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In geometry, an Template:Nowrap antiprism or Template:Nowrap is a polyhedron composed of two parallel direct copies (not mirror images) of an Template:Nowrap polygon, connected by an alternating band of Template:Math triangles. They are represented by the Conway notation Template: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 Template:Math triangles, rather than Template:Mvar quadrilaterals.
The dual polyhedron of an Template:Mvar-gonal antiprism is an Template:Mvar-gonal trapezohedron.
HistoryEdit
In his 1619 book Harmonices Mundi, Johannes Kepler observed the existence of the infinite family of antiprisms.<ref>Template:Cite book See also 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 of a hexagonal antiprism has been attributed to Hieronymus Andreae, who died in 1556.<ref>Template:Cite journal</ref>
The German form of the word "antiprism" was used for these shapes in the 19th century; Karl Heinze credits its introduction to Template:Ill.<ref>Template:Cite book</ref> Although the English "anti-prism" had been used earlier for an optical prism used to cancel the effects of a primary optical element,<ref>Template:Cite journal</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>Template:Cite journal</ref>
Special casesEdit
Right antiprismEdit
For an antiprism with [[Regular polygon|regular Template:Mvar-gon]] bases, one usually considers the case where these two copies are twisted by an angle of Template: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 Template:Mvar-gon bases, twisted by an angle of Template: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 Template:Math side faces are isosceles triangles.<ref name=oh>Template:Cite book</ref>
The symmetry group of a right Template:Mvar-antiprism is Template:Math of order Template:Math known as an antiprismatic symmetry, because it could be obtained by rotation of the bottom half of a prism by <math> \pi/n </math> in relation to the top half. A concave polyhedron created in this way would have this symmetry group, hence prefix "anti" before "prismatic".<ref>Template:Cite book</ref> There are two exceptions having groups different than Template:Math:
- Template:Math: the regular tetrahedron, which has the larger symmetry group Template:Math of order [[List of spherical symmetry groups#Polyhedral symmetry|Template:Math]], which has three versions of Template:Math as subgroups;
- Template:Math: the regular octahedron, which has the larger symmetry group Template:Math of order Template:Math, which has four versions of Template:Math as subgroups.<ref>Template:Cite book</ref>
If a right 2- or 3-antiprism is not uniform, then its symmetry group is Template:Math or Template:Math as usual.
The symmetry group contains inversion if and only if Template:Mvar is odd.
The rotation group is Template:Math of order Template:Math, except in the cases of:
- Template:Math: the regular tetrahedron, which has the larger rotation group Template:Math of order Template:Math, which has only one subgroup Template:Math;
- Template:Math: the regular octahedron, which has the larger rotation group Template:Math of order Template:Math, which has four versions of Template:Math as subgroups.
If a right 2- or 3-antiprism is not uniform, then its rotation group is Template:Math or Template:Math as usual.
The right Template:Mvar-antiprisms have congruent regular Template:Mvar-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform Template:Mvar-antiprism, for Template:Math.
Uniform antiprismEdit
A uniform Template:Mvar-antiprism has two congruent regular Template:Mvar-gons as base faces, and Template:Math equilateral triangles as side faces. As do uniform prisms, the uniform antiprisms form an infinite class of vertex-transitive polyhedra. For Template:Math, one has the digonal antiprism (degenerate antiprism), which is visually identical to the regular tetrahedron; for Template:Math, the regular octahedron is a triangular antiprism (non-degenerate antiprism).<ref name=oh/>
The Schlegel diagrams of these semiregular antiprisms are as follows:
Cartesian coordinatesEdit
Cartesian coordinates for the vertices of a right Template:Mvar-antiprism (i.e. with regular Template:Mvar-gon bases and Template:Math 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 Template:Math;
if the Template: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 areaEdit
Let Template:Mvar be the edge-length of a uniform Template:Mvar-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 [[#Right antiprism|right Template:Mvar-gonal antiprism]] with side length of its bases Template:Mvar and height Template:Mvar is given by:Template:Sfnp <math display=block>V = \frac{nhl^2}{12} \left( \csc\frac{\pi}{n} + 2\cot\frac{\pi}{n}\right).</math>
DerivationEdit
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>
The volume of a right Template:Mvar-gonal prism with the same Template:Mvar and Template: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.
