Duoprism

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

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	3-4	3-5	3-6	3-7	3-8
4-3	4-4	4-5	4-6	4-7	4-8
5-3	5-4	5-5	5-6	5-7	5-8
6-3	6-4	6-5	6-6	6-7	6-8
7-3	7-4	7-5	7-6	7-7	7-8
8-3	8-4	8-5	8-6	8-7	8-8

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]	[6]	[5]	[10]	[7]	[14]	[9]	[18]
Even
4-4 (tesseract)		6-6		8-8		10-10
[4]	[8]	[6]	[12]	[8]	[16]	[10]	[20]

p-q duoantiprism vertex figure, a gyrobifastigium

Great duoantiprism, 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 , t_0,1,2,3{p,2,q}, can be alternated into , 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 , t_0,1,2,3{2,2,2}, with its alternation as the 16-cell, , s{2}s{2}.

The only nonconvex uniform solution is p=5, q=5/3, ht_0,1,2,3{5,2,5/3}, , constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra, known as the great duoantiprism (gudap).[1][2]

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], ${\displaystyle {\bar {T}}_{7}}$. 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	A2A2	E6	${\displaystyle {\tilde {E}}_{6}}$=E6+	${\displaystyle {\bar {T}}_{7}}$=E6++
Coxeter diagram
Symmetry	[[3<sup>2,2,-1</sup>]]	[[3<sup>2,2,0</sup>]]	[[3<sup>2,2,1</sup>]]	[[3<sup>2,2,2</sup>]]	[[3<sup>2,2,3</sup>]]
Order	72	1440	103,680	∞
Graph				∞	∞
Name	−122	022	122	222	322