Prism (geometry)

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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
Faces2{n} + n {4}
Edges3n
Vertices2n
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
Propertiesconvex, 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.

Contents

Like many basic geometric terms, the word prism (Greek : πρίσμα, romanized: prisma, 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 rightn-prism is a rightn-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. 2.4.43.4.44.4.45.4.46.4.47.4.48.4.49.4.410.4.411.4.412.4.4...∞.4.4
Coxeter diagram CDel node 1.pngCDel 2.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 5.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 9.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 10.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel node 1.pngCDel 11.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel 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! TruncatedTriangularPrism.png
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-gonal4-gonal12-gonal
Schonhardt 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 Pentagonal frustum.svg
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 8D3h, order 12D5h, order 20D7h, order 28D8h, 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×{ }180ta{3}×{ }180{3}×{ }180{4}×{ }180{5}×{ }180{5/2}×{ }180{6}×{ }180
D2h, order 8D3d, order 12D4h, order 16D5d, order 20D6d, 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 16D6h, order 24
v=8, e=16, f=8v=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−1i-face elements. (With f−1 = 0, fn = 1.)

By dimension:

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:

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

Related Research Articles

Antiprism

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 2ntriangles.

Bipyramid Polyhedron formed by joining a pyramid and its mirror image base-to-base

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 2n triangle faces, 3n edges, and 2 + n vertices.

4-polytope Four-dimensional geometric object with flat sides

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.

Rhombicuboctahedron Archimedean solid with eight triangular and eighteen square faces

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares 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.

Schläfli symbol Notation that defines regular polytopes and tessellations

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

In geometry, a polytope is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.

Uniform 4-polytope

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.

Duoprism

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 2 (polygon) or higher.

Uniform polyhedron Polyhedron which has regular polygons as faces and is vertex-transitive

A uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

Triangular prism

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.

Hexagonal antiprism

In geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.

Pentagrammic prism

In geometry, the pentagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.

Pyramid (geometry) Conic solid with a polygonal base

In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. It is a conic solid with polygonal base. A pyramid with an n-sided base has n + 1 vertices, n + 1 faces, and 2n edges. All pyramids are self-dual.

Skew polygon

In geometry, a skew polygon is a polygon whose vertices are not all coplanar. Skew polygons must have at least four vertices. The interior surface of such a polygon is not uniquely defined.

Uniform polytope

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.

Uniform star polyhedron

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.

Prismatic uniform 4-polytope Type of uniform 4-polytope in four-dimensional geography

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.

8-8 duoprism

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.

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. 1 2 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