Antiprism

At the intersection of modern-day graph theory and coding theory, the triangulation of a set of points have interested mathematicians since Isaac Newton, who fruitlessly sought a mathematical proof of the kissing number problem in 1694.[2] The existence of antiprisms was discussed, and their name was coined by Johannes Kepler, though it is possible that they were previously known to Archimedes, as they satisfy the same conditions on faces and on vertices as the Archimedean solids. According to Ericson and Zinoviev, Harold Scott MacDonald Coxeter wrote at length on the topic,[2] and was among the first to apply the mathematics of Victor Schlegel to this field.

Knowledge in this field is "quite incomplete" and "was obtained fairly recently", i.e. in the 20th century. For example, as of 2001 it had been proven for only a limited number of non-trivial cases that the n-gonal antiprism is the mathematically optimal arrangement of 2n points in the sense of maximizing the minimum Euclidean distance between any two points on the set: in 1943 by László Fejes Tóth for 4 and 6 points (digonal and trigonal antiprisms, which are Platonic solids); in 1951 by Kurt Schütte and Bartel Leendert van der Waerden for 8 points (tetragonal antiprism, which is not a cube).[2]

The chemical structure of binary compounds has been remarked to be in the family of antiprisms;[3] especially those of the family of boron hydrides (in 1975) and carboranes because they are isoelectronic. This is a mathematically real conclusion reached by studies of X-ray diffraction patterns,[4] and stems from the 1971 work of Kenneth Wade,[5] the nominative source for Wade's rules of polyhedral skeletal electron pair theory.

Rare-earth metals such as the lanthanides form antiprismatic compounds with some of the halides or some of the iodides. The study of crystallography is useful here.[6] Some lanthanides, when arranged in peculiar antiprismatic structures with chlorine and water, can form molecule-based magnets.[7]

For an antiprism with regular n-gon bases, one usually considers the case where these two copies are twisted by an angle of 180/n 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 n-gon bases, twisted by an angle of 180/n 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 2n side faces are isosceles triangles.

A uniform n-antiprism has two congruent regular n-gons as base faces, and 2n equilateral triangles as side faces.

Uniform antiprisms form an infinite class of vertex-transitive polyhedra, as do uniform prisms. For n = 2, we have the regular tetrahedron as a digonal antiprism (degenerate antiprism); for n = 3, the regular octahedron as a triangular antiprism (non-degenerate antiprism).

Family of uniform n-gonal antiprisms

Antiprism name	Digonal antiprism	(Trigonal) Triangular antiprism	(Tetragonal) Square antiprism	Pentagonal antiprism	Hexagonal antiprism	Heptagonal antiprism	Octagonal antiprism	Enneagonal antiprism	Decagonal antiprism	Hendecagonal antiprism	Dodecagonal antiprism	...	Apeirogonal antiprism
Polyhedron image												...
Spherical tiling image												Plane tiling image
Vertex config.	2.3.3.3	3.3.3.3	4.3.3.3	5.3.3.3	6.3.3.3	7.3.3.3	8.3.3.3	9.3.3.3	10.3.3.3	11.3.3.3	12.3.3.3	...	∞.3.3.3

Schlegel diagrams

A3

A4

A5

A6

A7

A8

Cartesian coordinates for the vertices of a right n-antiprism (i.e. with regular n-gon bases and 2n isosceles triangle side faces) are:

${\displaystyle \left(\cos {\frac {k\pi }{n}},\sin {\frac {k\pi }{n}},(-1)^{k}h\right)}$

where 0 ≤ k ≤ 2n – 1;

if the n-antiprism is uniform (i.e. if the triangles are equilateral), then:

${\displaystyle 2h^{2}=\cos {\frac {\pi }{n}}-\cos {\frac {2\pi }{n}}.}$

Let a be the edge-length of a uniform n-gonal antiprism; then the volume is:

${\displaystyle V={\frac {n~{\sqrt {4\cos ^{2}{\frac {\pi }{2n}}-1}}\sin {\frac {3\pi }{2n}}}{12\sin ^{2}{\frac {\pi }{n}}}}~a^{3},}$

and the surface area is:

${\displaystyle A={\frac {n}{2}}\left(\cot {\frac {\pi }{n}}+{\sqrt {3}}\right)a^{2}.}$

Furthermore, the volume of a right n-gonal antiprism with side length of its bases l and height h is given by:

${\displaystyle V={\frac {nhl^{2}}{12}}\left(\csc \left({\frac {\pi }{n}}\right)+2\cot \left({\frac {\pi }{n}}\right)\right).}$

Note that the volume of a right n-gonal prism with the same l and h is:

${\displaystyle V_{\mathrm {prism} }={\frac {nhl^{2}}{4}}\cot \left({\frac {\pi }{n}}\right)}$

which is smaller than that of an antiprism.

