3-3 duoprism
3D perspective projection with two different rotations
Type	Uniform duoprism
Schläfli symbol	{3}×{3} = {3}2
Coxeter diagram
Dual	3-3 duopyramid
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. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.

It has 9 vertices, 18 edges, 15 faces (9 squares, and 6 triangles), in 6 triangular prism cells. It has Coxeter diagram , and symmetry [[3,2,3]], order 72. Its vertices and edges form a ${\displaystyle 3\times 3}$ rook's graph.

Description

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.

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, which has 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] 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 ${\displaystyle S=\{a,a^{2},b,b^{2}\}}$.

Symmetry

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

Symmetry	[[3,2,3]], order 72	[3,2], order 12
Coxeter diagram
Schlegel diagram
Name	t2α5	t03α5	t03γ5	t03β5

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

Symmetry	[3,2,3], order 36	[3,2], order 12	[3], order 6
Coxeter diagram
Skew orthogonal projection

Related complex polygons

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. ^[4]

Perspective projection

Orthogonal projection with coinciding central vertices

Orthogonal projection, offset view to avoid overlapping elements.

Related polytopes

k₂₂ figures in n dimensions

Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	A2A2	E6	${\displaystyle {\tilde {E}}_{6}}$=E6+	${\displaystyle {\bar {T}}_{7}}$=E6++
Coxeter diagram
Symmetry	[[32,2,-1]]	[[32,2,0]]	[[32,2,1]]	[[32,2,2]]	[[32,2,3]]
Order	72	1440	103,680	∞
Graph				∞	∞
Name	−122	022	122	222	322

3-3 duopyramid

3-3 duopyramid
Type	Uniform dual duopyramid
Schläfli symbol	{3}+{3} = 2{3}
Coxeter diagram
Cells	9 tetragonal disphenoids
Faces	18 isosceles triangles
Edges	15 (9+6)
Vertices	6 (3+3)
Symmetry	[[3,2,3]] = [6,2+,6], order 72
Dual	3-3 duoprism
Properties	convex, vertex-uniform, facet-transitive

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

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. ^[5]

The 2{4}3 with 6 vertices in blue and red connected by 9 2-edges as a complete bipartite graph.

It has 3 sets of 3 edges, seen here with colors.

See also

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

References

• 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

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