Doodle Discover
Put the paper in front of you and place a pen on it. Close your eyes. Now pick up the pen and draw sweepingly, randomly on the page without lifting the pen or running off the edge. Don’t draw too much, because later we’re going to ask you to count various parts of your work of art. Once you’ve had enough, stop drawing, lift the pen from the paper, and open your eyes. You have created your own masterpiece—you’re another Picasso. Shown at left is a sample of the genre.
We, the authors, will now attempt to demonstrate our ability to predict your actions across the void of time and space. Specifically, we will divine some features of the random drawing you just created. Please give us a moment to concentrate. Something is coming into focus . . . yes . . . yes . . . we see a squiggly object. Right? Amazing.
Please now accentuate each place where the curves cross each other in your drawing by placing a big dot at each crossing point. Also, draw dots at the points where you started and stopped. We now confidently assert that your drawing divides the paper into various regions.
Count on It—Detecting a Pattern
Okay, you may not yet be impressed with our clairvoyance, but hold on. We are detecting a pattern. We notice that each of your big dots has exactly four curves coming out from it—except for the starting and ending dots. Are we right? If we aren’t, then we must wonder if you really closed your eyes and drew randomly. If not, please start over again.
Please check our clairvoyance: Count the number of dots, regions, and edges (an edge is a segment, curved or not, connecting two dots). Don’t forget to count the one region on the outside—that is, the one containing the edge of the paper. (How did we know there was exactly one of these exterior regions?) Notice that the number of dots plus the number of regions is exactly two more than the num- ber of edges!
We have duly demonstrated our powers to read your mind and see through your eyes. However, these magical illusions are really mathematical feats. When you drew your squiggly curve, you unknowingly created some relationships among parts of your picture. These relationships are consequences of the topology of the paper and its idealized counterpart, the 2-dimensional plane. These relationships are topological in nature in that they are not associated with the particular size or shapes in your drawing. Let’s discover why the number of dots plus the number of regions is always two more than the number of edges. As always, we begin with an easy case.
2=V-E+F
V = Vertices
E = Edges
F = Faces
Euler Characteristic
Euler Primatives
