Sunday, April 17, 2011
Sunday Numbers 2.0, Vol. 7:
The Euler-Poincaré characteristic.
I have noted in earlier posts about Leonhard Euler, a.k.a. My Favorite Lenny, that if we named everything in math that was first studied by him after him, there would hardly be anything named for anyone else. One counterexample of this is the Euler-Poincaré characteristic, which is used to discuss the relationship between the parts of a three dimensional shape, the faces F (two dimensional), the edges E (one dimensional) and the points at which the edges meet, known most commonly as the vertices V, the plural of vertex (zero dimensional). For a shape like the cube shown here, which is does not have any holes going straight through like a donut does, the formula is given as F + V = E + 2.
Let's check to make sure. F is 6 and V is 8, while E is 12, so 6 + 8 = 12 + 2. Not surprisingly, it works. If it didn't work, it wouldn't be math, right?
Let's go to a less familiar shape, the dodecahedron, the twelve sided three dimensional shape whose faces are all pentagons. F = 12, V = 20 and E = 30, so once again the formula is correct.
There is a separate theory called the Angle Deficiency Theorem. Consider that if all the faces that meet at a vertex added up to 360°, that corner would be a flat surface. The only way to make a three dimensional shape is to have less than 360° at each vertex, and that is called the angle deficiency of the vertex. For any convex three dimensional shape without a hole going through it, the sum of all the angle deficiencies of all the vertices is 720°. Let's check on the cube and the dodecahedron.
Angle deficiency at each vertex; (360-270)° = 90°.
Since there are eight vertices, (8 × 90)° = 720°.
Dodecahedron: Each angle of a regular pentagon is 108° so (360-324)° = 36°.
There are 20 vertices, so (20 × 36)° = 720°.
This is a surprising result, but the simplest proof is an application of F + V = E + 2.
The proof itself is left an exercise to the motivated reader.
(I love typing that last sentence.)