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#hexagons#pentagons#edges#side#every#black#shares#face#white#number

Discussion (4 Comments)Read Original on HackerNews

pmdulaney•4 days ago
If every black pentagon shares a face with 5 hexagons, and every white hexagon shares a face with 3 pentagons, can you determine the ratio of hexagons to pentagons?

(I haven't solved this problem, but I think it is doable.)

gus_massa•2 days ago
Tip: Count the edges. Imagine each edge has two sides/ribons. One side/ribon is white and the other is black.
C-x_C-f•about 6 hours ago
You can even find the exact number of pentagons (or hexagons, or edges, or vertices) using the equation

V - E + F = 2

where V, E, F are the number of vertices, edges, and faces, respectively.

This holds for any polyhedron (and other shapes have similar equations possibly with a different right hand side) and the left hand side is called the Euler characteristic of the soccer ball (or any polyhedron).

Spoiler warning: the Wikipedia article for the Euler characteristic [0] has a worked out example specifically for the soccer ball.

[0] https://en.wikipedia.org/wiki/Euler_characteristic

vismit2000•about 6 hours ago
https://archive.ph/zvdwg