3-3 duoprism#Related complex polygons

The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons.{{r|coxeter-reg}} In the case of 3-3 duoprism is the simplest among them, and it can be constructed using Cartesian product of two triangles. The resulting duoprism has 9 vertices, 18 edges,{{r|ly}} and 15 faces—which include 9 squares and 6 triangles. Its cell has 6 triangular prism. It has Coxeter diagram {{CDD|branch_10|2|branch_10}}, and symmetry {{brackets|3,2,3}}, order 72.

The hypervolume of a uniform 3-3 duoprism with edge length $a$ is

$$V\_4\; =\; \{3\backslash over\; 16\}a^4.$$

This is the square of the area of an equilateral triangle,

$$A\; =\; \{\backslash sqrt3\backslash over\; 4\}a^2.$$

The 3-3 duoprism can be represented as a graph with the same number of vertices and edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique square. It is a toroidal graph, a locally linear graph, a strongly regular graph with parameters (9,4,1,2), the $3\backslash times\; 3$ rook's graph, and the Paley graph of order 9.{{r|f|mm}} This graph is also the Cayley graph of the group $G=\backslash langle\; a,b:a^3=b^3=1,\backslash \; ab=ba\backslash rangle\backslash simeq\; C\_3\backslash times\; C\_3$ with generating set $S=\backslash \{a,a^2,b,b^2\backslash \}$.

The minimal distance graph of a 3-3 duoprism may be ascertained by the Cartesian product of graphs between two identical both complete graphs $K\_3$.{{r|chen}}

File:3-3 duopyramid ortho.png

The dual polyhedron of a 3-3 duoprism is called a 3-3 duopyramid or triangular duopyramid.{{r|ma}}, page 45: "The dual of a p,q-duoprism is called a p,q-duopyramid." It has 9 tetragonal disphenoid cells, 18 triangular faces, 15 edges, and 6 vertices. It can be seen in orthogonal projection as a 6-gon circle of vertices, and edges connecting all pairs, just like a 5-simplex seen in projection.

The regular complex polygon ₂{4}₃, also ₃{ }+₃{ } has 6 vertices in $\backslash mathbb\{C\}^2$ with a real representation in $\backslash mathbb\{R\}^4$ matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the connecting edges of the 3-3 duopyramid, while the 6 edges connecting the two triangles are not included. It can be seen in a hexagonal projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete bipartite graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage.{{r|coxeter-com}}

• 3-4 duoprism
• Tesseract (4-4 duoprism)
• Duocylinder

{{reflist|refs=

{{citation

| last = Chen | first = Hao

| title = Apollonian Ball Packings and Stacked Polytopes

| doi = 10.1007/s00454-016-9777-3

| journal = Discrete & Computational Geometry

| year = 2016 | volume = 55 | issue = 4 | pages = 801–826

| doi-access = free| arxiv = 1306.2515}}

{{citation

| last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter

| year = 1974

| title = Regular Complex Polytopes

| url = https://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA110

| pages = 110, 114

| publisher = Cambridge University Press

}}

{{citation

| last = Coxeter | first = H. S. M. | author-link = Harold Scott MacDonald Coxeter

| title = Regular Polytopes

| publisher = Methuen & Co. Ltd. London

| year = 1948

| url = https://archive.org/details/regularpolytopes0000hsmc/page/124

| page = 124

}}

{{citation

| last = Fronček | first = Dalibor

| hdl = 10338.dmlcz/136481 | hdl-access = free

| issue = 1

| journal = Mathematica Slovaca

| mr = 1016323

| pages = 3–6

| title = Locally linear graphs

| volume = 39

| year = 1989

}}

{{citation

| last1 = Li | first1 = Ruiming

| last2 = Yao | first2 = Yan-An

| year = 2016

| title = Eversible duoprism mechanism

| journal = Frontiers of Mechanical Engineering

| volume = 11 | pages = 159–169

| doi = 10.1007/s11465-016-0398-6

}}

{{citation

| last = Mattheo | first = Nicholas

| year = 2015

| title = Convex polytopes and tilings with few flag orbits

| doi = 10.17760/D20194063

| publisher = Boston, Massachusetts : Northeastern University

}}

{{citation

| last1 = Makhnev | first1 = A. A.

| last2 = Minakova | first2 = I. M.

| date = January 2004

| doi = 10.1515/156939204872374

| issue = 2

| journal = Discrete Mathematics and Applications

| mr = 2069991

| title = On automorphisms of strongly regular graphs with parameters $\backslash lambda=1$, $\backslash mu=2$

| volume = 14| s2cid = 118034273

}}

}}

• 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)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• {{PolyCell | urlname =section6.html| title = Catalogue of Convex Polychora, section 6}}

• [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]—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
• [https://web.archive.org/web/20090829025933/http://geocities.com/os2fan2/gloss.htm Polygloss] – glossary of higher-dimensional terms
• [https://web.archive.org/web/20141201023452/http://www.bayarea.net/~kins/thomas_briggs/ Exploring Hyperspace with the Geometric Product]

Category:Uniform 4-polytopes