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Any convex polyhedron's surface has Euler characteristic = + = . This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. = ), and applies identically to spherical polyhedra. An illustration of the formula on all Platonic polyhedra is given below.
Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral combinatorics falls into two distinct areas.
The problem is that the 3-"cell" has a non-trivial second homology class (and so is not a cell). The right generalization is the inclusion–exclusion property. The Euler characteristic of the space between the two cubes is 2 because it's homeomorphic to S 2, so the Euler characteristic of the decomposition above is 4 − 2 = 2.
Euler's Gem: The Polyhedron Formula and the Birth of Topology is a book on the formula + = for the Euler characteristic of convex polyhedra and its connections to the history of topology. It was written by David Richeson and published in 2008 by the Princeton University Press , with a paperback edition in 2012.
The tetrahedron and the Szilassi polyhedron are the only two known polyhedra in which each face shares an edge with each other face.. If a polyhedron with f faces is embedded onto a surface with h holes, in such a way that each face shares an edge with each other face, it follows by some manipulation of the Euler characteristic that
The Euler characteristic for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula: + = where N k denotes the number of k-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).
A hemi-cuboctahedron is an abstract polyhedron, containing half the faces of a semiregular cuboctahedron. It has 4 triangular faces and 3 square faces, 12 edges, and 6 vertices. It can be seen as a rectified hemi-octahedron or rectified hemi-cube. Its skeleton matches 6 vertices and 12 edges of a regular octahedron.
Given two points s and t, say on the surface of a convex polyhedron, the problem is to compute a shortest path that never leaves the surface and connects s with t. This is a generalization of the problem from 2-dimension but it is much easier than the 3-dimensional problem.