View source for Duoprism

{{Short description|Cartesian product of two polytopes}}
{| class="wikitable" align="right" style="margin-left:10px" width="250"
|bgcolor=#e7dcc3 colspan=2 align=center|'''Set of uniform {{mvar|p-q}} duoprisms'''
|-
|bgcolor=#e7dcc3|Type||[[Uniform 4-polytope#Prismatic uniform 4-polytopes|Prismatic uniform 4-polytopes]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{{math|{''p''}×{''q''} }}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|p|node|2|node_1|q|node}}
|-
|bgcolor=#e7dcc3|Cells||{{mvar|p q}}-gonal [[prism (geometry)|prisms]],<BR>{{mvar|q p}}-gonal prisms
|-
|bgcolor=#e7dcc3|Faces||{{mvar|pq}} [[Square (geometry)|square]]s,<BR>{{mvar|p q}}-gons,<BR>{{mvar|q p}}-gons
|-
|bgcolor=#e7dcc3|Edges||{{math|2''pq''}}
|-
|bgcolor=#e7dcc3|Vertices||{{mvar|pq}}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Pq-duoprism verf.png|100px]]<BR>[[disphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||{{math|[''p'',2,''q'']}}, order {{math|4''pq''}}
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||{{mvar|p-q}} [[duopyramid]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[vertex-uniform]]
|-
|colspan=2|&nbsp;
|-
|bgcolor=#e7dcc3 colspan=2 align=center|'''Set of uniform p-p duoprisms'''
|-
|bgcolor=#e7dcc3|Type||[[Uniform 4-polytope#Prismatic uniform 4-polytopes|Prismatic uniform 4-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{{math|{''p''}×{''p''} }}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|p|node|2|node_1|p|node}}
|-
|bgcolor=#e7dcc3|Cells||{{math|2''p p''}}-gonal [[prism (geometry)|prisms]]
|-
|bgcolor=#e7dcc3|Faces||{{math|''p''{{sup|2}}}} [[Square (geometry)|square]]s,<BR>{{math|2''p p''}}-gons
|-
|bgcolor=#e7dcc3|Edges||{{math|2''p''{{sup|2}}}}
|-
|bgcolor=#e7dcc3|Vertices||{{math|''p''{{sup|2}}}}
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||{{math|1= [''p'',2,''p''] = [2''p'',2{{sup|+}},2''p''],}} order {{math|8''p''{{sup|2}}}}
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||{{mvar|p-p}} [[duopyramid]]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]], [[vertex-uniform]], [[Facet-transitive]]
|}
[[File:23,29-duoprism stereographic closeup.jpg|thumb|320px|A close up inside the 23-29 duoprism projected onto a 3-sphere, and perspective projected to 3-space. As {{mvar|m}} and {{mvar|n}} become large, a duoprism approaches the geometry of [[duocylinder]] just like a {{mvar|p}}-gonal prism approaches a [[Cylinder (geometry)|cylinder]].]]

In [[geometry]] of 4 dimensions or higher, a '''double prism'''<ref>''The Fourth Dimension Simply Explained'', Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: [https://web.archive.org/web/20030121092141/http://etext.lib.virginia.edu/etcbin/toccer-new2?id=ManFour.sgm&images=images%2Fmodeng&data=%2Ftexts%2Fenglish%2Fmodeng%2Fparsed&tag=public&part=all The Fourth Dimension Simply Explained]&mdash;contains a description of duoprisms (double prisms) and duocylinders (double cylinders). [https://books.google.com/books?id=Y7cEAAAAMAAJ&q=The+Fourth+Dimension+Simply+Explained Googlebook]</ref> or '''duoprism''' is a [[polytope]] resulting from the [[Cartesian product]] of two polytopes, each of two dimensions or higher. The Cartesian product of an {{mvar|n}}-polytope and an {{mvar|m}}-polytope is an {{math|(''n''+''m'')}}-polytope, where {{mvar|n}} and {{mvar|m}} are dimensions of 2 ([[polygon]]) or higher.

The lowest-dimensional '''duoprisms''' exist in [[4-dimensional space]] as [[4-polytope]]s being the Cartesian product of two polygons in 2-dimensional [[Euclidean space]]. More precisely, it is the [[Set (mathematics)|set]] of points:

:<math>P_1 \times P_2 = \{ (x,y,z,w) | (x,y)\in P_1, (z,w)\in P_2 \}</math>

where {{math|''P''{{sub|1}}}} and {{math|''P''{{sub|2}}}} are the sets of the points contained in the respective polygons. Such a duoprism is [[Convex polytope|convex]] if both bases are convex, and is bounded by [[prism (geometry)|prismatic cells]].

==Nomenclature==

Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two [[regular polygon]]s 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 ==
{|class=wikitable width=500
|- align=center
|[[Schlegel diagram]]<BR>[[File:16-16 duoprism.png|250px]]<BR>Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.
|[[Net (polytope)|net]]<BR>[[File:16-16 duoprism net.png|250px]]<BR>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 4-polytope|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 [[Prism (geometry)|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 2''n'' 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 ([[Cube (geometry)|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 ==

{| class=wikitable
|- align=center
|[[File:3-3_duoprism_net.png|100px]]<BR>[[3-3 duoprism|3-3]]
|- align=center
|[[File:4-3_duoprism_net.png|100px]]<BR>[[3-4 duoprism|3-4]]
|[[File:8-cell_net.png|100px]]<BR>[[4-4 duoprism|4-4]]
|-
|[[File:5-3_duoprism_net.png|100px]]<BR>[[3-5 duoprism|3-5]]
|[[File:5-4_duoprism_net.png|100px]]<BR>[[4-5 duoprism|4-5]]
|[[File:5-5_duoprism_net.png|100px]]<BR>[[5-5 duoprism|5-5]]
|-
|[[File:6-3_duoprism_net.png|100px]]<BR>[[3-6 duoprism|3-6]]
|[[File:6-4_duoprism_net.png|100px]]<BR>[[4-6 duoprism|4-6]]
|[[File:6-5_duoprism_net.png|100px]]<BR>[[5-6 duoprism|5-6]]
|[[File:6-6_duoprism_net.png|100px]]<BR>[[6-6 duoprism|6-6]]
|-
|[[File:7-3_duoprism_net.png|100px]]<BR>[[3-7 duoprism|3-7]]
|[[File:7-4_duoprism_net.png|100px]]<BR>[[4-7 duoprism|4-7]]
|[[File:7-5_duoprism_net.png|100px]]<BR>[[5-7 duoprism|5-7]]
|[[File:7-6_duoprism_net.png|100px]]<BR>[[6-7 duoprism|6-7]]
|[[File:7-7_duoprism_net.png|100px]]<BR>[[7-7 duoprism|7-7]]
|-
|[[File:8-3_duoprism_net.png|100px]]<BR>[[3-8 duoprism|3-8]]
|[[File:8-4_duoprism_net.png|100px]]<BR>[[4-8 duoprism|4-8]]
|[[File:8-5_duoprism_net.png|100px]]<BR>[[5-8 duoprism|5-8]]
|[[File:8-6_duoprism_net.png|100px]]<BR>[[6-8 duoprism|6-8]]
|[[File:8-7_duoprism_net.png|100px]]<BR>[[7-8 duoprism|7-8]]
|[[File:8-8_duoprism_net.png|100px]]<BR>[[8-8 duoprism|8-8]]
|-
|[[File:9-3_duoprism_net.png|100px]]<BR>[[3-9 duoprism|3-9]]
|[[File:9-4_duoprism_net.png|100px]]<BR>[[4-9 duoprism|4-9]]
|[[File:9-5_duoprism_net.png|100px]]<BR>[[5-9 duoprism|5-9]]
|[[File:9-6_duoprism_net.png|100px]]<BR>[[6-9 duoprism|6-9]]
|[[File:9-7_duoprism_net.png|100px]]<BR>[[7-9 duoprism|7-9]]
|[[File:9-8_duoprism_net.png|100px]]<BR>[[8-9 duoprism|8-9]]
|[[File:9-9_duoprism_net.png|100px]]<BR>[[9-9 duoprism|9-9]]
|-
|[[File:10-3_duoprism_net.png|100px]]<BR>[[3-10 duoprism|3-10]]
|[[File:10-4_duoprism_net.png|100px]]<BR>[[4-10 duoprism|4-10]]
|[[File:10-5_duoprism_net.png|100px]]<BR>[[5-10 duoprism|5-10]]
|[[File:10-6_duoprism_net.png|100px]]<BR>[[6-10 duoprism|6-10]]
|[[File:10-7_duoprism_net.png|100px]]<BR>[[7-10 duoprism|7-10]]
|[[File:10-8_duoprism_net.png|100px]]<BR>[[8-10 duoprism|8-10]]
|[[File:10-9_duoprism_net.png|100px]]<BR>[[9-10 duoprism|9-10]]
|[[File:10-10_duoprism_net.png|100px]]<BR>[[10-10 duoprism|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.
{| class=wikitable width=480
|+ [[Schlegel diagram]]s
|[[File:Hexagonal prism skeleton perspective.png|160px]]
|[[File:6-6 duoprism.png|160px]]
|-
!6-prism
![[6-6 duoprism]]
|-
|colspan=2|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.

