Prism (geometry)

In geometry, a prism is a polyhedron comprising an n-sided polygonal 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 for their bases, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.

Set of uniform prisms
(A hexagonal prism is shown)
Type	uniform polyhedron
Conway polyhedron notation	Pn
Faces	2+n total: 2 {n} n {4}
Edges	3n
Vertices	2n
Schläfli symbol	{n}×{}[1] or t{2, n}
Coxeter diagram
Vertex configuration	4.4.n
Symmetry group	D_nh, [n,2], (*n22), order 4n
Rotation group	D_n, [n,2]⁺, (n22), order 2n
Dual polyhedron	bipyramids
Properties	convex, semi-regular vertex-transitive
n-gonal prism net (n = 9 here)

Like many basic geometric terms, the word prism (Greek: πρίσμα, romanized: prisma, lit. '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 in nine subsequent propositions using the term he included examples of triangular-based prisms (i.e. with sides which were not parallelograms).[2] This inconsistency caused confusion amongst later geometricians.[3][4]

General, right and uniform prisms

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.[5] This applies if the joining faces are rectangular. If the joining edges and faces are not perpendicular to the base faces, it is called an oblique prism.

For 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.

A truncated triangular prism with its top face truncated at an oblique angle

A truncated prism is a prism with nonparallel top and bottom faces.[6]

Some texts may apply the term rectangular prism or square prism to both a right rectangular-sided prism and a right square-sided prism. A right p-gonal prism with rectangular sides has a Schläfli symbol { } × {p}.

A right rectangular prism is also called a cuboid, or informally a rectangular box. A right square prism is simply a square box, and may also be called a square cuboid. A right rectangular prism has Schläfli symbol { }×{ }×{ }.

An n-prism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity.

The term uniform prism or semiregular prism can be used for a right prism with square sides, since such prisms are in the set of uniform polyhedra. A uniform p-gonal prism has a Schläfli symbol t{2,p}. Right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra, the other series being the antiprisms.

The dual of a right prism is a bipyramid.

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 a n-gonal prism can be constructed via the geometrical truncation of a n-gonal hosohedron, as well as through the cantellation or expansion of a n-gonal dihedron.

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, and 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 is a cube, 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.
...

Higher order prismatic polytopes also exist as cartesian products of any two polytopes. The dimension of a polytope is the product of the dimensions of the elements. The first example of these exist in 4-dimensional space are called duoprisms as the product of two polygons. Regular duoprisms are represented as {p}×{q}.

Family of uniform prisms
Polyhedron
Coxeter
Tiling
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

Twisted prism

A twisted prism is a nonconvex prism polyhedron constructed by a uniform q-prism with the side faces bisected on the square diagonal, and twisting the top, usually by $π$ /q radians (180/q degrees) in the same direction, causing side triangles to be concave.[7][8]

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

A twisted prism is topologically identical to the antiprism, but has half the symmetry: D_n, [n,2]⁺, order 2n. It can be seen as a convex 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

Pentagonal frustum

A frustum is topologically identical to a prism, with trapezoid lateral faces and different 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 base vertices are inverted around the center (or rotated 180°). This transforms the side rectangular faces into crossed rectangles. For a regular polygon base, the appearance is an p-gonal hour glass, with all vertical edges passing through a single center, but no vertex is there. It is topologically identical to a p-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 prisms

A toroidal prism is a nonconvex polyhedron is like a crossed prism except instead of having base and top polygons, simple rectangular side faces are added to close 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 figure 4.4.4.4. A n-gonal toroidal prism has 2n vertices and faces, and 4n edges and 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

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References

N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3b
Elements: book 11, Def 13 and Prop 28, 29, 39; and book 12, Prop 3, 4, 5, 7, 8, 10
Thomas Malton (1774). A Royal Road to Geometry: Or, an Easy and Familiar Introduction to the Mathematics. ... By Thomas Malton. ... author, and sold. pp. 360–.
James Elliot (1845). Key to the Complete Treatise on Practical Geometry and Mensuration: Containing Full Demonstrations of the Rules ... Longman, Brown, Green, and Longmans. pp. 3–.
William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.28
William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.81
The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172

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

External links

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

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[1] N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, 11.3 Pyramids, Prisms, and Antiprisms, Figure 11.3b

[2] Elements: book 11, Def 13 and Prop 28, 29, 39; and book 12, Prop 3, 4, 5, 7, 8, 10

[Malton1774-3] Thomas Malton (1774). A Royal Road to Geometry: Or, an Easy and Familiar Introduction to the Mathematics. ... By Thomas Malton. ... author, and sold. pp. 360–.

[Elliot1845-4] James Elliot (1845). Key to the Complete Treatise on Practical Geometry and Mensuration: Containing Full Demonstrations of the Rules ... Longman, Brown, Green, and Longmans. pp. 3–.

[5] William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.28

[6] William F. Kern, James R Bland,Solid Mensuration with proofs, 1938, p.81

[7] The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172