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Prism

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Set of uniform prisms
Hexagonal Prism BC
Typeuniform polyhedron
Faces2 p-gons, p parallelograms
Edges3p
Vertices2p
Schläfli symbol{p}x{} or t{2, p}
Vertex configuration4.4.p
Symmetry groupDph
Dual polyhedronbipyramids
Propertiesconvex, semi-regular vertex-transitive

In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. Thus these joining faces are parallelograms. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoids.

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.

In the case of a rectangular or square prism there may be ambiguity because some texts may mean a right rectangular-sided prism and a right square-sided 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 n-prism, the floormade of regular polygons ends and rectangle 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 6 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 right cube, square box, and may also be called a square cuboid.

An equilateral square prism is simply a cube.

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=B\cdot h

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:

V=\frac{n}{4}s^3\cot\left(\frac{\pi}{n}\right)
V=s^3\frac{n}{(4n-8)\tan\left(\frac{180}{n}\right)}
V=\frac{s^3}{4}\sin\left(\frac{360}{n}\right)=\frac{s^3}{4\csc\left(\frac{360}{n}\right)}

Surface area

The surface area of a right prism is 2\cdot B+P\cdot 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 n-sided polygon with side length S is therefore:

A=2A_n+nA_4=S^2\left(\frac{n}{2}\cot\left(\frac{\pi}{n}\right)+n\right)
A=s^2\left(\frac{n}{(2n-4)\tan\left(\frac{180}{n}\right)}+3\right)

A=s^2\left(3+\frac{\sin\left(\frac{360}{n}\right)}{2}\right)=s^2\left(3+\frac{1}{2\csc\left(\frac{360}{n}\right)}\right)

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.

Prismatic polytope

A prismatic polytope is a 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 fi i-face elements (i = 0 , ... , n). Its (n + 1)-polytope prism will have 2fi + fi−1 i-face elements. (With f−1 = 0, fn = 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 {}.
    • Complete graph K2
  • 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 it: {}×{} = {4}.
    • Square diagonalsExample: 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: {2}×{} = {4,3}.
  • 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}.
  • ...

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}.

See also

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