3-3 duoprism

3-3 duoprism
3D perspective projection with two different rotations
Type	Uniform duoprism
Schläfli symbol	{3}×{3} = {3}2
Coxeter diagram
Properties	convex, vertex-uniform, facet-transitive

In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope.

The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons.^[1] 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,^[2] and 15 faces—which include 9 squares and 6 triangles. Its cell has 6 triangular prism. It has Coxeter diagram , and symmetry [[3,2,3]], order 72.

The hypervolume of a uniform 3-3 duoprism with edge length ${\displaystyle a}$ is $${\displaystyle V_{4}={3 \over 16}a^{4}.}$$ This is the square of the area of an equilateral triangle, $${\displaystyle A={{\sqrt {3}} \over 4}a^{2}.}$$

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 ${\displaystyle 3\times 3}$ rook's graph, and the Paley graph of order 9.^[3][4] This graph is also the Cayley graph of the group ${\displaystyle G=\langle a,b:a^{3}=b^{3}=1,\ ab=ba\rangle \simeq C_{3}\times C_{3}}$ with generating set ${\displaystyle S=\{a,a^{2},b,b^{2}\}}$.

The minimal distance graph of a 3-3 duoprism may be ascertained by the Cartesian product of graphs between two identical both complete graphs ${\displaystyle K_{3}}$.^[5]

The dual polyhedron of a 3-3 duoprism is called a 3-3 duopyramid or triangular duopyramid.^[6], page 45: "The dual of a p,q-duoprism is called a p,q-duopyramid."</ref> 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 ${\displaystyle \mathbb {C} ^{2}}$ with a real representation in ${\displaystyle \mathbb {R} ^{4}}$ 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.^[7]

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

• 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
• Catalogue of Convex Polychora, section 6, George Olshevsky.

• The Fourth Dimension Simply Explained—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
• Polygloss – glossary of higher-dimensional terms
• Exploring Hyperspace with the Geometric Product