Wednesday, July 21st, 2010...2:38 pm

Euler, Yoda, and Wildcats, oh my!

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In the last post Doodling with Euler, we explored the Euler Characteristic with a sweeping doodle! Now, let’s apply it to some other images!

First, consider this image of the Davidson College Wildcat logo.

The image is constructed first as a stippling of the standard logo. Then, an algorithm is run that forms a Traveling Salesman path over the vertices. So, we have one continuous line that forms a circuit over these points. A distance measure was also used so there are no intersecting lines in the image. So if we count vertices in the original image, there are 5,000. In the previous post, we included intersection points as vertices. Again, there are none in this image. How many edges? There is one edge for every vertex so there are 5,000 edges, too. So, how many faces are there in this image? What do you think?

Let’s lean on Euler for help. Remember

V – E + F = 2.

So, for the Wildcat image, we have:

5000 – 5000 + F = 2,

so, F = 2. Think about that. This means there are only 2 enclosed regions in this image! Believe it? Look below, and you can see it easier as we’ve enclosed one region in red and left the other its original white.

Let’s try another image. Now, let’s use for the force on an image of Yoda! This close-up of Yoda’s face is part of a model created by Kecskemeti B. Zoltan.

The larger image, which can be accessed from Using the Force of Math in Star Wars, contains 53,770 faces and 53,756 vertices. So, how many edges are there? Again, we simply apply Euler’s Characteristic to find:

E = V + F – 2 = 53770 + 53756 – 2 = 107,524.

Yoda’s feeling pretty on edge in this picture!

Hope you enjoyed using a powerful, among many, insight of Euler!

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