Tuesday, July 20th, 2010...6:53 pm

Doodling with Euler

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Let’s learn some math by doodling.  Let’s specifically learn the Euler Characteristic, which given it is attributed to Euler should lead to big expectations as to its power!  Why?  Let’s review a bit of mathematical history.

Leonard Euler

Euler was a famous mathematician who lived during the 1700’s.  He wrote hundreds and hundreds of pages on his mathematical discoveries!

Among Euler’s many works, he introduced the concept of a function and was the first to write f(x) to denote the function f applied to the argument x.  He also introduced the modern notation for e which is the base of the natural logarithm and the letter i to denote the imaginary unit.  The use of the Greek letter p to denote the ratio of a circle’s circumference to its diameter was also popularized by Euler, although it did not originate with him.

Euler’s powers of memory and concentration were legendary.  He easily dealt with interruptions or distractions, often working with his young children playing at his feet.  This became important later in life when he was blind.  There is a story of two of Euler’s students independently summing seventeen terms of a complicated series.  They disagreed on the 15th decimal place; Euler settled the dispute by recomputing the sum in his head!

Euler is often considered one of the greatest mathematicians and his discoveries are part of a lot of math today.

To noodle a doodle

Time to doodle! To begin, place a pencil on a piece of paper and maybe even close you eyes! Create your doodle be drawing in big sweeping motions, keeping the pencil on the paper at all times. Remember that the longer you doodle the harder the exercise. Here’s my doodle:

Now, place dots at every crossing point in your doodle. Also place a dot at both endpoints in your drawing, unless somehow you started and ended at the same spot. Count each dot, or vertex, and write that number on your doodle. Here are the 11 vertices for my doodle.

Now, we want to count the enclosed areas of the doodle. These are also called faces. Write this number on your paper. Here are the 9 faces I found in my doodle.

Notice that the enclosed area or face that is the entire outside area of the piece of paper is counted as a face. This will be important in a moment.

Finally, we count each segment between two vertices. These are called edges. Write this number on your paper. As you see below, my doodle has 19 edges.

Now, add the number of vertices and faces and subtract the number of faces. Here is my example with vertices, edges and faces all counted within the same figure.

Now, let me use my power of mathematical prediction! You got 2! In fact, if you didn’t, try counting your vertices, faces and edges again and check your addition and subtraction.

How did I know? If we call the number of vertices V, the number of edges E and the number of faces F, then the Euler Characteristic states that

V + F – E = 2.

Now, Euler didn’t simply show this to be true for every doodle he could draw. He proved that it would be true for ANY doodle. By proving it mathematically, he proved it would work for infinitely many doodles by writing one proof!

Try the activity again with a new doodle and see if you get the same answer. Again, V + F – E will always equal 2, for any doodle you draw!

Finally, as a teaser, we can extend Euler’s result to the dodecahedron we see below. Count the number of faces, edges and vertices on the shape and you’ll find that Euler’s Characteristic holds. Does it always? What do you think? Why?

Thanks to Austin Totty, a Davidson College math major, who helped develop this exercise, although slightly adapted for this blog. The exercise was used with middle schoolers at a Math Day at the University of North Carolina at Charlotte. The idea of doodling comes from teaching from The Heart of Mathematics by Edward Burger and Michael Starbird. Enjoy.

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