Prism (geometry)

An oblique prism is a prism in which the joining edges and faces are not perpendicular to the base faces.

Example: a parallelepiped is an oblique prism whose base is a parallelogram, or equivalently a polyhedron with six parallelogram faces.

The volume of a prism is the product of the area of the base by the height, i.e. the distance between the two base faces (in the case of a non-right prism, note that this means the perpendicular distance).

The volume is therefore:

${\displaystyle V=Bh,}$

where B is the base area and h is the height.

The volume of a prism whose base is an n-sided regular polygon with side length s is therefore:

${\displaystyle V={\frac {n}{4}}hs^{2}\cot \left({\frac {\pi }{n}}\right).}$

The surface area of a right prism is:

${\displaystyle 2B+Ph,}$

where B is the area of the base, h the height, and P the base perimeter.

The surface area of a right prism whose base is a regular n-sided polygon with side length s, and with height h, is therefore:

${\displaystyle A={\frac {n}{2}}s^{2}\cot \left({\frac {\pi }{n}}\right)+nsh.}$

The symmetry group of a right n-sided prism with regular base is D_nh of order 4n, except in the case of a cube, which has the larger symmetry group O_h of order 48, which has three versions of D_4h as subgroups. The rotation group is D_n of order 2n, except in the case of a cube, which has the larger symmetry group O of order 24, which has three versions of D₄ as subgroups.

The symmetry group D_nh contains inversion iff n is even.

The hosohedra and dihedra also possess dihedral symmetry, and an n-gonal prism can be constructed via the geometrical truncation of an n-gonal hosohedron, as well as through the cantellation or expansion of an n-gonal dihedron.

A truncated prism is a prism with non-parallel top and bottom faces.[5]

Example truncated triangular prism. Its top face is truncated at an oblique angle, but it is NOT an oblique prism!

A twisted prism is a nonconvex polyhedron constructed from a uniform n-prism with each side face bisected on the square diagonal, by twisting the top, usually by π/n radians (180/n degrees) in the same direction, causing sides to be concave.[6][7]

A twisted prism cannot be dissected into tetrahedra without adding new vertices. The smallest case: the triangular form, is called a Schönhardt polyhedron.

An n-gonal twisted prism is topologically identical to the n-gonal uniform antiprism, but has half the symmetry group: D_n, [n,2]⁺, order 2n. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.

3-gonal	4-gonal		12-gonal
Schönhardt polyhedron	Twisted square prism	Square antiprism	Twisted dodecagonal antiprism

A frustum is a similar construction to a prism, with trapezoid lateral faces and differently sized top and bottom polygons.

Example pentagonal frustum

A star prism is a nonconvex polyhedron constructed by two identical star polygon faces on the top and bottom, being parallel and offset by a distance and connected by rectangular faces. A uniform star prism will have Schläfli symbol {p/q} × { }, with p rectangle and 2 {p/q} faces. It is topologically identical to a p-gonal prism.

Examples

{ }×{ }180×{ }	ta{3}×{ }		{5/2}×{ }	{7/2}×{ }	{7/3}×{ }	{8/3}×{ }
D2h, order 8	D3h, order 12		D5h, order 20	D7h, order 28		D8h, order 32

Crossed prism

A crossed prism is a nonconvex polyhedron constructed from a prism, where the vertices of one base are inverted around the center of this base (or rotated by 180°). This transforms the side rectangular faces into crossed rectangles. For a regular polygon base, the appearance is an n-gonal hour glass. All oblique edges pass through a single body center. Note: no vertex is at this body centre. A crossed prism is topologically identical to an n-gonal prism.

Examples

{ }×{ }180×{ }180	ta{3}×{ }180		{3}×{ }180	{4}×{ }180	{5}×{ }180	{5/2}×{ }180	{6}×{ }180
D2h, order 8	D3d, order 12			D4h, order 16	D5d, order 20		D6d, order 24

Toroidal prism

A toroidal prism is a nonconvex polyhedron like a crossed prism, but without bottom and top base faces, and with simple rectangular side faces closing the polyhedron. This can only be done for even-sided base polygons. These are topological tori, with Euler characteristic of zero. The topological polyhedral net can be cut from two rows of a square tiling (with vertex configuration 4.4.4.4): a band of n squares, each attached to a crossed rectangle. An n-gonal toroidal prism has 2n vertices, 2n faces: n squares and n crossed rectangles, and 4n edges. It is topologically self-dual.

Examples

D4h, order 16	D6h, order 24
v=8, e=16, f=8	v=12, e=24, f=12

A prismatic polytope is a higher-dimensional generalization of a prism. An n-dimensional prismatic polytope is constructed from two (n − 1)-dimensional polytopes, translated into the next dimension.

The prismatic n-polytope elements are doubled from the (n − 1)-polytope elements and then creating new elements from the next lower element.

Take an n-polytope with f_i i-face elements (i = 0, ..., n). Its (n + 1)-polytope prism will have 2f_i + f_i−1 i-face elements. (With f₋₁ = 0, f_n = 1.)

By dimension:

• Take a polygon with n vertices, n edges. Its prism has 2n vertices, 3n edges, and 2 + n faces.
• Take a polyhedron with v vertices, e edges, and f faces. Its prism has 2v vertices, 2e + v edges, 2f + e faces, and 2 + f cells.
• Take a polychoron with v vertices, e edges, f faces, and c cells. Its prism has 2v vertices, 2e + v edges, 2f + e faces, 2c + f cells, and 2 + c hypercells.

Uniform prismatic polytope

A regular n-polytope represented by Schläfli symbol {p,q,...,t} can form a uniform prismatic (n + 1)-polytope represented by a Cartesian product of two Schläfli symbols: {p,q,...,t}×{ }.

By dimension:

• A 0-polytopic prism is a line segment, represented by an empty Schläfli symbol { }.

  

• A 1-polytopic prism is a rectangle, made from 2 translated line segments. It is represented as the product Schläfli symbol { }×{ }. If it is square, symmetry can be reduced: { }×{ } = {4}.

  Example: , Square, { }×{ }, two parallel line segments, connected by two line segment sides.

• A polygonal prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon {p} can construct a uniform n-gonal prism represented by the product {p}×{ }. If p = 4, with square sides symmetry it becomes a cube: {4}×{ } = {4,3}.

  Example: , Pentagonal prism, {5}×{ }, two parallel pentagons connected by 5 rectangular sides.

• A polyhedral prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron {p,q} can construct the uniform polychoric prism, represented by the product {p,q}×{ }. If the polyhedron and the sides are cubes, it becomes a tesseract: {4,3}×{ } = {4,3,3}.

  Example: , Dodecahedral prism, {5,3}×{ }, two parallel dodecahedra connected by 12 pentagonal prism sides.

• ...

A {23}×{29} duoprism, showing edges in stereographic projection. The squares make a 23×29 grid flat torus.

Higher order prismatic polytopes also exist as cartesian products of any two or more polytopes. The dimension of a product polytope is the sum of the dimensions of its elements. The first examples of these exist in 4-dimensional space; they are called duoprisms as the product of two polygons in 4-dimensions.

Regular duoprisms are represented as {p}×{q}, with pq vertices, 2pq edges, pq square faces, p q-gon faces, q p-gon faces, and bounded by p q-gonal prisms and q p-gonal prisms.

For example, {4}×{4}, a 4-4 duoprism is a lower symmetry form of a tesseract, as is {4,3}×{ }, a cubic prism. {4}×{4}×{ } (4-4 duoprism prism), {4,3}×{4} (cube-4 duoprism) and {4,3,3}×{ } (tesseractic prism) are lower symmetry forms of a 5-cube.

• Apeirogonal prism
• Rectified prism
• Prismanes
• List of shapes

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

Wikisource has the text of the 1911 Encyclopædia Britannica article "Prism".

• Weisstein, Eric W. "Prism". MathWorld.
• Paper models of prisms and antiprisms Free nets of prisms and antiprisms
• Paper models of prisms and antiprisms Using nets generated by Stella