duoprism

Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two regular polygons of the same edge length 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, shortened from double prism. John Horton Conway proposed a similar name proprism for product prism, a Cartesian product of two or more polytopes of dimension at least two. The duoprisms are proprisms formed from exactly two polytopes.

A 4-dimensional uniform duoprism is created by the product of a regular n-sided polygon and a regular m-sided polygon with the same edge length. It is bounded by n m-gonal prisms 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.

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. As such, duoprisms are useful as non-quadric approximations of the duocylinder.

A cell-centered perspective projection makes a duoprism look like a torus, with two sets of orthogonal cells, p-gonal and q-gonal prisms.

The p-q duoprisms are identical to the q-p duoprisms, but look different in these projections because they are projected in the center of different cells.

Vertex-centered orthogonal projections of p-p duoprisms project into [2n] symmetry for odd degrees, and [n] for even degrees. There are n vertices projected into the center. For 4,4, it represents the A₃ Coxeter plane of the tesseract. The 5,5 projection is identical to the 3D rhombic triacontahedron.

File:Duocylinder ridge animated.gif of a rotating duocylinder, divided into a checkerboard surface of squares from the {4,4{{pipe}}n} skew polyhedron]]

The regular skew polyhedron, {4,4|n}, exists in 4-space as the n² square faces of a n-n duoprism, using all 2n² edges and n² vertices. The 2n n-gonal faces can be seen as removed. (skew polyhedra can be seen in the same way by a n-m duoprism, but these are not regular.)

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File:Snub p2q verf.png, a gyrobifastigium]]

File:great duoantiprism.png, stereographic projection, centred on one pentagrammic crossed-antiprism]]

Like the antiprisms as alternated prisms, there is a set of 4-dimensional duoantiprisms: 4-polytopes that can be created by an alternation operation applied to a duoprism. The alternated vertices create nonregular tetrahedral cells, except for the special case, the 4-4 duoprism (tesseract) which creates the uniform (and regular) 16-cell. The 16-cell is the only convex uniform duoantiprism.

The duoprisms {{CDD|node_1|p|node_1|2|node_1|q|node_1}}, t_0,1,2,3{p,2,q}, can be alternated into {{CDD|node_h|p|node_h|2x|node_h|q|node_h}}, ht_0,1,2,3{p,2,q}, the "duoantiprisms", which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the tesseract {{CDD|node_1|2|node_1|2|node_1|2|node_1}}, t_0,1,2,3{2,2,2}, with its alternation as the 16-cell, {{CDD|node_h|2x|node_h|2x|node_h|2x|node_h}}, s{2}s{2}.

The only nonconvex uniform solution is p=5, q=5/3, ht_0,1,2,3{5,2,5/3}, {{CDD|node_h|5|node_h|2x|node_h|5|rat|3x|node_h}}, constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra, known as the great duoantiprism (gudap).[http://www.polytope.net/hedrondude/misc.htm Jonathan Bowers - Miscellaneous Uniform Polychora] 965. Gudaphttp://www.polychora.com/12GudapsMovie.gif {{Webarchive|url=https://web.archive.org/web/20140222055922/http://www.polychora.com/12GudapsMovie.gif |date=2014-02-22 }} Animation of cross sections

Also related are the ditetragoltriates or octagoltriates, formed by taking the octagon (considered to be a ditetragon or a truncated square) to a p-gon. The octagon of a p-gon can be clearly defined if one assumes that the octagon is the convex hull of two perpendicular rectangles; then the p-gonal ditetragoltriate is the convex hull of two p-p duoprisms (where the p-gons are similar but not congruent, having different sizes) in perpendicular orientations. The resulting polychoron is isogonal and has 2p p-gonal prisms and p² rectangular trapezoprisms (a cube with D_2d symmetry) but cannot be made uniform. The vertex figure is a triangular bipyramid.

Like the duoantiprisms as alternated duoprisms, there is a set of p-gonal double antiprismoids created by alternating the 2p-gonal ditetragoltriates, creating p-gonal antiprisms and tetrahedra while reinterpreting the non-corealmic triangular bipyramidal spaces as two tetrahedra. The resulting figure is generally not uniform except for two cases: the grand antiprism and its conjugate, the pentagrammic double antiprismoid (with p = 5 and 5/3 respectively), represented as the alternation of a decagonal or decagrammic ditetragoltriate. The vertex figure is a variant of the sphenocorona.

The 3-3 duoprism, -1₂₂, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k₂₂ series. The 3-3 duoprism is the vertex figure for the second, the birectified 5-simplex. The fourth figure is a Euclidean honeycomb, 2₂₂, and the final is a paracompact hyperbolic honeycomb, 3₂₂, with Coxeter group [3^2,2,3], $\{\backslash bar\{T\}\}\_7$. Each progressive uniform polytope is constructed from the previous as its vertex figure.

{{k 22 polytopes}}

• Polytope and 4-polytope
• Convex regular 4-polytope
• Duocylinder
• Tesseract

{{reflist}}

• Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, {{ISBN|0-486-40919-8}} (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
• Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

Category:Uniform 4-polytopes