{| class="wikitable"
|+ [[Schlegel diagram]]s
|- align=center
|[[File:3-3 duoprism.png|75px]]<BR>[[3-3 duoprism|3-3]]<!--<BR>(triddip)-->
|[[File:3-4 duoprism.png|75px]]<BR>[[3-4 duoprism|3-4]]<!--<BR>(tisdip)-->
|[[File:3-5 duoprism.png|75px]]<BR>[[3-5 duoprism|3-5]]<!--<BR>(trapedip)-->
|[[File:3-6 duoprism.png|75px]]<BR>[[3-6 duoprism|3-6]]<!--<BR>(thiddip)-->
|[[File:3-7 duoprism.png|75px]]<BR>3-7<!--<BR>(theddip)-->
|[[File:3-8 duoprism.png|75px]]<BR>[[3-8 duoprism|3-8]]<!--<BR>(todip)-->
|- align=center
|[[File:4-3 duoprism.png|75px]]<BR>[[3-4 duoprism|4-3]]<!--<BR>(tisdip)-->
|[[File:4-4 duoprism.png|75px]]<BR>[[4-4 duoprism|4-4]]<!--<BR>(tes)-->
|[[File:4-5 duoprism.png|75px]]<BR>[[4-5 duoprism|4-5]]<!--<BR>(squipdip)-->
|[[File:4-6 duoprism.png|75px]]<BR>[[4-6 duoprism|4-6]]<!--<BR>(shiddip)-->
|[[File:4-7 duoprism.png|75px]]<BR>4-7<!--<BR>(shedip)-->
|[[File:4-8 duoprism.png|75px]]<BR>[[4-8 duoprism|4-8]]<!--<BR>(sodip)-->
|- align=center
|[[File:5-3 duoprism.png|75px]]<BR>[[3-5 duoprism|5-3]]<!--<BR>(trapedip)-->
|[[File:5-4 duoprism.png|75px]]<BR>[[4-5 duoprism|5-4]]<!--<BR>(squipdip)-->
|[[File:5-5 duoprism.png|75px]]<BR>[[5-5 duoprism|5-5]]<!--<BR>(pedip)-->
|[[File:5-6 duoprism.png|75px]]<BR>5-6<!--<BR>(phiddip)-->
|[[File:5-7 duoprism.png|75px]]<BR>5-7<!--<BR>(pheddip)-->
|[[File:5-8 duoprism.png|75px]]<BR>5-8<!--<BR>(podip)-->
|- align=center
|[[File:6-3 duoprism.png|75px]]<BR>[[3-6 duoprism|6-3]]<!--<BR>(thiddip)-->
|[[File:6-4 duoprism.png|75px]]<BR>[[4-6 duoprism|6-4]]<!--<BR>(shiddip)-->
|[[File:6-5 duoprism.png|75px]]<BR>6-5<!--<BR>(phiddip)-->
|[[File:6-6 duoprism.png|75px]]<BR>[[6-6 duoprism|6-6]]<!--<BR>(hiddip)-->
|[[File:6-7 duoprism.png|75px]]<BR>6-7<!--<BR>(hahedip)-->
|[[File:6-8 duoprism.png|75px]]<BR>[[6-8 duoprism|6-8]]<!--<BR>(hodip)-->
|- align=center
|[[File:7-3 duoprism.png|75px]]<BR>7-3<!--<BR>(theddip)-->
|[[File:7-4 duoprism.png|75px]]<BR>7-4<!--<BR>(shedip)-->
|[[File:7-5 duoprism.png|75px]]<BR>7-5<!--<BR>(pheddip)-->
|[[File:7-6 duoprism.png|75px]]<BR>7-6<!--<BR>(hahedip)-->
|[[File:7-7 duoprism.png|75px]]<BR>7-7<!--<BR>(hedip)-->
|[[File:7-8 duoprism.png|75px]]<BR>7-8<!--<BR>(heodip)-->
|- align=center
|[[File:8-3 duoprism.png|75px]]<BR>[[3-8 duoprism|8-3]]<!--<BR>(todip)-->
|[[File:8-4 duoprism.png|75px]]<BR>[[4-8 duoprism|8-4]]<!--<BR>(sodip)-->
|[[File:8-5 duoprism.png|75px]]<BR>8-5<!--<BR>(podip)-->
|[[File:8-6 duoprism.png|75px]]<BR>[[6-8 duoprism|8-6]]<!--<BR>(hodip)-->
|[[File:8-7 duoprism.png|75px]]<BR>8-7<!--<BR>(heodip)-->
|[[File:8-8 duoprism.png|75px]]<BR>[[8-8 duoprism|8-8]]<!--<BR>(odip)-->
|}

