3-3 Duoprism

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 , and symmetry , order 72.

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

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 $G=\langle a,b:a^3=b^3=1,\ ab=ba\rangle\simeq C_3\times C_3$ with generating set $S=\$.

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

The dual polyhedron of a 3-3 duoprism is called a 3-3 duopyramid or triangular duopyramid., 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 ₂₃, also ₃+₃ has 6 vertices in $\mathbb^2$ with a real representation in $\mathbb^4$ 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
* {{PolyCell , urlname =section6.html, title = Catalogue of Convex Polychora, section 6

External links

The Fourth Dimension Simply Explained
mdash;describes duoprisms as "double prisms" and duocylinders as "double cylinders"

– glossary of higher-dimensional terms
Exploring Hyperspace with the Geometric Product

Uniform 4-polytopes