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 regular icosahedron.

Antiprisms
				...
s{2,4}	s{2,6}	s{2,8}	s{2,10}	s{2,2n}
Truncated antiprisms
				...
ts{2,4}	ts{2,6}	ts{2,8}	ts{2,10}	ts{2,2n}
Snub antiprisms
J84	Icosahedron	J85	Irregular faces...
				...
ss{2,4}	ss{2,6}	ss{2,8}	ss{2,10}	ss{2,2n}

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

The symmetry group of a right n-antiprism (i.e. with regular bases and isosceles side faces) is D_nd = D_nv of order 4n, except in the cases of:

• n = 2: the regular tetrahedron, which has the larger symmetry group T_d of order 24 = 3×(4×2), which has three versions of D_2d as subgroups;

• n = 3: the regular octahedron, which has the larger symmetry group O_h of order 48 = 4×(4×3), 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 cases of:

• n = 2: the regular tetrahedron, which has the larger rotation group T of order 12 = 3×(2×2), which has three versions of D₂ as subgroups;

• n = 3: the regular octahedron, which has the larger rotation group O of order 24 = 4×(2×3), which has four versions of D₃ as subgroups.

Note: The right n-antiprisms have congruent regular n-gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform n-antiprism, for n ≥ 4.

5/2-antiprism			5/3-antiprism
9/2-antiprism		9/4-antiprism		9/5-antiprism

This shows all the non-star and star antiprisms up to 15 sides, together with those of a 29-gon.

Uniform star antiprisms are named by their star polygon bases, {p/q}, and exist in prograde and in retrograde (crossed) solutions. Crossed forms have intersecting vertex figures, and are denoted by "inverted" fractions: p/(p – q) instead of p/q; example: 5/3 instead of 5/2.

A right star antiprism has two congruent coaxial regular convex or star polygon base faces, and 2n 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 p/q-gon bases can be constructed if p and q have common factors. Example: a star 10/4-antiprism is the compound of two star 5/2-antiprisms.

Star p/q-antiprisms by symmetry, for p ≤ 12

Symmetry group	Uniform stars				Right stars
D4d [2+,8] (2*4)					3.3/2.3.4
D5h [2,5] (*225)	3.3.3.5/2				3.3/2.3.5
D5d [2+,10] (2*5)	3.3.3.5/3
D6d [2+,12] (2*6)					3.3/2.3.6
D7h [2,7] (*227)	3.3.3.7/2	3.3.3.7/4
D7d [2+,14] (2*7)	3.3.3.7/3
D8d [2+,16] (2*8)	3.3.3.8/3	3.3.3.8/5
D9h [2,9] (*229)	3.3.3.9/2	3.3.3.9/4
D9d [2+,18] (2*9)	3.3.3.9/5
D10d [2+,20] (2*10)	3.3.3.10/3
D11h [2,11] (*2.2.11)	3.3.3.11/2	3.3.3.11/4	3.3.3.11/6
D11d [2+,22] (2*11)	3.3.3.11/3	3.3.3.11/5	3.3.3.11/7
D12d [2+,24] (2*12)	3.3.3.12/5	3.3.3.12/7
...	...

• Apeirogonal antiprism
• Grand antiprism – a four-dimensional polytope
• One World Trade Center, a building consisting primarily of an elongated square antiprism
• Skew polygon

• Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. ISBN 0-520-03056-7. Chapter 2: Archimedean polyhedra, prisms and antiprisms

• Media related to Antiprisms at Wikimedia Commons
• Weisstein, Eric W. "Antiprism". MathWorld.
• Nonconvex Prisms and Antiprisms
• Paper models of prisms and antiprisms