3-3 duoprism

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 , order 72. Its vertices and edges form a rook's graph. The hypervolume of a uniform 3-3 duoprism, with edge length a, is . This is the square of the area of an equilateral triangle, . 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 rook's graph, and the Paley graph of order 9. 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: 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 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. 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.

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3-4 duoprism

In geometry of 4 dimensions, a 3-4 duoprism, the second smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of a triangle and a square. The 3-4 duoprism exists in some of the uniform 5-polytopes in the B5 family. The quasiregular complex polytope 3{}×4{}, , in has a real representation as a 3-4 duoprism in 4-dimensional space. It has 12 vertices, and 4 3-edges and 3 4-edges. Its symmetry is 3[2]4, order 12. The birectified 5-cube, has a uniform 3-4 duoprism vertex figure: The dual of a 3-4 duoprism is called a 3-4 duopyramid.

Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Precise definitions exist only for the regular complex polytopes, which are configurations.

Uniform 5-polytope

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets. The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.

Object-Centered Surface Reconstruction: Combining Multi-Image Stereo and Shading

Pascal Fua

Our goal is to reconstruct both the shape and reflectance properties of surfaces from multiple images. To this end, we introduce an object-centered representation, specifically, a hexagonally-connected 3-D mesh of vertices with triangular facets. Such a re ...

Springer Verlag1995