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Prism (geometry)

From Wikipedia, the free encyclopedia

Set of uniform n-gonal prisms
Hexagonal Prism BC.svg

Example uniform hexagonal prism
Type uniform in the sense of semiregular polyhedron
Conway polyhedron notation Pn
Faces 2{n} + n{4}
Edges 3n
Vertices 2n
Schläfli symbol {n}×{}[1] or t{2, n}
Coxeter diagram CDel node 1.pngCDel 2.pngCDel node 1.pngCDel n.pngCDel node.png
Vertex configuration 4.4.n
Symmetry group Dnh, [n,2], (*n22), order 4n
Rotation group Dn, [n,2]+, (n22), order 2n
Dual polyhedron convex dual-uniform n-gonal bipyramid
Properties convex, regular polygon faces, vertex-transitive, translated bases, sides ⊥ bases[2]
Generalized prisim net.svg

Example uniform enneagonal prism net (n = 9)

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; example: a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.

Like many basic geometric terms, the word prism (Greek: πρίσμα, 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, 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.[3][4]

Oblique prism

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.

Right prism, uniform prism

Right prism

A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.[2] This applies iff 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.

Special cases

  • 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.

Uniform 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
Yellow square.gif
Triangular prism.png
Tetragonal prism.png
Pentagonal prism.png
Hexagonal prism.png
Prism 7.png
Octagonal prism.png
Prism 9.png
Decagonal prism.png
Hendecagonal prism.png
Dodecagonal prism.png
...
Spherical tiling image
Tetragonal dihedron.png
Spherical triangular prism.png
Spherical square prism.png
Spherical pentagonal prism.png
Spherical hexagonal prism.png
Spherical heptagonal prism.png
Spherical octagonal prism.png
Spherical decagonal prism.png
Plane tiling image
Infinite prism.svg
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 CDel node 1.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 9.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 10.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 11.pngCDel node.pngCDel 2.pngCDel node 1.png CDel node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel node 1.png ... CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.png

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:

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:

Surface area

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:

Schlegel diagrams

Triangular prismatic graph.png

P3
Cubical graph.png

P4
Pentagonal prismatic graph.png

P5
Hexagonal prismatic graph.png

P6
Heptagonal prismatic graph.png

P7
Octagonal prismatic graph.png

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 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.

Truncated prism

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

Example truncated triangular prism. Its top face is truncated at an oblique angle, but it is NOT an oblique prism!
Example truncated triangular prism. Its top face is truncated at an oblique angle, but it is NOT an oblique prism!

Twisted prism

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: Dn, [n,2]+, order 2n. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles.

3-gonal 4-gonal 12-gonal
Schönhardt polyhedron.svg

Schönhardt polyhedron
Twisted square antiprism.png

Twisted square prism
Square antiprism.png

Square antiprism
Twisted dodecagonal antiprism.png

Twisted dodecagonal antiprism

Frustum

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

Example pentagonal frustum
Example pentagonal frustum

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-square prism.png
Crossed hexagonal prism.png
Crossed-unequal hexagonal prism.png
Pentagrammic prism.png
Heptagrammic prism 7-2.png
Heptagrammic prism 7-3.png
Prism 8-3.png

Crossed prism

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.

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
Crossed crossed-square prism.png
Crossed crossed-hexagonal prism.png
Crossed crossed-unequal hexagonal prism.png
Crossed triangular prism.png
Crossed cube.png
Crossed pentagonal prism.png
Crossed pentagrammic prism.png
Crossed2 hexagonal prism.png

Toroidal prism

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 2n vertices, 2n faces: n squares and n crossed rectangles, and 4n edges. It is topologically self-dual.

Examples
D4h, order 16 D6h, order 24
v=8, e=16, f=8 v=12, e=24, f=12
Toroidal square prism.png
Toroidal hexagonal prism.png

Prismatic polytope

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 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, 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 {}.
  • 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}.
    • Square diagonals.svg
      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 p = 4, with square sides symmetry it becomes a cube: {4}×{} = {4, 3}.
  • 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}×{} = {4, 3, 3}.
  • ...

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}.

See also

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. ^ a b William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p.28
  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.81
  6. ^ The facts on file: Geometry handbook, Catherine A. Gorini, 2003, ISBN 0-8160-4875-4, p.172
  7. ^ [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

External links

This page was last edited on 8 June 2021, at 23:50
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