Duoprism

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Duoprism

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....
, a duoprism is a 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 ....
resulting from the Cartesian product

Cartesian product

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Encyclopedia

Set of uniform p,q-duoprisms Example 16,16-duoprism Schlegel diagram Schlegel diagram In geometry, a Schlegel diagram is a projection of a polytope from into through a point beyond one of its facets. The resulting entity is a polytopal subdivision of the facet in that is combinatorially equivalent to the original polytope.... Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.
Type	Prismatic uniform polychoron Uniform polychoron In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
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....	x
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter-Dynkin diagram is a Graph with labelled edges. It represents the spatial relations between a collection of mirrors , and describes a Kaleidoscope construction....
Cells	p q-gonal prisms Prism (geometry) In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides.... , q p-gonal prisms
Faces	pq 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 .... s, p q-gons, q p-gons
Edges	2pq
Vertices	pq
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off....	disphenoid tetrahedron Example for 16-16 duoprism (Each edge is part of a face at the central vertex with a given number of sides)
Symmetry group Coxeter group In mathematics, a Coxeter group, named after Harold Scott MacDonald Coxeter, is an group that admits a group presentation in terms of mirror symmetries....	[p]x[q] or [p,2,q]
Properties	convex Convex set In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object.... if both bases are convex

In geometry

Geometry

Polytope

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 polytopes 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

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 ....
) or higher.

The lowest dimensional duoprisms exist in 4-dimensional space as polychora

Polychoron

Cartesian product

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 ....
s in 2-dimensional Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
. More precisely, it is the set of points:

where P₁ and P₂ are the sets of the points contained in the respective polygons. Such a duoprism is convex

Convex set

In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object....
if both bases are convex, and is bounded by prismatic cells

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
.

Nomenclature

Four-dimensional duoprisms are considered to be prismatic polychora. A duoprism constructed from two 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 of the same size is a uniform duoprism.

A duoprism made of n-polygons and m-polygons is named by prefixing 'duoprism' with the names of the base polygons, for example: a triangular-pentagonal duoprism is the Cartesian product of a triangle and a pentagon.

An alternative, more concise way of specifying a particular duoprism is by prefixing with numbers denoting the base polygons, for example: 3,5-duoprism for the triangular-pentagonal duoprism.

Other alternative names:

q-gonal-p-gonal prism
q-gonal-p-gonal double prism
q-gonal-p-gonal hyperprism

The term duoprism is coined by George Olshevsky

George Olshevsky

George Olshevsky is a freelance editing, writer, publisher, paleontologist, and mathematician living in San Diego, California.Olshevsky maintains the comprehensive online Dinosaur Genera List....
. Conway

Conway

Conway may refer to:...
proposed a similar name proprism for product prism.

Geometry of 4-dimensional duoprisms

A 4-dimensional uniform
Uniform polychoron

In geometry, a Uniform polytope polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedron.This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms....
duoprism is created by the product of a regular n-sided 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 ....
and a regular m-sided polygon with the same edge length. It is bounded by n m-gonal prisms

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
and m n-gonal prisms. For example, the Cartesian product of a triangle and a hexagon is a duoprism bounded by 6 triangular prisms and 3 hexagonal prisms.

When m and n are identical, the resulting duoprism is bounded by 2n identical n-gonal prisms. For example, the Cartesian product of two triangles is a duoprism bounded by 6 triangular prisms.

When m and n are identically 4, the resulting duoprism is bounded by 8 square prisms (cubes), and is identical to the 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 ....
.

The m-gonal prisms are attached to each other via their m-gonal faces, and form a closed loop. Similarly, the n-gonal prisms are attached to each other via their n-gonal faces, and form a second loop perpendicular to the first. These two loops are attached to each other via their square faces, and are mutually perpendicular.

As m and n approach infinity, the corresponding duoprisms approach the duocylinder

Duocylinder

The duocylinder, or double cylinder, is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two disk s of radius r:...
. As such, duoprisms are useful as non-quadric

Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface defined as the locus of root of a quadratic polynomial....
approximations of the duocylinder.

Polychoral duoantiprisms

Like 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 as alternated prism

Prism (geometry)

In geometry, an n-sided prism is a polyhedron made of an n-sided polygon base, a Translation copy, and n faces joining corresponding sides....
s, there is a set of 4-dimensional duoantiprisms 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"....
s that can be created by an alternation operation applied to a duoprism. However most are not uniform. The alternated vertices create nonregular tetrahedral cells, except for the special case, the 4-4 duoprism (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 ....
) which creates the uniform (and regular) 16-cell

16-cell

In Fourth dimension geometry, a 16-cell, is a regular convex polychora, or polytope existing in four dimensions. It is also known as the hexadecachoron....
. The 16-cell is the only convex uniform duoantiprism.

See also grand antiprism

Grand antiprism

In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform polychoron bounded by 320 cell : 20 pentagonal antiprisms, and 300 tetrahedron....
.

Images of uniform polychoral duoprisms

All of these images are Schlegel diagram

Schlegel diagram

In geometry, a Schlegel diagram is a projection of a polytope from into through a point beyond one of its facets. The resulting entity is a polytopal subdivision of the facet in that is combinatorially equivalent to the original polytope....
s with one cell shown. The p-q duoprisms are identical to the q-p duoprisms, but look different because they are projected in the center of different cells.

3-3	3-4	3-5	3-6
4-3	4-4	4-5	4-6
5-3	5-4	5-5	5-6
6-3	6-4	6-5	6-6

External links

- —describes duoprisms as "double prisms" and duocylinders as "double cylinders"
- glossary of higher-dimensional terms
The word Duoprism is also the name of an . It has no relation to the mathematical use of the term as described here.