Prism (geometry)

A prism, in geometry, is a polyhedron that consists of two equal and parallel faces called bases, and lateral faces that are parallelograms. Prisms are named for the shape of their base, so a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

Definition

A prism is a polyhedron that meets the following two properties:

1. There are exactly two congruent faces on parallel planes, they are named bases.
2. All the other faces are parallel.

Right 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-sided rectangular prism and a right-sided square prism. The term uniform prism can be used for a right prism with square sides, since such prisms are in the uniform set of polyhedra.

A prism with n lateral faces with endpoints of regular polygons and rectangular faces approaches a cylindrical solid as n approaches infinity.

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

The dual of a right prism is a dipyramid.

A parallelepiped is a prism whose base is a parallelogram, or equivalently a polyhedron with six faces that are all parallelograms.

A right rectangular prism is also known as a cuboid, or informally a rectangular box. A right square prism is simply a square box, and may also be called a square cuboid. Prisms are polyhedra that consist of two equal and parallel faces called bases, and side faces that are parallelograms.

Each prism consists of the following elements:

• Bases: are the two equal and parallel faces of the prism, one on which it rests and the other on the opposite.
• Side faces: are the faces that share two sides with the bases. The sum of its areas is the side surface of the prism.
• Arists: are the sides of the bases and side faces.
• Vetices: are the points where each pair of edges are found.
• Height: is the distance between the bases.
• Diagonal: are the segments that unite two non-secure vertices of the prism. The diagonals of one face or between two faces can be traced.

Volume

The volume of a prism is the product of the area of the base times the distance or height between the two bases. Its value is expressed as:

V=B⋅ ⋅ h{displaystyle V=Bcdot h}

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

V=n4hs2cot π π n{displaystyle V={frac {n}{4}}}hs^{2}{cot {frac {pi }{n}}}}}

Symmetry

The symmetry group of an n-sided right prism with the regular base is Dnh of order 4n, except in the case of a cube, which has the largest octahedral symmetry group, 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 largest symmetry group O of the order 2₄, which has three versions of D₄ as subgroups.

The symmetry group D_nh contains inversion if n is even.

Prismatic polytope

A prismatic polytope is a generalization of prisms to dimensions other than 3. A dimensional prismatic polytope is defined recursively as a figure created from two polytopes (n − 1)-dimensional congruents in parallel hyperplanes, whose corresponding facets are connected by (n − 1)-dimensional prisms.

Given an n-polytope with f_i elements of dimension i (i = 0,..., n), the prism generated from it will have 2f_i + f_{i− 1} elements of dimension i (taking f₋₁ = 0, f_n = 1).

By dimension:

• If we leave a polygon with n vertices and n aristas, your prisma will have 2n vertices, 3n edges and 2 + n faces.
• If we leave a polyhedron with v Vertices, e arists and f faces, your prism will have 2v Vertices, 2e + v arists, 2f + e faces, and 2 + f cells.
• If we leave a polycorum with v Vertices, e Aristas, f faces and c cells, your prism will have 2v Vertices, 2e + v edges, 2f + e faces, and 2 + c hypercells.

Uniform prismatic polytope

A regular n-polytope of represented by the 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-politopic prism is a straight segment, represented by a symbol of empty Schläfli {}.
  • 
• A 1-politopic prism is a rectangle, formed from the translation of 2 line segments. It is represented as the symbol Schläfli product {} × {}. If it is a square, you can reduce symmetry to: {} x {} = {4}.
  • Example: square, {} x {}, two parallel straight segments, connected by two sides of straight segments.
• A polygonal prism is a 3-dimensional prism made from two polygons moved, connected by rectangles. A regular polygon {pYou can build the prism n-gonal uniform represented by the product {p*** Yeah. p = 4, with symmetrical square sides, becomes a cube: {4}×{} = {4,3}.
  • Example: pentagonal prism, {5}×{}, two parallel pentagonals connected by 5 rectangular sides.
• A polyhedral prism is a 4-dimensional prism made by two moved polyhedrons connected by three-dimensional prism cells. A regular polyhedron {p, q} can build the uniform polychromatic prism, represented by the product {p,q}×{}. If the polyhedron is a cube, and the sides are cubes, it becomes a teseracto: {4, 3}×} = {4, 3, 3}.
  • Example: Dodecaédrico prisma {5, 3} × {}, two parallel dodecaedros connected by 12 sides of pentagonal prisms.

Higher-order prismatic polytopes also exist as Cartesian products of two polytopes. The dimension of a polytope is the product of the dimensions of the elements. The first example of this exists in a 4-dimensional space called a duoprism 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
Image

Prism Trunk

It is a part of a prism limited between the base and the section originated by a plane not parallel to the base and that intersects all the lateral edges.