In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
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 triangles, rather than quadrilaterals.
The dual polyhedron of an -gonal antiprism is an -gonal trapezohedron.
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 .[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]
For an antiprism with regular -gon bases, one usually considers the case where these two copies are twisted by an angle of 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 -gon bases, twisted by an angle of 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 side faces are isosceles triangles.[6]
The symmetry group of a right -antiprism is of order known as an antiprismatic symmetry, because it could be obtained by rotation of the bottom half of a prism by
\pi/n
the regular tetrahedron, which has the larger symmetry group of order, which has three versions of as subgroups;
the regular octahedron, which has the larger symmetry group of order, which has four versions of as subgroups.[8] If a right 2- or 3-antiprism is not uniform, then its symmetry group is or as usual.
The symmetry group contains inversion if and only if is odd.
The rotation group is of order, except in the cases of:
the regular tetrahedron, which has the larger rotation group of order, which has only one subgroup ;
the regular octahedron, which has the larger rotation group of order, which has four versions of as subgroups.If a right 2- or 3-antiprism is not uniform, then its rotation group is or as usual.
The right -antiprisms have congruent regular -gon bases and congruent isosceles triangle side faces, thus have the same (dihedral) symmetry group as the uniform -antiprism, for .
Cartesian coordinates for the vertices of a right -antiprism (i.e. with regular -gon bases and isosceles triangle side faces, circumradius of the bases equal to 1) are:
\left(\cos
| k\pi | |
| n |
,\sin
| k\pi | |
| n |
,(-1)kh\right)
where ;
if the -antiprism is uniform (i.e. if the triangles are equilateral), then:
A uniform -antiprism has two congruent regular -gons as base faces, and equilateral triangles as side faces. As do uniform prisms, the uniform antiprisms form an infinite class of vertex-transitive polyhedra. For, one has the digonal antiprism (degenerate antiprism), which is visually identical to the regular tetrahedron; for, the regular octahedron is a triangular antiprism (non-degenerate antiprism).[6]
The Schlegel diagrams of these semiregular antiprisms are as follows:
Let be the edge-length of a uniform -gonal antiprism; then the volume is:and the surface area is:Furthermore, the volume of a regular right -gonal antiprism with side length of its bases and height is given by:
The circumradius of the horizontal circumcircle of the regular
n
R(0)=
| l | |||
|
.
| \left(\begin{array}{c}R(0)\cos | 2\pim | \ R(0)\sin |
| n |
| 2\pim | |
| n |
\ 0\end{array}\right), m=0..n-1;
| \left(\begin{array}{c}R(0)\cos | 2\pi(m+1/2) | \\R(0)\sin |
| n |
| 2\pi(m+1/2) | |
| n |
\\h\end{array}\right), m=0..n-1.
| \left(\begin{array}{c} | R(0) | [(h-z)\cos |
| h |
| 2\pim | +z\cos | |
| n |
| \pi(2m+1) | ]\\ | |
| n |
| R(0) | [(h-z)\sin | |
| h |
| 2\pim | +z\sin | |
| n |
| \pi(2m+1) | |
| n |
]\\ \\z\end{array}\right), 0\lez\leh,m=0..n-1
| \left(\begin{array}{c} | R(0) | [(h-z)\cos |
| h |
| 2\pi(m+1) | +z\cos | |
| n |
| \pi(2m+1) | ]\\ | |
| n |
| R(0) | [(h-z)\sin | |
| h |
| 2\pi(m+1) | +z\sin | |
| n |
| \pi(2m+1) | |
| n |
]\\ \\z\end{array}\right), 0\lez\leh,m=0..n-1.
x
y
z
R(z)2=
| R(0)2 | |
| h2 |
[h2-2hz+2z
| ||||
].
0\lez\leh
2n
n
n
l1(z)=l(1-z/h)
n
l2(z)=lz/h
n
R(z),R(z)
l1
R(z)+l1(z)/2
n
R(z),R(z)
l2(z)
R(z)+l2(z)/2
Q1(z)=
| R(0)2 | (h-z)\left[(h-z)\cos | |
| h2 |
| \pi | |
| n |
+z\right]\sin
| \pi | |
| n |
Q2(z)=
| R(0)2 | z\left[z\cos | |
| h2 |
| \pi | |
| n |
+h-z\right]\sin
| \pi | |
| n |
.
n[Q1(z)+Q2(z)]
V=
| h | |
| n\int | |
| 0 |
[Q1(z)+Q2(z)]dz=
| nh | |
| 3 |
| |||||
| R(0) | (1+2\cos |
| \pi | |
| n |
) =
| nh | |
| 12 |
| ||||||||||
| l |
.
The volume of a right -gonal prism with the same and is:which is smaller than that of an antiprism.
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]
A right star -antiprism has two congruent coaxial regular convex or star polygon base faces, and 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:
Also, star antiprism compounds with regular star -gon bases can be constructed if and have common factors. Example: a star (10/4)-antiprism is the compound of two star (5/2)-antiprisms.
The dual of a antiprism with is a -trapezohedron, and the dual of a antiprism with, i.e. crossed antiprism, is a concave trapezohedron, where "concave" refers to 2-dimensional faces of the 3D solid.[11]
If the notation is used for an antiprism, then for the antiprism is crossed (by definition) and for is not. In this section all antiprisms are assumed to be non-degenerate, i.e., . Also, the condition (and are relatively prime) holds, as compounds are excluded from counting. The number of uniform crossed antiprisms for fixed can be determined using simple inequalities. The condition on possible is
and
Examples:
In the first column of the following table, the symbols are Schoenflies, Coxeter, and orbifold notation, in this order.
