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steiniu
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There are 7 letters in STEINIU ( E1I1N1S1T1U1 )
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In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex-connected planar graphs (with at least four * vertices). That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.”The theorem appears in a 1922 paper of Ernst Steinitz, after whom it is named. It can be proven by mathematical induction (as Steinitz did), by finding the minimum-energy state of a two-dimensional spring system into three dimensions, or by using the circle packing theorem. * Several extensions of the theorem are known, in which the polyhedron that realizes a given graph has additional constraints; for instance, every polyhedral graph is the graph of a convex polyhedron with integer coordinates, or the graph of a convex polyhedron all of whose edges are tangent to a common midsphere. * In higher dimensions, the problem of characterizing the graphs of convex polytopes remains open. |