Duoprism

From HandWiki
Short description: Cartesian product of two polytopes
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 double prism[1] or 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 dimensions of 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:

[math]\displaystyle{ P_1 \times P_2 = \{ (x,y,z,w) | (x,y)\in P_1, (z,w)\in P_2 \} }[/math]

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

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.

Example 16-16 duoprism

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

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

3-3 duoprism net.png
3-3
4-3 duoprism net.png
3-4
8-cell net.png
4-4
5-3 duoprism net.png
3-5
5-4 duoprism net.png
4-5
5-5 duoprism net.png
5-5
6-3 duoprism net.png
3-6
6-4 duoprism net.png
4-6
6-5 duoprism net.png
5-6
6-6 duoprism net.png
6-6
7-3 duoprism net.png
3-7
7-4 duoprism net.png
4-7
7-5 duoprism net.png
5-7
7-6 duoprism net.png
6-7
7-7 duoprism net.png
7-7
8-3 duoprism net.png
3-8
8-4 duoprism net.png
4-8
8-5 duoprism net.png
5-8
8-6 duoprism net.png
6-8
8-7 duoprism net.png
7-8
8-8 duoprism net.png
8-8
9-3 duoprism net.png
3-9
9-4 duoprism net.png
4-9
9-5 duoprism net.png
5-9
9-6 duoprism net.png
6-9
9-7 duoprism net.png
7-9
9-8 duoprism net.png
8-9
9-9 duoprism net.png
9-9
10-3 duoprism net.png
3-10
10-4 duoprism net.png
4-10
10-5 duoprism net.png
5-10
10-6 duoprism net.png
6-10
10-7 duoprism net.png
7-10
10-8 duoprism net.png
8-10
10-9 duoprism net.png
9-10
10-10 duoprism net.png
10-10

Perspective projections

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

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 60px 3-3 duoprism ortho-Dih3.png 5-5 duoprism ortho-5.png 60px 5-5 duoprism ortho-Dih5.png 7-7 duopism ortho-7.png 60px 7-7 duoprism ortho-Dih7.png 9-9 duoprism-ortho-9.png 60px 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 60px 4-cube t0.svg 6-6 duoprism ortho-Dih6.png 60px 6-6 duoprism ortho-3.png 8-8 duoprism ortho-Dih8.png 60px 8-8 duoprism ortho-3.png 10-10 duoprism ortho-Dih10.png 60px 10-10 duoprism ortho-3.png
[4] [8] [6] [12] [8] [16] [10] [20]

Related polytopes

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

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, s{2}s{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).[2][3]

Ditetragoltriates

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 D2d symmetry) but cannot be made uniform. The vertex figure is a triangular bipyramid.

Double antiprismoids

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.

k_22 polytopes

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], [math]\displaystyle{ {\bar{T}}_7 }[/math]. Each progressive uniform polytope is constructed from the previous as its vertex figure.

See also

Notes

  1. The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: The Fourth Dimension Simply Explained—contains a description of duoprisms (double prisms) and duocylinders (double cylinders). Googlebook
  2. Jonathan Bowers - Miscellaneous Uniform Polychora 965. Gudap
  3. http://www.polychora.com/12GudapsMovie.gif Animation of cross sections

References

  • 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