Prism (geometry)
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Prism Geometry
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 Dnh, [n,2], (*n22), order 4n
Rotation group Dn, [n,2]+, (n22), order 2n
Dual polyhedron bipyramids
Properties convex, semi-regular vertex-transitive

n-gonal prism net

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.

Like many basic geometric terms, the word prism (Greek: , romanizedprisma, 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:

${\displaystyle V=Bh}$

where B is the base area and h is the height. The volume of a prism whose base is a regular n-sided polygon with side length s is therefore:

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

## Surface area

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 height h is therefore:

${\displaystyle A={\frac {n}{2}}s^{2}\cot {\frac {\pi }{n}}+nsh}$

## Schlegel diagrams

 P3 P4 P5 P6 P7 P8

## Symmetry

The symmetry group of a right n-sided prism with regular base is Dnh of order 4n, except in the case of a cube, which has the larger symmetry group Oh of order 48, which has three versions of D4h as subgroups. The rotation group is Dn 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 Dnh 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 -dimensional polytopes, translated into the next dimension.

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

Take an n-polytope with fi i-face elements . Its -polytope prism will have i-face elements. (With , .)

By dimension:

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

### Uniform prismatic polytope

A regular n-polytope represented by Schläfli symbol  t} can form a uniform prismatic -polytope represented by a Cartesian product of two Schläfli symbols:  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:
• 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 , with square sides symmetry it becomes a cube:
• A polyhedral prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron {pq} can construct the uniform polychoric prism, represented by the product {pq}×{}. If the polyhedron is a cube, and the sides are cubes, it becomes a tesseract: {4, 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 4-dimensional space are called duoprisms as the product of two polygons. Regular duoprisms are represented as {p}×{q}.

## 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 radians ( 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: Dn, [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×{ } 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 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 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 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.

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

## References

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
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-.
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
8. ^ [1]
• 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