Set of uniform prisms

(A hexagonal prism is shown)

Type

uniform polyhedron

Faces

2+n total:
2 {n}
n {4}

Edges

3n

Vertices

2n

Schläfli symbol

{n}×{} or t{2, n}

CoxeterDynkin 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, semiregular vertextransitive

ngonal prism net (n = 9 here)

In geometry, a prism is a polyhedron with an nsided polygonal base, another congruent parallel base (with the same rotational orientation), and n other faces (necessarily all parallelograms) joining corresponding sides of the two bases. All crosssections parallel to the base faces are congruent to the bases. Prisms are named for their base, so a prism with a pentagonal base is called a pentagonal prism. The prisms are a subclass of the prismatoids.
Contents

General, right and uniform prisms 1

Volume 2

Surface area 3

Symmetry 4

Prismatic polytope 5

Uniform prismatic polytope 5.1

See also 6

References 7

External links 8
General, right and uniform prisms
A right prism is a prism in which the joining edges and faces are perpendicular to the base faces. 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.
Some texts may apply the term rectangular prism or square prism to both a right rectangularsided prism and a right squaresided prism. The term uniform prism can be used for a right prism with square sides, since such prisms are in the set of uniform polyhedra.
An nprism, having regular polygon ends and rectangular sides, approaches a cylindrical solid as n approaches infinity.
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.
A parallelepiped is a prism of which the base is a parallelogram, or equivalently a polyhedron with six faces which are all parallelograms.
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.
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 nonright prism, note that this means the perpendicular distance).
The volume is therefore:

V = B \cdot h
where B is the base area and h is the height. The volume of a prism whose base is a regular nsided polygon with side length s is therefore:

V = \frac{n}{4}hs^2 \cot\frac{\pi}{n}.
Surface area
The surface area of a right prism is 2 · B + P · h, 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 nsided polygon with side length s and height h is therefore:

A = \frac{n}{2} s^2 \cot{\frac{\pi}{n}} + n s h.
Symmetry
The symmetry group of a right nsided 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 D_{4} as subgroups.
The symmetry group D_{nh} contains inversion iff n is even.
Prismatic polytope
A prismatic polytope is a higherdimensional generalization of a prism. An ndimensional prismatic polytope is constructed from two (n − 1)dimensional polytopes, translated into the next dimension.
The prismatic npolytope elements are doubled from the (n − 1)polytope elements and then creating new elements from the next lower element.
Take an npolytope with f_{i} iface elements (i = 0, ..., n). Its (n + 1)polytope prism will have 2f_{i} + f_{i−1} iface 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 npolytope 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 0polytopic prism is a line segment, represented by an empty Schläfli symbol {}.

A 1polytopic 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 it: {}×{} = {4}.

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

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

A polyhedral prism is a 4dimensional prism made from two translated polyhedra connected by 3dimensional 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}.

...
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 4dimensional space are called duoprisms as the product of two polygons. Regular duoprisms are represented as {p}×{q}.
Family of uniform prisms
Symmetry

3

4

5

6

7

8

9

10

11

12

[2n,2]
[n,2]
[2n,2^{+}]











Image











As spherical polyhedra

Image











See also
References

Anthony Pugh (1976). Polyhedra: A visual approach. California: University of California Press Berkeley. Chapter 2: Archimedean polyhedra, prisma and antiprisms
External links

Weisstein, Eric W., "Prism", MathWorld.

Olshevsky, George, Prismatic polytope at Glossary for Hyperspace.

Nonconvex Prisms and Antiprisms

Surface Area MATHguide

Volume MATHguide

Paper models of prisms and antiprisms Free nets of prisms and antiprisms

Paper models of prisms and antiprisms Using nets generated by Stella.

Stella: Polyhedron Navigator: Software used to create the 3D and 4D images on this page.


Platonic solids (regular)









Dihedral regular



Dihedral uniform



Dihedral others



Degenerate polyhedra are in italics.


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