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Coxeter, Longuet-Higgins & Miller (1954) define uniform polyhedra to be vertex-transitive polyhedra with regular faces. They define a polyhedron to be a finite set of polygons such that each side of a polygon is a side of just one other polygon, such that no non-empty proper subset of the polygons has the same property.
The rhombicosidodecahedron shares its vertex arrangement with three nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters: [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron: A vertex is placed at the point A. This produces a polyhedron with Wythoff symbol a|b c, where a equals π divided by the angle of the triangle at A, and similarly for b and c. A vertex is placed at a point on line AB so that it bisects the angle at C.
In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U 4. It has 7 faces (4 triangles and 3 squares ), 12 edges, and 6 vertices. [ 1 ] Its vertex figure is a crossed quadrilateral .
A 4-polytope is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra. The faces of a uniform 4-polytope must be regular. A 4-polytope is scaliform if it is vertex-transitive, and has all equal length edges. This allows cells which are not uniform, such as the regular-faced convex ...
In geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U 15. It has 10 faces (6 squares and 4 regular hexagons), 24 edges and 12 vertices. [1] Its vertex figure is a crossed quadrilateral. It is given Wythoff symbol 4 ⁄ 3 4 | 3, although that is a double-covering of this figure.
The topology of any given 10-polytope is defined by its Betti numbers and torsion coefficients. [1]The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, and is zero for all 10-polytopes, whatever their underlying topology.