Prism (geometry)
Template:Short description Template:More sources needed {{#invoke:Infobox|infobox}}Template:Template other
In geometry, a prism is a polyhedron comprising an Template:Nowrap polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and Template:Mvar 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.<ref name="prismatoid">Template:Cite journal</ref>
Like many basic geometric terms, the word prism (Template:Ety) 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 regard to the nature of the bases (a cause of some confusion amongst generations of later geometry writers).<ref name="malton-1774">Template:Cite book</ref><ref name="elliot-1845">Template:Cite book</ref>
Oblique vs rightEdit
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 whose base is a parallelogram, or equivalently a polyhedron with six parallelogram faces.
A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.<ref name="mensuration">Template:Cite book</ref> This applies if and only if all the joining faces are rectangular.
The dual of a right Template:Mvar-prism is a right Template:Mvar-bipyramid.
A right prism (with rectangular sides) with [[Regular polygon|regular Template:Mvar-gon]] bases has Schläfli symbol Template:Math It approaches a cylinder as Template:Mvar approaches infinity.<ref name="cylinder">Template:Cite book</ref>
Special casesEdit
- 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 Template:Math
- 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.
TypesEdit
Regular prismEdit
A regular prism is a prism with regular bases.
Uniform prismEdit
A uniform prism or semiregular prism is a right prism with regular bases and all edges of the same length.
Thus all the side faces of a uniform prism are squares.
Thus all the faces of a uniform prism are regular polygons. Also, such prisms are isogonal; thus they are uniform polyhedra. They form one of the two infinite series of semiregular polyhedra, the other series being formed by the antiprisms.
A uniform Template:Mvar-gonal prism has Schläfli symbol Template:Math
PropertiesEdit
VolumeEdit
The volume of a prism is the product of the area of the base by the height, i.e. the distance between the two base faces (in the case of a non-right prism, note that this means the perpendicular distance).
The volume is therefore:
- <math>V = Bh,</math>
where Template:Mvar is the base area and Template:Mvar is the height.
The volume of a prism whose base is an Template:Mvar-sided regular polygon with side length Template:Mvar is therefore: <math display=block>V = \frac{n}{4} h s^2 \cot\frac{\pi}{n}.</math>
Surface areaEdit
The surface area of a right prism is:
- <math>2B + Ph,</math>
where Template:Mvar is the area of the base, Template:Mvar the height, and Template:Mvar the base perimeter.
The surface area of a right prism whose base is a regular Template:Mvar-sided polygon with side length Template:Mvar, and with height Template:Mvar, is therefore:
- <math>A = \frac{n}{2} s^2 \cot\frac{\pi}{n} + nsh.</math>
SymmetryEdit
The symmetry group of a right Template:Mvar-sided prism with regular base is Template:Math of order Template:Math, except in the case of a cube, which has the larger symmetry group Template:Math of order 48, which has three versions of Template:Math as subgroups. The rotation group is Template:Math of order Template:Math, except in the case of a cube, which has the larger symmetry group Template:Math of order 24, which has three versions of Template:Math as subgroups.
The symmetry group Template:Math contains inversion iff Template:Mvar is even.
The hosohedra and dihedra also possess dihedral symmetry, and an Template:Mvar-gonal prism can be constructed via the geometrical truncation of an Template:Mvar-gonal hosohedron, as well as through the cantellation or expansion of an Template:Mvar-gonal dihedron.
Schlegel diagramsEdit
Similar polytopesEdit
Truncated prismEdit
A truncated prism is formed when prism is sliced by a plane that is not parallel to its bases. A truncated prism's bases are not congruent, and its sides are not parallelograms.Template:Sfnp
Twisted prismEdit
A twisted prism is a nonconvex polyhedron constructed from a uniform Template:Mvar-prism with each side face bisected on the square diagonal, by twisting the top, usually (but not necessarily) by Template:Sfrac radians (Template:Sfrac degrees). If the bisectors are slanted to the left, then twisting the top base in the right direction (looking at the top of the prism) by a small angle gives nonconvex polyhedron and twisting it in the left direction, a convex polyhedron (see twisted square prism on the image). If the bisectors are slanted to the right, then twisting the top base in the left direction gives nonconvex polyhedron, in the right direction, convex one (see twisted dodecagonal prism).<ref name="gorini-2003">Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
A twisted prism cannot be dissected into tetrahedra without adding new vertices. The simplest twisted prism has triangle bases and is called a Schönhardt polyhedron.
