7/17/2026 at 5:19:47 PM
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.)
by pmdulaney
7/19/2026 at 3:50:54 PM
Tip: Count the edges. Imagine each edge has two sides/ribons. One side/ribon is white and the other is black.by gus_massa
7/21/2026 at 5:16:06 AM
You can even find the exact number of pentagons (or hexagons, or edges, or vertices) using the equationV - 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.
by C-x_C-f
7/21/2026 at 3:22:01 PM
Side note: From Wikipedia:> The surfaces of nonconvex polyhedra can have various Euler characteristics:
It's strange because the examples use weird faces, but in most (all?) of them it is possible to split the weird faces into a few poligonal faces and get V-E+F=2. For example https://en.wikipedia.org/wiki/Small_stellated_dodecahedron use faces that are stars that intersect other faces and the intersection is not an edge. Replacing each star with 5 triangles the Euler characteristic is 2.
by gus_massa