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Topological classification

Some polyhedra have two distinct sides to their surface. For example, the inside and outside of a convex polyhedron paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable. The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as the tetrahemihexahedron, it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours.In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it is orientable or non-orientable by considering a topological cell complex with the same incidences between its vertices, edges, and faces.

A more subtle distinction between polyhedron surfaces is given by their Euler characteristic, which combines the numbers of vertices V {\displaystyle V}, edges E {\displaystyle E}, and faces F {\displaystyle F} of a polyhedron into a single number χ {\displaystyle \chi } defined by the formula

The same formula is also used for the Euler characteristic of other kinds of topological surfaces. It is an invariant of the surface, meaning that when a single surface is subdivided into vertices, edges, and faces in more than one way, the Euler characteristic will be the same for these subdivisions. For a convex polyhedron, or more generally any simply connected polyhedron with surface a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles or cross-caps in the surface and will be less than 2. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example, the one-holed toroid and the Klein bottle both have χ = 0 {\displaystyle \chi =0}, with the first being orientable and the other not.

For many (but not all) ways of defining polyhedra, the surface of the polyhedron is required to be a manifold. This means that every edge is part of the boundary of exactly two faces (disallowing shapes like the union of two cubes that meet only along a shared edge) and that every vertex is incident to a single alternating cycle of edges and faces (disallowing shapes like the union of two cubes sharing only a single vertex). For polyhedra defined in these ways, the classification of manifolds implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability. For example, every polyhedron whose surface is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere.

A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle.