Prism (geometry): Difference between revisions

Set of uniform $n$ -gonal prisms
Set of uniform n-gonal prisms
	Example uniform hexagonal prism
Type	uniform in the sense of semiregular polyhedron
Faces	2 n-gonal regular polygons; n squares
Edges	3n
Vertices	2n
Vertex configuration	4.4.n
Schläfli symbol	{n}×{} ; t{2, n}
Conway notation	Pn
Coxeter diagram
Symmetry group	Dnh, [n,2], (*n22), order 4n
Rotation group	Dn, [n,2]+, (n22), order 2n
Dual polyhedron	convex dual-uniform n-gonal bipyramid
Properties	convex, regular polygon faces, vertex-transitive, translated bases, sides ⊥ bases
Net

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In geometry, a prism is a polyhedron comprising an $n$ -sided polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and $n$ other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

Like many basic geometric terms, the word prism (from Greek πρίσμα (prisma) 'something sawed') was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers.^[2]^[3]

Oblique prism

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 of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms.

Right prism, uniform prism

Right prism

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.^[4] This applies if all the joining faces are rectangular.

The dual of a right n-prism is a right n-bipyramid.

A right prism (with rectangular sides) with regular n-gon bases has Schläfli symbol { }×{n}. It approaches a cylindrical solid as n approaches infinity; a cylinder is considered a circular prism.

Special cases

A right rectangular prism (with a rectangular base) is also called a cuboid, or informally a rectangular box. A right rectangular prism has Schläfli symbol { }×{ }×{ }.

A right square prism (with a square base) is also called a square cuboid, or informally a square box.

Note: some texts may apply the term rectangular prism or square prism to both a right rectangular-based prism and a right square-based prism.

Uniform prism

A uniform prism or semiregular prism is a right prism with regular bases and square sides, since such prisms are in the set of uniform polyhedra.

A uniform n-gonal prism has Schläfli symbol t{2,n}.

Right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being antiprisms.

Family of uniform n-gonal prisms v t e
Prism name	Digonal prism	(Trigonal) Triangular prism	(Tetragonal) Square prism	Pentagonal prism	Hexagonal prism	Heptagonal prism	Octagonal prism	Enneagonal prism	Decagonal prism	Hendecagonal prism	Dodecagonal prism	...	Apeirogonal prism
Polyhedron image												...
Spherical tiling image												Plane tiling image
Vertex config.	2.4.4	3.4.4	4.4.4	5.4.4	6.4.4	7.4.4	8.4.4	9.4.4	10.4.4	11.4.4	12.4.4	...	∞.4.4
Coxeter diagram												...

Volume

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

The volume is therefore:

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:

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

Surface area

The surface area of a right prism is:

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 height h is therefore:

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

Schlegel diagrams

P3	P4	P5	P6	P7	P8

Symmetry

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.

Truncated prism

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!

Twisted 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

Frustum

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

Star prism

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×{ }	t_a{3}×{ }		{5/2}×{ }	{7/2}×{ }	{7/3}×{ }	{8/3}×{ }
D_2h, order 8	D_3h, order 12		D_5h, order 20	D_7h, order 28		D_8h, 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	t_a{3}×{ }₁₈₀		{3}×{ }₁₈₀	{4}×{ }₁₈₀	{5}×{ }₁₈₀	{5/2}×{ }₁₈₀	{6}×{ }₁₈₀
D_2h, order 8	D_3d, order 12			D_4h, order 16	D_5d, order 20		D_6d, 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
D_4h, order 16	D_6h, order 24
v=8, e=16, f=8	v=12, e=24, f=12

Prismatic polytope

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}×{}.