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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.
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.
Any convex polyhedron's surface has Euler characteristic + =, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of faces is 2 more than the excess of the number of edges over the number of vertices.
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.).
In this case the equation derived from Euler's formula is not affected by the number of quadrilaterals, and for every assignment to the numbers of faces of other types that obeys this equation it is possible to choose a number of quadrilaterals that allows a 4-regular polyhedron to be realized. [1]
From the fact that each facet of a three-dimensional polyhedron has at least three edges, it follows by double counting that 2e ≥ 3f, and using this inequality to eliminate e and f from Euler's formula leads to the further inequalities e ≤ 3v − 6 and f ≤ 2v − 4. By duality, e ≤ 3f − 6 and v ≤ 2f − 4.
Euler's formula is also valid for convex polyhedra. This is no coincidence: every convex polyhedron can be turned into a connected, simple, planar graph by using the Schlegel diagram of the polyhedron, a perspective projection of the polyhedron onto a plane with the center of perspective chosen near the center of one of the polyhedron's faces ...
The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.