Euler's FormulaDate: 11/28/2001 at 13:33:13 From: Amanda Cornett Subject: Euler's formula I have to find Euler's formula for two-dimensional figures and explain it at a university level and at an elementary-school level. Date: 11/28/2001 at 13:37:31 From: Doctor Tom Subject: Re: Euler's formula Hi Amanda, Here's what I'd do: First, go through a bunch of simple examples where the kids can count vertices, edges, and faces, and verify that V - E + F = 2. Or maybe start with four or five examples where you don't even calculate V - E + F, but just make a table of those values. Then have the kids look at the table and look for patterns. You should be able to lead them to show that the above sum/difference is two by noticing things like the fact that if V or F goes up, so does E. Next, after they've guessed the formula (with or without your help), try making some more drawings to test the formula. What I would do here would be to draw a new configuration, count the items, and check it. Then make the item a bit more complicated by adding vertices in the middles of edges and by adding edges that connect two existing vertices (or make a loop from a vertex to itself). You're doing this to secretly convince the kids that arbitrarily complex connected configurations can be made from simple ones by adding vertices to edges or edges connecting existing vertices. Finally, for the proof, show that it's true for a single vertex in the plane (V=1, F=1, and E=0). Next show that if you have ANY configuration, adding a vertex to an edge increases V by 1 and E by 1, leaving V - E + F the same, and that adding an edge between two vertices increases F by 1 and E by 1, again preserving the Euler characteristic. So you have a trivial situation where the formula holds, and two operations that are guaranteed to preserve the characteristic. Finally, begin with the dot on the plane and show how to construct a few of the examples you've already done one step at a time. Perhaps this isn't a rock-solid mathematical proof, but it should certainly be enough to convince the kids that the theorem is true, and shows them (in a secret sort of way) the ideas of mathematical induction and the idea of using an invariant for a proof. I might even end by showing that exactly the same formula holds for a 3-D cube and an assertion that the formula is also true in 3-D, to give some of the brighter kids something to think about and play with. I hope this helps. - Doctor Tom, The Math Forum http://mathforum.org/dr.math/ |
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