3-3 duoprism | |
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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 rook's graph.
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 is This is the square of the area of an equilateral triangle,
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 rook's graph, and the Paley graph of order 9.[3] This graph is also the Cayley graph of the group with generating set .
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 | ||
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Coxeter diagram |
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Schlegel diagram |
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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 |
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Coxeter diagram |
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Skew orthogonal projection |
The regular complex polytope 3{4}2, , in 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. |
Space | Finite | Euclidean | Hyperbolic | ||
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n | 4 | 5 | 6 | 7 | 8 |
Coxeter group |
A2A2 | E6 | =E6+ | =E6++ | |
Coxeter diagram |
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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 | |
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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.
The regular complex polygon 2{4}3, also 3{ }+3{ } has 6 vertices in with a real representation in 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. |