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Prism (geometry)
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an n-sided prism is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 made of an n-sided polygonal
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 base, a translated
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
 copy, and n faces joining corresponding sides. Thus these joining faces are parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
s. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoid
Prismatoid

A prismatoid is a polyhedron where all vertices lie in two parallel planes. If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height is h, then the volume of the prismatoid i...
s.

General, right and uniform prisms
A right prism is a prism in which the joining edges and faces are perpendicular to the base faces.






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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, an n-sided prism is a polyhedron
Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
 made of an n-sided polygonal
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 base, a translated
Translation (geometry)

In Euclidean geometry, a translation is moving every point a constant distance in a specified direction. It is one of the Euclidean groups . A translation can also be interpreted as the addition of a constant vector space to every point, or as shifting the Origin of the coordinate system....
 copy, and n faces joining corresponding sides. Thus these joining faces are parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
s. All cross-sections parallel to the base faces are the same. A prism is a subclass of the prismatoid
Prismatoid

A prismatoid is a polyhedron where all vertices lie in two parallel planes. If the areas of the two parallel faces are A1 and A3, the cross-sectional area of the intersection of the prismatoid with a plane midway between the two parallel faces is A2, and the height is h, then the volume of the prismatoid i...
s.

General, right and uniform 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.

In the case of a rectangular or square prism there may be ambiguity because some texts may mean a right rectangular-sided prism and a right square-sided prism.

The term uniform prism can be used for a right prism with square sides since such prisms are in the set of uniform polyhedra
Uniform polyhedron

A Uniform polytope polyhedron is a polyhedron which has regular polygons as Face and is transitive on its vertex . It follows that all vertices are Congruence , and the polyhedron has a high degree of reflectional and rotational symmetry....
.

An n-prism, the floormade of regular polygon
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
s ends and rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
 sides approaches a cylindrical
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
 solid as n approaches infinity
Infinity

Infinity comes from the Latin infinitas or "unboundedness." It refers to several distinct concepts – usually linked to the idea of "without end" – which arise in philosophy, mathematics, and theology....
.

Right prisms with regular bases and equal edge lengths form one of the two infinite series of semiregular polyhedra
Semiregular polyhedron

A semiregular polyhedron is a polyhedron with regular polygon faces and a symmetry group which is transitive on its vertices. Or at least, that is what follows from Thorold Gosset's 1900 definition of the more general semiregular polytope....
, the other series being the antiprism
Antiprism

An n-sided antiprism is a polyhedron composed of 2 parallel copies of some particular n-sided polygon, connected by an alternating band of triangles....
s.

The dual
Dual polyhedron

In geometry, polyhedron are associated into pairs called duals, where the wikt:vertex of one correspond to the face s of the other. The dual of the dual is the original polyhedron....
 of a right prism is a bipyramid
Bipyramid

An n-agonal bipyramid or dipyramid is a polyhedron formed by joining an n-agonal Pyramid and its mirror image base-to-base.The referenced n-agon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the 2 pyramid halves....
.

A parallelepiped
Parallelepiped

In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. It is to a parallelogram as a cube is to a square : Euclidean geometry supports all four notions but affine geometry admits only parallelograms and parallelepipeds....
 is a prism of which the base is a parallelogram
Parallelogram

In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite or facing sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are of equal size....
, or equivalently a polyhedron with 6 faces which are all parallelograms.

A right rectangular prism is also called a cuboid
Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
, or informally a rectangular box. A right square prism is simply a square box, and may also be called a square cuboid.

An equilateral square prism is simply a cube
Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
.

Area and volume


The volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
 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).

Symmetry


The symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 of a right n-sided prism with regular base is Dnh
Dihedral group

In mathematics, a dihedral group is the group of symmetry of a regular polygon, including both rotational symmetry and reflection symmetry. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry....
 of order 4n, except in the case of a cube, which has the larger symmetry group Oh
Octahedral symmetry

A regular octahedron has 24 rotational symmetries, and a total of 48 symmetries including transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the dual polyhedron of an octahedron....
 of order 48, which has three versions of D4h as subgroups. The rotation group
Rotation group

In classical mechanics and geometry, the rotation group is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of functional composition....
 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
Inversion in a point

In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*....
 iff
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 n is even.

