A Mathematical Tribute to the Soccer Ball

#1

Post by igonvalue » Fri, Jul 17, 2026, 3:30 PM UTC

A Mathematical Tribute to the Soccer Ball
nytimes.com

Re: A Mathematical Tribute to the Soccer Ball

#2

Post by pmdulaney » Fri, Jul 17, 2026, 5:19 PM UTC

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.)

Re: A Mathematical Tribute to the Soccer Ball

#3

Post by gus_massa » Sun, Jul 19, 2026, 3:50 PM UTC

pmdulaney wrote: post #2

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.)

Tip: Count the edges. Imagine each edge has two sides/ribons. One side/ribon is white and the other is black.

Re: A Mathematical Tribute to the Soccer Ball

#4

Post by C-x_C-f » Tue, Jul 21, 2026, 5:16 AM UTC

gus_massa wrote: post #3

pmdulaney wrote: post #2

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.)

Tip: Count the edges. Imagine each edge has two sides/ribons. One side/ribon is white and the other is black.

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

Re: A Mathematical Tribute to the Soccer Ball

#5

Post by vismit2000 » Tue, Jul 21, 2026, 5:24 AM UTC

https://archive.ph/zvdwg

Re: A Mathematical Tribute to the Soccer Ball

#6

Post by gus_massa » Tue, Jul 21, 2026, 3:22 PM UTC

C-x_C-f wrote: post #4

gus_massa wrote: post #3

Earlier quoted context omitted.

Tip: Count the edges. Imagine each edge has two sides/ribons. One side/ribon is white and the other is black.

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:…

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.