Antiprism

In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies (not mirror images) of an n-sided polygon, connected by an alternating band of 2n triangles.

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 2n triangles, rather than n quadrilaterals.

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 Theodor Wittstein [de].

[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.

[5] 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.

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.

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.

The Schlegel diagrams of these semiregular antiprisms are as follows: Cartesian coordinates for the vertices of a right n-antiprism (i.e. with regular n-gon bases and 2n isosceles triangle side faces, circumradius of the bases equal to 1) are: where 0 ≤ k ≤ 2n – 1; if the n-antiprism is uniform (i.e. if the triangles are equilateral), then:

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

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

The circumradius of the horizontal circumcircle of the regular

-gon at the base is The vertices at the base are at the vertices at the top are at 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 and at By building the sums of the squares of the

coordinates in one of the previous two vectors, the squared circumradius of this section at altitude

(These are derived from the length of the difference of the previous two vectors.)

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

The symmetry group of a right n-antiprism (i.e. with regular bases and isosceles side faces) is Dnd = Dnv of order 4n, except in the cases of: The symmetry group contains inversion if and only if n is odd.

The rotation group is Dn of order 2n, except in the cases of: 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.

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 polychora that cannot be combined with their duals to form five-dimensional antiprisms.

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: Also, star antiprism compounds with regular star p/q-gon bases can be constructed if p and q have common factors.