An Template:Mvar-gonal twisted prism is topologically identical to the Template:Mvar-gonal uniform antiprism, but has half the symmetry group: Template:Math, order Template:Math. It can be seen as a nonconvex antiprism, with tetrahedra removed between pairs of triangles. Any twisted Template:Mvar-gonal prism is an antiprism, so the twisted square prism and twisted dodecagonal prism shown on the image are both antiprisms.
| 3-gonal | 4-gonal | 12-gonal | |
|---|---|---|---|
| File:Schönhardt polyhedron.svg | File:Twisted square antiprism.png | File:Square antiprism.png | File:Twisted dodecagonal antiprism.png |
| Schönhardt polyhedron | Twisted square prism | Square antiprism | Twisted dodecagonal prism |
FrustumEdit
A frustum is a similar construction to a prism, with trapezoid lateral faces and differently sized top and bottom polygons.
Star prismEdit
Template:Further 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 Template:Math with Template:Mvar rectangles and 2 Template:Math faces. It is topologically identical to a Template:Mvar-gonal prism.
Crossed prismEdit
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 Template:Mvar-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 Template:Mvar-gonal prism.
Toroidal prismEdit
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 Template:Math): a band of Template:Mvar squares, each attached to a crossed rectangle. An Template:Mvar-gonal toroidal prism has Template:Math vertices, Template:Math faces: Template:Mvar squares and Template:Mvar crossed rectangles, and Template:Math edges. It is topologically self-dual.
| Template:Math, order 16 | Template:Math, order 24 |
| Template:Math, Template:Math, Template:Math | Template:Math, Template:Math, Template:Math |
| File:Toroidal square prism.png | File:Toroidal hexagonal prism.png |
Prismatic polytopeEdit
A prismatic polytope is a higher-dimensional generalization of a prism. An Template:Mvar-dimensional prismatic polytope is constructed from two (Template:Math)-dimensional polytopes, translated into the next dimension.
The prismatic Template:Mvar-polytope elements are doubled from the (Template:Math)-polytope elements and then creating new elements from the next lower element.
Take an Template:Mvar-polytope with Template:Mvar [[Face (geometry)|Template:Mvar-face]] elements (Template:Math). Its (Template:Math)-polytope prism will have Template:Math Template:Mvar-face elements. (With Template:Math, Template:Math.)
By dimension:
- Take a polygon with Template:Mvar vertices, Template:Mvar edges. Its prism has Template:Math vertices, Template:Math edges, and Template:Math faces.
- Take a polyhedron with Template:Mvar vertices, Template:Mvar edges, and Template:Mvar faces. Its prism has Template:Math vertices, Template:Math edges, Template:Math faces, and Template:Math cells.
- Take a polychoron with Template:Mvar vertices, Template:Mvar edges, Template:Mvar faces, and Template:Mvar cells. Its prism has Template:Math vertices, Template:Math edges, Template:Math faces, Template:Math cells, and Template:Math hypercells.
Uniform prismatic polytopeEdit
Template:See also Template:See also A regular Template:Mvar-polytope represented by Schläfli symbol Template:Math can form a uniform prismatic (Template:Math)-polytope represented by a Cartesian product of two Schläfli symbols: Template:Math
By dimension:
- A 0-polytopic prism is a line segment, represented by an empty Schläfli symbol Template:Math
- A 1-polytopic prism is a rectangle, made from 2 translated line segments. It is represented as the product Schläfli symbol Template:Math If it is square, symmetry can be reduced: Template:Math
- Example: File:Square diagonals.svg, Square, Template:Math 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 Template:Math can construct a uniform Template:Mvar-gonal prism represented by the product Template:Math If Template:Math, with square sides symmetry it becomes a cube: Template:Math
- Example: File:Pentagonal prism.png, Pentagonal prism, Template:Math two parallel pentagons connected by 5 rectangular sides.
- A polyhedral prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron Template:Math can construct the uniform polychoric prism, represented by the product Template:Math If the polyhedron and the sides are cubes, it becomes a tesseract: Template:Math
- Example: File:Dodecahedral prism.png, Dodecahedral prism, Template:Math two parallel dodecahedra connected by 12 pentagonal prism sides.
- ...
Higher order prismatic polytopes also exist as cartesian products of any two or more polytopes. The dimension of a product polytope is the sum 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 in 4-dimensions.
Regular duoprisms are represented as Template:Math with Template:Mvar vertices, Template:Math edges, Template:Mvar square faces, Template:Mvar Template:Mvar-gon faces, Template:Mvar Template:Mvar-gon faces, and bounded by Template:Mvar Template:Mvar-gonal prisms and Template:Mvar Template:Mvar-gonal prisms.
For example, Template:Math a 4-4 duoprism is a lower symmetry form of a tesseract, as is Template:Math a cubic prism. Template:Math (4-4 duoprism prism), Template:Math (cube-4 duoprism) and Template:Math (tesseractic prism) are lower symmetry forms of a 5-cube.
See alsoEdit
ReferencesEdit
- Template:Cite book Chapter 2: Archimedean polyhedra, prisma and antiprisms
External linksEdit
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Prism%7CPrism.html}} |title = Prism |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}
- Paper models of prisms and antiprisms Free nets of prisms and antiprisms
- Paper models of prisms and antiprisms Using nets generated by Stella