Английская Википедия:Antiprism
Шаблон:Short description Шаблон:More footnotes Шаблон:Infobox polyhedron
In geometry, an Шаблон:Nowrap antiprism or Шаблон:Nowrap is a polyhedron composed of two parallel direct copies (not mirror images) of an Шаблон:Nowrap polygon, connected by an alternating band of Шаблон:Math triangles. They are represented by the Conway notation Шаблон:Math.
Antiprisms are a subclass of prismatoids, and are a (degenerate) type of snub polyhedron.
Antiprisms are similar to prisms, except that the bases are twisted relatively to each other, and that the side faces (connecting the bases) are Шаблон:Math triangles, rather than Шаблон:Mvar quadrilaterals.
The dual polyhedron of an Шаблон:Mvar-gonal antiprism is an Шаблон:Mvar-gonal trapezohedron.
History
In his 1619 book Harmonices Mundi, Johannes Kepler observed the existence of the infinite family of antiprisms.[1] 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 of a hexagonal antiprism has been attributed to Hieronymus Andreae, who died in 1556.[2]
The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to Шаблон:Ill.[3] Although the English "anti-prism" had been used earlier for an optical prism used to cancel the effects of a primary optimal element,[4] 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.[5]
Special cases
Right antiprism
For an antiprism with [[Regular polygon|regular Шаблон:Mvar-gon]] bases, one usually considers the case where these two copies are twisted by an angle of Шаблон:Math 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 congruent regular Шаблон:Mvar-gon bases, twisted by an angle of Шаблон:Math degrees, more regularity is obtained if the bases have the same axis: are coaxial; i.e. (for non-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 Шаблон:Math side faces are isosceles triangles.
Uniform antiprism
A uniform Шаблон:Mvar-antiprism has two congruent regular Шаблон:Mvar-gons as base faces, and Шаблон:Math equilateral triangles as side faces.
Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For Шаблон:Math, we have the regular tetrahedron as a digonal antiprism (degenerate antiprism); for Шаблон:Math, the regular octahedron as a triangular antiprism (non-degenerate antiprism).
The Schlegel diagrams of these semiregular antiprisms are as follows:
Cartesian coordinates
Cartesian coordinates for the vertices of a right Шаблон:Mvar-antiprism (i.e. with regular Шаблон:Mvar-gon bases and Шаблон:Math isosceles triangle side faces) are:
- <math>\left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^k h \right)</math>
where Шаблон:Math;
if the Шаблон:Mvar-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 Шаблон:Mvar be the edge-length of a uniform Шаблон:Mvar-gonal antiprism; then the volume is: <math display=block>V = ~n\frac{\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 [[Antiprism#Right_antiprism|right Шаблон:Mvar-gonal antiprism]] with side length of its bases Шаблон:Mvar and height Шаблон:Mvar is given by:
<math display=block>V = \frac{nhl^2}{12} \left( \csc\frac{\pi}{n} + 2\cot\frac{\pi}{n}\right).</math>
Note that the volume of a right Шаблон:Mvar-gonal prism with the same Шаблон:Mvar and Шаблон:Mvar 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 Шаблон:Mvar-antiprism (i.e. with regular bases and isosceles side faces) is Шаблон:Math of order Шаблон:Math, except in the cases of:
- Шаблон:Math: the regular tetrahedron, which has the larger symmetry group Шаблон:Math of order Шаблон:Math, which has three versions of Шаблон:Math as subgroups;
- Шаблон:Math: the regular octahedron, which has the larger symmetry group Шаблон:Math of order Шаблон:Math, which has four versions of Шаблон:Math as subgroups.
The symmetry group contains inversion if and only if Шаблон:Mvar is odd.
The rotation group is Шаблон:Math of order Шаблон:Math, except in the cases of:
- Шаблон:Math: the regular tetrahedron, which has the larger rotation group Шаблон:Math of order Шаблон:Math, which has three versions of Шаблон:Math as subgroups;
- Шаблон:Math: the regular octahedron, which has the larger rotation group Шаблон:Math of order Шаблон:Math, which has four versions of Шаблон:Math as subgroups.
Note: The right Шаблон:Mvar-antiprisms have congruent regular Шаблон:Mvar-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform Шаблон:Mvar-antiprism, for Шаблон:Math.
Generalizations
In higher dimensions
Four-dimensional antiprisms can be defined as having two dual polyhedra as parallel opposite faces, so that each 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.[6] However, there exist four-dimensional polyhedra that cannot be combined with their duals to form five-dimensional antiprisms.[7]
Self-crossing polyhedra
| Файл:Pentagrammic antiprism.png 5/2-antiprism |
Файл:Pentagrammic crossed antiprism.png 5/3-antiprism | ||||
| Файл:Antiprism 9-2.png 9/2-antiprism |
Файл:Antiprism 9-4.png 9/4-antiprism |
Файл:Antiprism 9-5.png 9/5-antiprism | |||
Шаблон:See Uniform star antiprisms are named by their star polygon bases, Шаблон:Math and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: Шаблон:Math instead of Шаблон:Math; example: 5/3 instead of 5/2.
A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and Шаблон:Math 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, 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, 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 Шаблон:Math-gon bases can be constructed if Шаблон:Mvar and Шаблон:Mvar have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.
See also
- Grand antiprism, a four-dimensional polytope
- Skew polygon, a three-dimensional polygon whose convex hull is an antiprism
References
Further reading
- Шаблон:Cite book Chapter 2: Archimedean polyhedra, prisms and antiprisms
External links
- Шаблон:Commonscatinline
- Шаблон:MathWorld
- Nonconvex Prisms and Antiprisms
- Paper models of prisms and antiprisms
- ↑ Шаблон:Cite book See also illustration A, of a heptagonal antiprism.
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
