Antiprism

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Octagonal antiprism

In geometry, an Template:Mvar-gonal antiprism or Template:Mvar-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an Template:Mvar-sided polygon, connected by an alternating band of $2 n$ Script error: No such module "Check for unknown parameters". triangles. They are represented by the Conway notation $A n$ Script error: No such module "Check for unknown parameters"..

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 $2 n$ Script error: No such module "Check for unknown parameters". triangles, rather than Template:Mvar quadrilaterals.

The dual polyhedron of an Template:Mvar-gonal antiprism is an Template: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 Template:Ill.^[3] Although the English "anti-prism" had been used earlier for an optical prism used to cancel the effects of a primary optical 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 Template:Mvar-gon]] bases, one usually considers the case where these two copies are twisted by an angle of $Template:Sfrac$ Script error: No such module "Check for unknown parameters". 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 $Template:Sfrac$ Script error: No such module "Check for unknown parameters". 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 $2 n$ Script error: No such module "Check for unknown parameters". side faces are isosceles triangles.^[6]

The symmetry group of a right Template:Mvar-antiprism is $D n d$ Script error: No such module "Check for unknown parameters". of order $4 n$ Script error: No such module "Check for unknown parameters". known as an antiprismatic symmetry, because it could be obtained by rotation of the bottom half of a prism by $π / n$ in relation to the top half. A concave polyhedron created in this way would have this symmetry group, hence prefix "anti" before "prismatic".^[7] There are two exceptions having groups different than $D n d$ Script error: No such module "Check for unknown parameters".:

$n = 2$ Script error: No such module "Check for unknown parameters".: the regular tetrahedron, which has the larger symmetry group $T d$ Script error: No such module "Check for unknown parameters". of order $24$ Script error: No such module "Check for unknown parameters"., which has three versions of $D 2d$ Script error: No such module "Check for unknown parameters". as subgroups;
$n = 3$ Script error: No such module "Check for unknown parameters".: the regular octahedron, which has the larger symmetry group $O h$ Script error: No such module "Check for unknown parameters". of order $48$ Script error: No such module "Check for unknown parameters"., which has four versions of $D 3d$ Script error: No such module "Check for unknown parameters". as subgroups.^[8]

If a right 2- or 3-antiprism is not uniform, then its symmetry group is $D 2d$ Script error: No such module "Check for unknown parameters". or $D 3d$ Script error: No such module "Check for unknown parameters". as usual.
The symmetry group contains inversion if and only if Template:Mvar is odd.

The rotation group is $D n$ Script error: No such module "Check for unknown parameters". of order $2 n$ Script error: No such module "Check for unknown parameters"., except in the cases of:

$n = 2$ Script error: No such module "Check for unknown parameters".: the regular tetrahedron, which has the larger rotation group $T$ Script error: No such module "Check for unknown parameters". of order $12$ Script error: No such module "Check for unknown parameters"., which has only one subgroup $D 2$ Script error: No such module "Check for unknown parameters".;
$n = 3$ Script error: No such module "Check for unknown parameters".: the regular octahedron, which has the larger rotation group $O$ Script error: No such module "Check for unknown parameters". of order $24$ Script error: No such module "Check for unknown parameters"., which has four versions of $D 3$ Script error: No such module "Check for unknown parameters". as subgroups.

If a right 2- or 3-antiprism is not uniform, then its rotation group is $D 2$ Script error: No such module "Check for unknown parameters". or $D 3$ Script error: No such module "Check for unknown parameters". 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 $n \geq 4$ Script error: No such module "Check for unknown parameters"..

Uniform antiprism

A uniform Template:Mvar-antiprism has two congruent regular Template:Mvar-gons as base faces, and $2 n$ Script error: No such module "Check for unknown parameters". equilateral triangles as side faces. As do uniform prisms, the uniform antiprisms form an infinite class of vertex-transitive polyhedra. For $n = 2$ Script error: No such module "Check for unknown parameters"., one has the digonal antiprism (degenerate antiprism), which is visually identical to the regular tetrahedron; for $n = 3$ Script error: No such module "Check for unknown parameters"., the regular octahedron is a triangular antiprism (non-degenerate antiprism).^[6]

Template:UniformAntiprisms

The Schlegel diagrams of these semiregular antiprisms are as follows:

File:Triangular antiprismatic graph.png A3	File:Square antiprismatic graph.png A4	File:Pentagonal antiprismatic graph.png A5	File:Hexagonal antiprismatic graph.png A6	File:Heptagonal antiprism graph.png A7	File:Octagonal antiprismatic graph.png A8

Cartesian coordinates

Cartesian coordinates for the vertices of a right Template:Mvar-antiprism (i.e. with regular Template:Mvar-gon bases and $2 n$ Script error: No such module "Check for unknown parameters". isosceles triangle side faces, circumradius of the bases equal to 1) are:

(\cos \frac{k π}{n}, \sin \frac{k π}{n}, (- 1)^{k} h)

where $0 \leq k \leq 2 n - 1$ Script error: No such module "Check for unknown parameters".;

if the Template:Mvar-antiprism is uniform (i.e. if the triangles are equilateral), then: $2 h^{2} = \cos \frac{π}{n} - \cos \frac{2 π}{n} .$

Volume and surface area

Let Template:Mvar be the edge-length of a uniform Template:Mvar-gonal antiprism; then the volume is: $V = \frac{n \sqrt{4 \cos^{2} \frac{π}{2 n} - 1} \sin \frac{3 π}{2 n}}{12 \sin^{2} \frac{π}{n}} a^{3},$ and the surface area is: $A = \frac{n}{2} (\cot \frac{π}{n} + \sqrt{3}) a^{2} .$ 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 $V = \frac{n h l^{2}}{12} (\csc \frac{π}{n} + 2 \cot \frac{π}{n}) .$

Derivation

The circumradius of the horizontal circumcircle of the regular $n$ -gon at the base is

R (0) = \frac{l}{2 \sin \frac{π}{n}} .

The vertices at the base are at

(\begin{array}{c} R (0) \cos \frac{2 π m}{n} \\ R (0) \sin \frac{2 π m}{n} \\ 0 \end{array}), m = 0 . . n - 1;

the vertices at the top are at

(\begin{array}{c} R (0) \cos \frac{2 π (m + 1 / 2)}{n} \\ R (0) \sin \frac{2 π (m + 1 / 2)}{n} \\ h \end{array}), m = 0 . . n - 1 .

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

(\begin{array}{c} \frac{R (0)}{h} [(h - z) \cos \frac{2 π m}{n} + z \cos \frac{π (2 m + 1)}{n}] \\ \frac{R (0)}{h} [(h - z) \sin \frac{2 π m}{n} + z \sin \frac{π (2 m + 1)}{n}] \\ z \end{array}), 0 \leq z \leq h, m = 0 . . n - 1

and at

(\begin{array}{c} \frac{R (0)}{h} [(h - z) \cos \frac{2 π (m + 1)}{n} + z \cos \frac{π (2 m + 1)}{n}] \\ \frac{R (0)}{h} [(h - z) \sin \frac{2 π (m + 1)}{n} + z \sin \frac{π (2 m + 1)}{n}] \\ z \end{array}), 0 \leq z \leq h, m = 0 . . n - 1 .

By building the sums of the squares of the $x$ and $y$ coordinates in one of the previous two vectors, the squared circumradius of this section at altitude $z$ is

R (z)^{2} = \frac{R (0)^{2}}{h^{2}} [h^{2} - 2 h z + 2 z^{2} + 2 z (h - z) \cos \frac{π}{n}] .

The horizontal section at altitude $0 \leq z \leq h$ above the base is a $2 n$ -gon (truncated $n$ -gon) with $n$ sides of length $l_{1} (z) = l (1 - z / h)$ alternating with $n$ sides of length $l_{2} (z) = l z / h$ . (These are derived from the length of the difference of the previous two vectors.) It can be dissected into $n$ isoceless triangles of edges $R (z), R (z)$ and $l_{1}$ (semiperimeter $R (z) + l_{1} (z) / 2$ ) plus $n$ isoceless triangles of edges $R (z), R (z)$ and $l_{2} (z)$ (semiperimeter $R (z) + l_{2} (z) / 2$ ). According to Heron's formula the areas of these triangles are

Q_{1} (z) = \frac{R (0)^{2}}{h^{2}} (h - z) [(h - z) \cos \frac{π}{n} + z] \sin \frac{π}{n}

and

Q_{2} (z) = \frac{R (0)^{2}}{h^{2}} z [z \cos \frac{π}{n} + h - z] \sin \frac{π}{n} .

The area of the section is $n [Q_{1} (z) + Q_{2} (z)]$ , and the volume is

V = n \int_{0}^{h} [Q_{1} (z) + Q_{2} (z)] d z = \frac{n h}{3} R (0)^{2} \sin \frac{π}{n} (1 + 2 \cos \frac{π}{n}) = \frac{n h}{12} l^{2} \frac{1 + 2 \cos \frac{π}{n}}{\sin \frac{π}{n}} .

The volume of a right Template:Mvar-gonal prism with the same Template:Mvar and Template:Mvar is: $V_{p r i s m} = \frac{n h l^{2}}{4} \cot \frac{π}{n}$ which is smaller than that of an antiprism.

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.^[9] However, there exist four-dimensional polychora that cannot be combined with their duals to form five-dimensional antiprisms.^[10]

Self-crossing polyhedra

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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

File:Antiprisms.pdf

All the non-star and star uniform antiprisms up to 15 sides, together with those of a 29-gon (or icosaenneagon). For example, the icosaenneagrammic crossed antiprism (

29/ q

Script error: No such module "Check for unknown parameters".) with the greatest

q

Script error: No such module "Check for unknown parameters"., such that it can be uniform, has

q = 19

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q \geq 20

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28

Script error: No such module "Check for unknown parameters". the crossed antiprism cannot be uniform.
Note: Octagrammic crossed antiprism (8/5) is missing.

