An Even Dozen

2020 08 20 an even dozen

https://commons.wikimedia.org/wiki/File:Football_(soccer_ball).svg

The floor of a regular soccer ball is roofed with 20 hexagons and 12 pentagons. Apparently, whereas we’d differ the variety of hexagons, the variety of pentagons should at all times be 12.

That’s as a result of the Euler characteristic of a sphere is 2, so VE + F = 2, the place V is the variety of vertices, or corners, E is the variety of edges, and F is the variety of faces. If P is the variety of pentagons and H is the variety of hexagons, then the full variety of faces is F = P + H; the full variety of vertices is V = (5P + 6H) / 3 (we divide by 3 as a result of three faces meet at every vertex); and the full variety of edges is E = (5P + 6H) / 2 (dividing by 2 as a result of two faces meet at every edge). Placing these collectively offers

displaystyle V-E+F={frac {5P+6H}{3}}-{frac {5P+6H}{2}}+P+H={frac {P}{6}},

and for the reason that Euler attribute is 2, this implies P should at all times be 12.

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