}<BR>{ } {{Unicode|⊗}} {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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{{UniformAntiprisms}} | {{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== | ||
Line 55: | Line 66: | ||
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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== Star antiprism == | == Star antiprism == | ||
+ | {| 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" | {| class="wikitable" | ||
+ | |+ Uniform star antiprisms by symmetry, up to 12 | ||
![[List of spherical symmetry groups|Symmetry group]] | ![[List of spherical symmetry groups|Symmetry group]] | ||
!colspan=4|Star forms | !colspan=4|Star forms |
Latest revision as of 08:05, 11 January 2015
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.
Schlegel diagrams
A3 |
A4 |
A5 |
A6 |
A7 |
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.
... | ||||
ts{2,4} | ts{2,6} | ts{2,8} | ts{2,10} | ts{2,2n} |
Snub antiprisms | ||||
J84 | Icosahedron | J85 | Irregular... | |
... | ||||
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 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
5/2-antiprism |
5/3-antiprism | ||||
9/2-antiprism |
9/4-antiprism |
9/5-antiprism |
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.
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