GeneralizationsEdit
In higher dimensionsEdit
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.<ref>Template:Cite journal</ref> However, there exist four-dimensional polychora that cannot be combined with their duals to form five-dimensional antiprisms.<ref>Template:Cite journal</ref>
Self-crossing polyhedraEdit
{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Ambox }} }} Template:Further
| File:Crossed-triangular prism.png 3/2-antiprism nonuniform |
File:Crossed pentagonal antiprism.png 5/4-antiprism nonuniform |
File:Pentagrammic antiprism.png 5/2-antiprism |
File:Pentagrammic crossed antiprism.png 5/3-antiprism |
| File:Antiprism 9-2.png 9/2-antiprism |
File:Antiprism 9-4.png 9/4-antiprism |
File:Antiprism 9-5.png 9/5-antiprism |
Note: Octagrammic crossed antiprism (8/5) is missing.
Uniform star antiprisms are named by their star polygon bases, Template:Math and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: Template:Math instead of Template:Math; example: (5/3) instead of (5/2).
A right star Template:Math-antiprism has two congruent coaxial regular convex or star polygon base faces, and Template: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, 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 Template:Mset-gon bases (vertex configuration: 3.3.3.7/5) cannot be uniform.
Also, star antiprism compounds with regular star Template:Math-gon bases can be constructed if Template:Mvar and Template:Mvar have common factors. Example: a star (10/4)-antiprism is the compound of two star (5/2)-antiprisms.
Number of uniform crossed antiprismsEdit
If the notation Template:Math is used for an antiprism, then for Template:Math the antiprism is crossed (by definition) and for Template:Math is not. In this section all antiprisms are assumed to be non-degenerate, i.e. Template:Math, Template:Math. Also, the condition Template:Math (Template:Mvar and Template:Mvar are relatively prime) holds, as compounds are excluded from counting. The number of uniform crossed antiprisms for fixed Template:Mvar can be determined using simple inequalities. The condition on possible Template:Mvar is
Examples:
- Template:Mvar = 3: 2 ≤ Template:Mvar ≤ 1 – a uniform triangular crossed antiprism does not exist.
- Template:Mvar = 5: 3 ≤ Template:Mvar ≤ 3 – one antiprism of the type (5/3) can be uniform.
- Template:Mvar = 29: 15 ≤ Template:Mvar ≤ 19 – there are five possibilities (15 thru 19) shown in the rightmost column, below the (29/1) convex antiprism, on the image above.
- Template:Mvar = 15: 8 ≤ Template:Mvar ≤ 9 – antiprism with Template:Mvar = 8 is a solution, but Template:Mvar = 9 must be rejected, as (15,9) = 3 and Template:Sfrac = Template:Sfrac. The antiprism (15/9) is a compound of three antiprisms (5/3). Since 9 satisfies the inequalities, the compound can be uniform, and if it is, then its parts must be. Indeed, the antiprism (5/3) can be uniform by example 2.
In the first column of the following table, the symbols are Schoenflies, Coxeter, and orbifold notation, in this order.
See alsoEdit
- Antiprism graph, graph of an antiprism
- Grand antiprism, a four-dimensional polytope
- Skew polygon, a three-dimensional polygon whose convex hull is an antiprism
ReferencesEdit
Further readingEdit
- Template:Cite book Chapter 2: Archimedean polyhedra, prisms and antiprisms
External linksEdit
- Template:Commons category-inline
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Antiprism%7CAntiprism.html}} |title = Antiprism |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}