Duoprism

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Set of uniform p-q duoprisms
Type Prismatic uniform 4-polytopes
Schläfli symbol {p}×{q}
Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node.png
Cells p,q-gonal prisms,
q p-gonal prisms
Faces pq squares,
p q-gons,
q p-gons
Edges 2pq
Vertices pq
Vertex figure Pq-duoprism verf.png
disphenoid
Symmetry [p,2,q], order 4pq
Dual p,q-duopyramid
Properties convex, vertex-uniform
 
Set of uniform p-p duoprisms
Type Prismatic uniform 4-polytope
Schläfli symbol {p}×{p}
Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel p.pngCDel node.png
Cells 2p p-gonal prisms
Faces p2 squares,
2p p-gons
Edges 2p2
Vertices p2
Symmetry [[p,2,p]] = [2p,2+,2p], order 8p2
Dual p-p duopyramid
Properties convex, vertex-uniform, Facet-transitive
A close up inside the 23-29 duoprism projected onto a 3-sphere, and perspective projected to 3-space. As m and n become large, a duoprism approaches the geometry of duocylinder just like a p-gonal prism approaches a cylinder.

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.

The lowest-dimensional duoprisms exist in 4-dimensional space as 4-polytopes being the Cartesian product of two polygons in 2-dimensional Euclidean space. More precisely, it is the set of points:

P_1 \times P_2 = \{ (x,y,z,w) | (x,y)\in P_1, (z,w)\in P_2 \}

where P1 and P2 are the sets of the points contained in the respective polygons. Such a duoprism is convex if both bases are convex, and is bounded by prismatic cells.

Nomenclature[edit]

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. Conway proposed a similar name proprism for product prism.

Example 16-16 duoprism[edit]

Schlegel diagram
16-16 duoprism.png
Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.
net
16-16 duoprism net.png
The two sets of 16-gonal prisms are shown. The top and bottom faces of the vertical cylinder are connected when folded together in 4D.

Geometry of 4-dimensional duoprisms[edit]

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.

Nets[edit]

3,3 duoprism net.png
3-3
4,4 duoprism net.png
4-4
5,5 duoprism net.png
5-5
6,6 duoprism net.png
6-6
8,8 duoprism net.png
8-8
10-10 duoprism net.png
10-10
3,4 duoprism net.png
3-4
3,5 duoprism net.png
3-5
3,6 duoprism net.png
3-6
4,5 duoprism net.png
4-5
4,6 duoprism net.png
4-6
3-8 duoprism net.png
3-8

Perspective projections[edit]

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

Schlegel diagrams
Hexagonal prism skeleton perspective.png 6-6 duoprism.png
6-prism 6-6 duoprism
A hexagonal prism, projected into the plane by perspective, centered on a hexagonal face, looks like a double hexagon connected by (distorted) squares. Similarly a 6-6 duoprism projected into 3D approximates a torus, hexagonal both in plan and in section.

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.

Schlegel diagrams
3-3 duoprism.png
3-3
3-4 duoprism.png
3-4
3-5 duoprism.png
3-5
3-6 duoprism.png
3-6
3-7 duoprism.png
3-7
3-8 duoprism.png
3-8
4-3 duoprism.png
4-3
4-4 duoprism.png
4-4
4-5 duoprism.png
4-5
4-6 duoprism.png
4-6
4-7 duoprism.png
4-7
4-8 duoprism.png
4-8
5-3 duoprism.png
5-3
5-4 duoprism.png
5-4
5-5 duoprism.png
5-5
5-6 duoprism.png
5-6
5-7 duoprism.png
5-7
5-8 duoprism.png
5-8
6-3 duoprism.png
6-3
6-4 duoprism.png
6-4
6-5 duoprism.png
6-5
6-6 duoprism.png
6-6
6-7 duoprism.png
6-7
6-8 duoprism.png
6-8
7-3 duoprism.png
7-3
7-4 duoprism.png
7-4
7-5 duoprism.png
7-5
7-6 duoprism.png
7-6
7-7 duoprism.png
7-7
7-8 duoprism.png
7-8
8-3 duoprism.png
8-3
8-4 duoprism.png
8-4
8-5 duoprism.png
8-5
8-6 duoprism.png
8-6
8-7 duoprism.png
8-7
8-8 duoprism.png
8-8

Orthogonal projections[edit]

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.

Orthogonal projection wireframes of p-p duoprisms
Odd
3-3 5-5 7-7 9-9
3-3 duoprism ortho-dih3.png 3-3 duoprism ortho-Dih3.png 5-5 duoprism ortho-5.png 5-5 duoprism ortho-Dih5.png 7-7 duopism ortho-7.png 7-7 duoprism ortho-Dih7.png 9-9 duoprism-ortho-9.png 9-9 duoprism ortho-Dih9.png
[3] [6] [5] [10] [7] [14] [9] [18]
Even
4-4 (tesseract) 6-6 8-8 10-10
4-cube t0 A3.svg 4-cube t0.svg 6-6 duoprism ortho-Dih6.png 6-6 duoprism ortho-3.png 8-8 duoprism ortho-Dih8.png 8-8 duoprism ortho-3.png 10-10 duoprism ortho-Dih10.png 10-10 duoprism ortho-3.png
[4] [8] [6] [12] [8] [16] [10] [20]

Related polytopes[edit]

A stereographic projection of a rotating duocylinder, divided into a checkerboard surface of squares from the {4,4|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.)


Duoantiprism[edit]

p-q duoantiprism vertex figure, a gyrobifastigium

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 CDel node 1.pngCDel p.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel q.pngCDel node 1.png, t0,1,2,3{p,2,q}, can be alternated into CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel q.pngCDel node h.png, ht0,1,2,3{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 CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel 2.pngCDel node 1.png, t0,1,2,3{2,2,2}, with its alternation as the 16-cell, CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png, ht0,1,2,3{2,2,2}.

The only nonconvex uniform solution is p=5, q=5/3, ht0,1,2,3{5,2,5/3}, CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 5.pngCDel rat.pngCDel 3x.pngCDel node h.png, constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra, known as the great duoantiprism (gudap).[1][2]

k_22 polytopes[edit]

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 [32,2,3], {\bar{T}}_7. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
2A2 A5 E6 {\tilde{E}}_{6}=E6+ {\bar{T}}_7=E6++
Coxeter
diagram
CDel nodes.pngCDel 3ab.pngCDel nodes 11.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry [[32,2,-1]] [[32,2,0]] [[32,2,1]] [[32,2,2]] [[32,2,3]]
Order 72 1440 103,680
Graph 3-3 duoprism ortho-skew.png 5-simplex t2.svg Up 1 22 t0 E6.svg
Name −122 022 122 222 322

See also[edit]

Notes[edit]

References[edit]

External links[edit]