=== 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 A<sub>3</sub> Coxeter plane of the [[tesseract]]. The 5,5 projection is identical to the 3D [[Rhombic triacontahedron#Orthogonal projections|rhombic triacontahedron]]. 
{| class=wikitable
|+ Orthogonal projection wireframes of p-p duoprisms
|- 
!colspan=12|Odd
|- 
!colspan=3|[[3-3 duoprism|3-3]]
!colspan=3|[[5-5 duoprism|5-5]]
!colspan=3|7-7
!colspan=3|9-9
|- 
|[[File:3-3 duoprism ortho-dih3.png|60px]]||[[File:3-3_duoprism-isotoxal.svg|60px]]||[[File:3-3_duoprism_ortho-Dih3.png|60px]]
|[[File:5-5 duoprism ortho-5.png|60px]]||[[File:5-5_duoprism-isotoxal.svg|60px]]||[[File:5-5_duoprism_ortho-Dih5.png|60px]]
|[[File:7-7_duopism_ortho-7.png|60px]]||[[File:7-7_duoprism-isotoxal.svg|60px]]||[[File:7-7_duoprism_ortho-Dih7.png|60px]]
|[[File:9-9_duoprism-ortho-9.png|60px]]||[[File:9-9_duoprism-isotoxal.svg|60px]]||[[File:9-9_duoprism_ortho-Dih9.png|60px]]
|-  align=center
|colspan=2|[3]
|[6]
|colspan=2|[5]
|[10]
|colspan=2|[7]
|[14]
|colspan=2|[9]
|[18]
|- 
!colspan=12|Even
|-
!colspan=3|[[4-4 duoprism|4-4]] (tesseract)
!colspan=3|[[6-6 duoprism|6-6]]
!colspan=3|[[8-8 duoprism|8-8]]
!colspan=3|[[10-10 duoprism|10-10]]
|-
|[[File:4-cube t0 A3.svg|60px]]||[[File:4-4 duoprism-isotoxal.svg|60px]]||[[File:4-cube t0.svg|60px]]
|[[File:6-6_duoprism_ortho-Dih6.png|60px]]||[[File:6-6_duoprism-isotoxal.svg|60px]]||[[File:6-6_duoprism_ortho-3.png|60px]]
||[[File:8-8_duoprism_ortho-Dih8.svg|60px]]||[[File:8-8 duoprism-isotoxal.svg|60px]]||[[File:8-8 duoprism ortho-3.png|60px]]
||[[File:10-10_duoprism_ortho-Dih10.png|60px]]||[[File:10-10 duoprism-isotoxal.svg|60px]]||[[File:10-10 duoprism ortho-3.png|60px]]
|- align=center
|colspan=2|[4]
|[8]
|colspan=2|[6]
|[12]
|colspan=2|[8]
|[16]
|colspan=2|[10]
|[20]
|}

