Euler's Formula and Graph Duality
A description of planar graph duality, and how it can be applied in a particularly elegant proof of Euler's Characteristic Formula. Music: Wyoming 307 by Time For ...
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In my video on the circle division problem, I referenced Euler’s Characteristic Formula, and here I would like to share a particularly nice proof of this fact. It’s very different from the inductive proof typically given, but I’m not trying to argue that this is somehow better or easier to understand than other proofs. Instead, I chose this topic to illustrate one example of the incredible notion of duality, and how it can produce wonderfully elegant math. First, let’s go over what the theorem states. If you draw some dots with some lines between them, that is, a graph, and if none of the lines intersect, which is to say you have a planar graph, and if your drawing is connected, then Euler’s Formula states that the number of dots, minus the number of lines, plus the number ...
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In my video on the circle division problem, I referenced Euler’s Characteristic Formula, and here I would like to share a particularly nice proof of this fact. It’s very different from the inductive proof typically given, but I’m not trying to argue that this is somehow better or easier to understand than other proofs. Instead, I chose this topic to illustrate one example of the incredible notion of duality, and how it can produce wonderfully elegant math. First, let’s go over what the theorem states. If you draw some dots with some lines between them, that is, a graph, and if none of the lines intersect, which is to say you have a planar graph, and if your drawing is connected, then Euler’s Formula states that the number of dots, minus the number of lines, plus the number ...
