Set of uniform n-gonal prisms | |
---|---|

Type | uniform in the sense of semiregular polyhedron |

Faces | 2 n-gonal regular polygons n squares |

Edges | 3n |

Vertices | 2n |

Vertex configuration | 4.4.n |

Schläfli symbol | {n}×{} ^{ [1] }t{2, n} |

Coxeter diagram | |

Symmetry group | D_{nh}, [n,2], (*n22), order 4n |

Rotation group | D_{n}, [n,2]^{+}, (n22), order 2n |

Dual polyhedron | convex dual-uniform n-gonal bipyramid |

Properties | convex, regular polygon faces, vertex-transitive, translated bases, sides ⊥ bases |

Net | |

In geometry, a **prism** is a polyhedron comprising an n-sided polygon 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 after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

- Oblique prism
- Right prism, uniform prism
- Right prism
- Uniform prism
- Volume
- Surface area
- Schlegel diagrams
- Symmetry
- Truncated prism
- Twisted prism
- Frustum
- Star prism
- Crossed prism
- Toroidal prism
- Prismatic polytope
- Uniform prismatic polytope
- See also
- References
- External links

Like many basic geometric terms, the word *prism* (from Greek πρίσμα* (prisma)* '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, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers.^{ [2] }^{ [3] }

An **oblique prism** is a prism in which the joining edges and faces are * not perpendicular * to the base faces.

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 **right prism** is a prism in which the joining edges and faces are * perpendicular * to the base faces.^{ [4] } This applies if all the joining faces are * rectangular *.

The dual of a *right**n*-prism is a *right**n*-bipyramid.

A right prism (with rectangular sides) with regular *n*-gon bases has Schläfli symbol { }×{*n*}. It approaches a cylindrical solid as *n* approaches infinity.

- A right rectangular prism (with a rectangular base) is also called a
*cuboid*, or informally a*rectangular box*. A right rectangular prism has Schläfli symbol { }×{ }×{ }.

- A right square prism (with a square base) is also called a
*square cuboid*, or informally a*square box*.

Note: some texts may apply the term *rectangular prism* or *square prism* to both a right rectangular-based prism and a right square-based prism.

A **uniform prism** or **semiregular prism** is a *right prism* with * regular bases* and * square sides*, since such prisms are in the set of uniform polyhedra.

A uniform *n*-gonal prism has Schläfli symbol t{2,*n*}.

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

Family of uniform n-gonal prisms | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Prism name | Digonal prism | (Trigonal) Triangular prism | (Tetragonal) Square prism | Pentagonal prism | Hexagonal prism | Heptagonal prism | Octagonal prism | Enneagonal prism | Decagonal prism | Hendecagonal prism | Dodecagonal prism | ... | Apeirogonal prism |

Polyhedron image | ... | ||||||||||||

Spherical tiling image | Plane tiling image | ||||||||||||

Vertex 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 | ... | ∞.4.4 |

Coxeter diagram | ... |

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:

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:

The surface area of a right prism is:

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:

P3 | P4 | P5 | P6 | P7 | P8 |

The symmetry group of a right *n*-sided prism with regular base is D_{nh} of order 4*n*, 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 2*n*, 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.

The hosohedra and dihedra also possess dihedral symmetry, and an *n*-gonal prism can be constructed via the geometrical truncation of an *n*-gonal hosohedron, as well as through the cantellation or expansion of an *n*-gonal dihedron.

A **truncated prism** is a prism with non-parallel top and bottom faces.^{ [5] }

A **twisted prism** is a nonconvex polyhedron constructed from a uniform *n*-prism with each side face bisected on the square diagonal, by twisting the top, usually by π/*n* radians (180/*n* degrees) in the same direction, causing sides to be concave.^{ [6] }^{ [7] }

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

An *n*-gonal *twisted prism* is topologically identical to the *n*-gonal uniform antiprism, but has half the symmetry group: D_{n}, [*n*,2]^{+}, order 2*n*. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.

3-gonal | 4-gonal | 12-gonal | |
---|---|---|---|

Schönhardt polyhedron | Twisted square prism | Square antiprism | Twisted dodecagonal antiprism |

A frustum is a similar construction to a prism, with trapezoid lateral faces and differently sized top and bottom polygons.

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.

{ }×{ }_{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 | ||

A **crossed prism** is a nonconvex polyhedron constructed from a prism, where the vertices of one base are inverted around the center of this base (or rotated by 180°). This transforms the side rectangular faces into crossed rectangles. For a regular polygon base, the appearance is an *n*-gonal hour glass. All oblique edges pass through a single body center. Note: no vertex is at this body centre. A crossed prism is topologically identical to an *n*-gonal prism.

{ }×{ }_{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 | |||

A **toroidal prism** is a nonconvex polyhedron like a *crossed prism*, but without bottom and top base faces, and with simple rectangular side faces closing 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 configuration *4.4.4.4*): a band of *n* squares, each attached to a crossed rectangle. An *n*-gonal toroidal prism has 2*n* vertices, 2*n* faces: *n* squares and *n* crossed rectangles, and 4*n* edges. It is topologically self-dual.

D_{4h}, order 16 | D_{6h}, order 24 |

v=8, e=16, f=8 | v=12, e=24, f=12 |

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}*

By dimension:

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

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}.
- 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*.

- Example: Pentagonal prism, {5}×{}, two parallel pentagons connected by 5 rectangular
- 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*.

- Example: Dodecahedral prism, {5, 3}×{}, two parallel dodecahedra connected by 12 pentagonal prism
- ...

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

In geometry, an **n-gonal antiprism** or **n-antiprism** is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2*n* triangles. They are represented by the Conway notation A*n*.

A (symmetric) n-gonal **bipyramid** or **dipyramid** is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2*n* triangle faces, 3*n* edges, and 2 + *n* vertices.

In geometry, every polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

In geometry, a **polyhedron** is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.

In geometry, a **4-polytope** is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

In geometry, the **rhombicuboctahedron**, or **small rhombicuboctahedron**, is an Archimedean solid with eight triangular, six square, and twelve rectangular faces. There are 24 identical vertices, with one triangle, one square, and two rectangles meeting at each one. The polyhedron has octahedral symmetry, like the cube and octahedron. Its dual is called the deltoidal icositetrahedron or trapezoidal icositetrahedron, although its faces are not really true trapezoids.

In geometry, the **Schläfli symbol** is a notation of the form that defines regular polytopes and tessellations.

In geometry, a **uniform 4-polytope** is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry of 4 dimensions or higher, a **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (*n*+*m*)-polytope, where n and m are dimensions of 2 (polygon) or higher.

In geometry, a **uniform polyhedron** has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, a **triangular prism** is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A **right triangular prism** has rectangular sides, otherwise it is *oblique*. A **uniform triangular prism** is a right triangular prism with equilateral bases, and square sides.

In geometry, the **hexagonal prism** is a prism with hexagonal base. This polyhedron has 8 faces, 18 edges, and 12 vertices.

The **cubic honeycomb** or **cubic cellulation** is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a **cubille**.

In geometry, a **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a **uniform star polyhedron** is a self-intersecting uniform polyhedron. They are also sometimes called **nonconvex polyhedra** to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, or both.

In geometry, a **uniform tiling** is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

In four-dimensional geometry, a **prismatic uniform 4-polytope** is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons.

In geometry of 4 dimensions, a **8-8 duoprism** or **octagonal duoprism** is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons.

- ↑ 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 - ↑ 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
- ↑ "Pictures of Twisted Prisms".

- 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

Wikisource has the text of the 1911 Encyclopædia Britannica article " Prism ". |

- 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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