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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
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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