== Related polytopes ==
[[File:Duocylinder ridge animated.gif|right|frame|A [[stereographic projection]] 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 n<sup>2</sup> square faces of a ''n-n duoprism'', using all 2n<sup>2</sup> edges and n<sup>2</sup> vertices. The 2''n'' ''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''.)

{{Clear}}

=== Duoantiprism ===
[[File:Snub p2q verf.png|thumb|left|120px|p-q duoantiprism [[vertex figure]], a [[gyrobifastigium]]]]
[[File:great duoantiprism.png|thumb|[[Great duoantiprism]], [[stereographic projection]], centred on one [[pentagrammic crossed-antiprism]]]]
Like the [[antiprism]]s as alternated [[Prism (geometry)|prism]]s, there is a set of 4-dimensional duoantiprisms: [[4-polytopes]] that can be created by an [[Alternation (geometry)|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<sub>0,1,2,3</sub>{p,2,q}, can be alternated into {{CDD|node_h|p|node_h|2x|node_h|q|node_h}}, ht<sub>0,1,2,3</sub>{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<sub>0,1,2,3</sub>{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<sub>0,1,2,3</sub>{5,2,5/3}, {{CDD|node_h|5|node_h|2x|node_h|5|rat|3x|node_h}}, constructed from 10 [[pentagonal antiprism]]s, 10 [[pentagrammic crossed-antiprism]]s, and 50 tetrahedra, known as the [[great duoantiprism]] (gudap).<ref>[http://www.polytope.net/hedrondude/misc.htm Jonathan Bowers - Miscellaneous Uniform Polychora] 965. Gudap</ref><ref>http://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</ref>

=== 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 [[rectangle]]s; 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 p<sup>2</sup> rectangular trapezoprisms (a [[cube]] with ''D<sub>2d</sub>'' 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]], -1<sub>22</sub>, is first in a dimensional series of uniform polytopes, expressed by [[Coxeter]] as k<sub>22</sub> series. The 3-3 duoprism is the vertex figure for the second, the [[birectified 5-simplex]]. The fourth figure is a Euclidean honeycomb, [[2 22 honeycomb|2<sub>22</sub>]], and the final is a paracompact hyperbolic honeycomb, 3<sub>22</sub>, with Coxeter group [3<sup>2,2,3</sup>], <math>{\bar{T}}_7</math>. Each progressive [[uniform polytope]] is constructed from the previous as its [[vertex figure]].
{{k 22 polytopes}}

==See also==

*[[Polytope]] and [[4-polytope]]
*[[Convex regular 4-polytope]]
*[[Duocylinder]]
*[[Tesseract]]

== Notes==
{{reflist}}

==References==
*''Regular Polytopes'', [[Harold Scott MacDonald Coxeter|H. S. M. Coxeter]], Dover Publications, Inc., 1973, New York, p.&nbsp;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 Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26)
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966

[[Category:Uniform 4-polytopes]]