The hypervolume of a uniform 3-3 duoprism, with edge length 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 graph of vertices and edges of the 3-3 duoprism has 9 vertices and 18 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.^[1] 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 ${\displaystyle S=\{a,a^{2},b,b^{2}\}}$ .

In 5-dimensions, some uniform 5-polytopes have 3-3 duoprism vertex figures, some with unequal edge-lengths and therefore lower symmetry:

The birectified 16-cell honeycomb also has a 3-3 duoprism vertex figure. There are three constructions for the honeycomb with two lower symmetries.

The regular complex polytope ₃{4}₂,    , in ${\displaystyle \mathbb {C} ^{2}}$  has a real representation as a 3-3 duoprism in 4-dimensional space. ₃{4}₂ has 9 vertices, and 6 3-edges. Its symmetry is ₃[4]₂, order 18. It also has a lower symmetry construction,    , or ₃{}×₃{}, with symmetry ₃[2]₃, order 9. This is the symmetry if the red and blue 3-edges are considered distinct.^[2]

3-3 duopyramid edit

The dual of a 3-3 duoprism is called a 3-3 duopyramid or triangular 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.

 
orthogonal projection

Related complex polygon edit

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.^[3]

• 3-4 duoprism
• Tesseract (4-4 duoprism)
• 5-5 duoprism
• Convex regular 4-polytope
• Duocylinder

• 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)
• 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.
• Apollonian Ball Packings and Stacked Polytopes Discrete & Computational Geometry, June 2016, Volume 55, Issue 4, pp 801–826

• 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