Polyhedron ProblemDate: 10/29/96 at 23:54:2 From: Lottie English Subject: Geometry As a homework problem in geometry, we have to answer the following question: Let P be a polyhedron with v vertices, e edges, and f faces. Show that e>=(3/2)f and e>=(3/2)v. I know Euler's formula but the professor said that this formula is not necessary when solving this problem. I only know the faces, edges, and vertices of the platonic solids, and I know there are more polyhedra than those. How do I go about solving this problem? Date: 12/06/96 at 16:12:21 From: Doctor Lorenzo Subject: Re: Geometry Ask yourself how many edges each face has. How many faces share each edge? If ownership of each edge were divided equally between the faces that share the edge, each triangle would own 3/2 edges, each quadrilateral would own 4/2 edges, and so on. So what can you say about how many edges the *average* face would own? This average is another way of saying e/f. A similar argument with counting edges at each vertex and vertices at each edge gives you a lower bound on e/v. -Doctor Lorenzo, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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