Uniform star antiprisms are named by their star polygon bases, ${p / q},$ Script error: No such module "Check for unknown parameters". and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: $p /(p - q)$ Script error: No such module "Check for unknown parameters". instead of $p / q$ Script error: No such module "Check for unknown parameters".; example: (5/3) instead of (5/2).

A right star $n$ Script error: No such module "Check for unknown parameters".-antiprism has two congruent coaxial regular convex or star polygon base faces, and $2 n$ Script error: No such module "Check for unknown parameters". 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 $Template:Mset$ Script error: No such module "Check for unknown parameters".-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 antiprisms

If the notation $(p / q)$ Script error: No such module "Check for unknown parameters". is used for an antiprism, then for $q > p /2$ Script error: No such module "Check for unknown parameters". the antiprism is crossed (by definition) and for $q < p /2$ Script error: No such module "Check for unknown parameters". is not. In this section all antiprisms are assumed to be non-degenerate, i.e. $p \geq 3$ Script error: No such module "Check for unknown parameters"., $q \neq p /2$ Script error: No such module "Check for unknown parameters".. Also, the condition $(p, q) = 1$ Script error: No such module "Check for unknown parameters". (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

Template:Sfrac < q < Template:Sfrac p

Script error: No such module "Check for unknown parameters". and

(p, q) = 1.

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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.

Star ( $p / q$ **Script error: No such module "Check for unknown parameters".**)-antiprisms by symmetry, for $p \leq 12$ **Script error: No such module "Check for unknown parameters".**
Symmetry group	Uniform stars			Right stars
$D 3h [2,3] (23)$ Script error: No such module "Check for unknown parameters".*				File:Crossed triangular antiprism.svg 3.3/2.3.3 Crossed triangular antiprism
$D 4d [2 +,8] (24)$ Script error: No such module "Check for unknown parameters".*				File:Crossed square antiprism.png 3.3/2.3.4 Crossed square antiprism
$D 5h [2,5] (225)$ Script error: No such module "Check for unknown parameters".*	File:Pentagrammic antiprism.png 3.3.3.5/2 Pentagrammic antiprism			File:Crossed pentagonal antiprism.png 3.3/2.3.5 Crossed pentagonal antiprism
$D 5d [2 +,10] (25)$ Script error: No such module "Check for unknown parameters".*	File:Pentagrammic crossed antiprism.png 3.3.3.5/3 Pentagrammic crossed-antiprism
$D 6d [2 +,12] (26)$ Script error: No such module "Check for unknown parameters".*				File:Crossed hexagonal antiprism.png 3.3/2.3.6 Crossed hexagonal antiprism
$D 7h [2,7] (227)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 7-2.png 3.3.3.7/2 Heptagrammic antiprism (7/2)	File:Antiprism 7-4.png 3.3.3.7/4 Heptagrammic crossed antiprism (7/4)
$D 7d [2 +,14] (27)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 7-3.png 3.3.3.7/3 Heptagrammic antiprism (7/3)
$D 8d [2 +,16] (28)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 8-3.png 3.3.3.8/3 Octagrammic antiprism	File:Antiprism 8-5.png 3.3.3.8/5 Octagrammic crossed-antiprism
$D 9h [2,9] (229)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 9-2.png 3.3.3.9/2 Enneagrammic antiprism (9/2)	File:Antiprism 9-4.png 3.3.3.9/4 Enneagrammic antiprism (9/4)
$D 9d [2 +,18] (29)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 9-5.png 3.3.3.9/5 Enneagrammic crossed-antiprism
$D 10d [2 +,20] (210)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 10-3.png 3.3.3.10/3 Decagrammic antiprism
$D 11h [2,11] (2.2.11)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 11-2.png 3.3.3.11/2 Undecagrammic (11/2)	File:Antiprism 11-4.png 3.3.3.11/4 Undecagrammic (11/4)	File:Antiprism 11-6.png 3.3.3.11/6 Undecagrammic crossed (11/6)
$D 11d [2 +,22] (211)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 11-3.png 3.3.3.11/3 Undecagrammic (11/3)	File:Antiprism 11-5.png 3.3.3.11/5 Undecagrammic (11/5)	File:Antiprism 11-7.png 3.3.3.11/7 Undecagrammic crossed (11/7)
$D 12d [2 +,24] (212)$ Script error: No such module "Check for unknown parameters".*	File:Antiprism 12-5.png 3.3.3.12/5 Dodecagrammic	File:Antiprism 12-7.png 3.3.3.12/7 Dodecagrammic crossed
...	...

References

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External links

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Template:Polyhedron navigator

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Antiprism

Contents

History

Special cases

Right antiprism

Uniform antiprism

Cartesian coordinates

Volume and surface area

Derivation

Generalizations

In higher dimensions

Self-crossing polyhedra

Number of uniform crossed antiprisms

See also

References

Further reading

External links

Navigation menu

Antiprism

History

Special cases

Right antiprism

Uniform antiprism

Cartesian coordinates

Volume and surface area

Derivation

Generalizations

In higher dimensions

Self-crossing polyhedra

Number of uniform crossed antiprisms

See also

References

Further reading

External links

Navigation menu

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