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In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
The tetrahemihexahedron, a non-orientable self-intersecting polyhedron with four triangular faces (red) and three square faces (yellow). As with a Möbius strip or Klein bottle , a continuous path along the surface of this polyhedron can reach the point on the opposite side of the surface from its starting point, making it impossible to ...
The three square faces of the tetrahemihexahedron are, like the three facial orientations of the cube, mutually perpendicular. The "half-as-many" characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other.
The relations can be made apparent by examining the vertex figures obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example, the cube has vertex figure 4.4.4, which is to say, three adjacent square faces. The possible faces are 3 - equilateral ...
The hemi-cube is a regular projective polyhedron with 3 square faces, 6 edges, and 4 vertices. The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra: [4] Hemi-cube, {4,3}/2; Hemi-octahedron, {3,4}/2
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).
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.
They are defined by three properties: each face is either a pentagon or hexagon, exactly three faces meet at each vertex, and they have rotational icosahedral symmetry. They are not necessarily mirror-symmetric; e.g. GP(5,3) and GP(3,5) are enantiomorphs of each other. A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron.