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Euler's Formula


Date: 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/   
    
Associated Topics:
Elementary Geometry
Elementary Polyhedra
Elementary Two-Dimensional Geometry
High School Euclidean/Plane Geometry
High School Geometry
High School Polyhedra
Middle School Geometry
Middle School Polyhedra
Middle School Two-Dimensional Geometry

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