Prismatic polytope


A prismatic polytope
Polytope

In geometry, polytope is a generic term that can refer to a two-dimensional polygon, a three-dimensional polyhedron, or any of the various generalizations thereof, including generalizations to higher dimensions and other abstractions ....
 is a dimensional generalization of a prism. An n-dimensional prismatic polytope is constructed from 2 (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
Face

The term face refers to the central sense organ complex, for those animals that have one, normally on the ventral surface of the head and can depending on the definition in the human case, include the hair, forehead, eyebrow, eyes, nose, ears, cheeks, mouth, lips, philtrum, tooth, skin, and chin....
 elements (i=0..n). Its (n+1)-polytope prism will have 2*fi+fi-1 i-face elements. (With f-1=0, fn=1.)

By dimension:
  • Take a polygon
    Polygon

    In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
     with n vertices, n edges. Its prism has 2n vertices, 2n+n edges, and 2+n faces.
  • Take a polyhedron
    Polyhedron

    |}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
     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
    Polychoron

    In geometry, a four-dimensional polytope is sometimes called a polychoron , from the Greek language root poly, meaning "many", and choros meaning "room" or "space"....
     with v vertices, e edges, f faces and c cells. Its prism has 2v vertices, 2e+v edges, 2f+e faces, and 2c+f cells, and 2+c hypercells.


Uniform prismatic polytope


A regular n-polytope represented by Schläfli symbol
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
  can form a uniform prismatic (n+1)-polytope represented by a Cartesian product
Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
 of two Schläfli symbols
Schläfli symbol

In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
: x.

By dimension:
  • A 0-polytopic prism is a line segment
    Line segment

    In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
    , represented by an empty Schläfli symbol
    Schläfli symbol

    In mathematics, the Schl?fli symbol is a notation of the form that defines regular polytopes and tessellations.The Schl?fli symbol is named after the 19th-century mathematician Ludwig Schl?fli who made important contributions in geometry and other areas....
     .
    • Complete Graph K2
      * A 1-polytopic prism is a rectangle
      Rectangle

      In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
      , made from two translated line segments. It is represented as the product Schläfli symbol x. If it is square
      Square (geometry)

      In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
      , symmetry can be reduced it: x = .
    • Example: Square, x, two parallel line segments, connected by two line segment sides.
  • A polygonal
    Polygon

    In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
     prism is a 3-dimensional prism made from two translated polygons connected by rectangles. A regular polygon can construct a uniform n-gonal prism represented by the product x. If p=4, with square sides symmetry it becomes a cube
    Cube

    A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
    : x = .
    • Pentagonal Prism
      Example: Pentagonal prism
      Pentagonal prism

      In geometry, the pentagonal prism is a Prism with a pentagonal base. It is a type of heptahedron.If faces are all regular, the pentagonal prism is a semiregular polyhedron, and the third in an infinite set of prisms formed by square sides and two regular polygon caps....
      , x, two parallel pentagon
      Pentagon

      In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
      s connected by 5 rectangular sides.
  • A polyhedral
    Polyhedron

    |}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
     prism is a 4-dimensional prism made from two translated polyhedra connected by 3-dimensional prism cells. A regular polyhedron can construct the uniform polychoric prism, represented by the product x. If the polyhedron is a cube, and the sides are cubes, it becomes a tesseract
    Tesseract

    In geometry, the tesseract, also called an 8-cell or regular octachoron, is the Fourth dimension analog of the cube. The tesseract is to the cube as the cube is to the square ....
    : x = .
    • Example: Dodecahedral prism, x, two parallel dodecahedra
      Dodecahedron

      A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
       connected by 12 pentagonal prism sides.
  • ...


Higher order prismatic polytopes also exist as Cartesian product
Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
s of any two polytopes. The dimension of a polytope is the product of the dimensions of the elements. The first example of these exist in 4-dimensional space are called duoprism
Duoprism

In geometry, a duoprism is a polytope resulting from the Cartesian product of two polytopes of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an -polytope, where n and m are 2 or higher....
s as the product of two polygons. Regular duoprisms are represented as x.

See also

  • Prismatic uniform polyhedron
    Prismatic uniform polyhedron

    A prismatic uniform polyhedron is a uniform polyhedron with Dihedral symmetry in three dimensions. They exist in two infinite families, the uniform Prism and the uniform antiprisms....
    .
  • Cylinder (geometry)
    Cylinder (geometry)

    A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
  • Apeirogonal prism
    Apeirogonal prism

    In geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of Prism ; it can be considered an infinite polyhedron or a tiling of the plane....


External links

  • MATHguide
  • MATHguide
  • Free nets of prisms and antiprisms
  • Using nets generated by Stella
    Stella (software)

    Stella, a computer program available in three versions was created by Robert Webb of Australia. The programs contain a large library of polyhedra which can be manipulated and altered in various ways....
    .
  • : Software used to create the 3D and 4D images on this page.