3-3 duoprism

3-3 duoprism
File:Triangular Duoprism YW and ZW Rotations.gif 3D perspective projection with two different rotations
Type	Uniform duoprism
Schläfli symbol	{3}×{3} = {3}2
Coxeter diagram	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png
Dual	3-3 duopyramid = cyclic polytope C(6,4)
Properties	convex, vertex-uniform, facet-transitive, projectively unique

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

Descriptions

The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. 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, and 15 faces—which include 9 squares and 6 triangles. Its cell has 6 triangular prism. It has Coxeter diagram File:CDel branch 10.pngFile:CDel 2.pngFile:CDel branch 10.png, and symmetry [[3,2,3]], order 72.

The hypervolume of a uniform 3-3 duoprism with edge length $a$ is $$V_4=\frac{3}{16}{a}^4.$$ This is the square of the area of an equilateral triangle, $$A=\frac{\sqrt{3}}{4}{a}^2.$$

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 $3\times 3$ rook's graph, and the Paley graph of order 9. This graph is also the Cayley graph of the group $G=⟨a,b:a^3=b^3=1,ab=ba⟩\simeq C_3\times C_3$ with generating set $S=\{a,a^2,b,b^2\}$.

The minimal distance graph of a 3-3 duoprism may be ascertained by the Cartesian product of graphs between two identical both complete graphs $K_3$.

3-3 duopyramid

File:3-3 duopyramid ortho.png

The orthogonal projection of a 3-3 duopyramid

The dual polyhedron of a 3-3 duoprism is called a 3-3 duopyramid or triangular duopyramid. It is also known as the cyclic polytope C(6,4). 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 ${ℂ}^2$ with a real representation in ${ℝ}^4$ 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.

See also

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

References

• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (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, (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.

External links

• 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