How many sewings are there on a soccer ball?

The issue is that the picture depicts not the conventional soccer ball (a truncated icosahedron) but rather something a little different, the chamfered dodecahedron, also known as a truncated rhombic triacontahedron. This actually does have 120 edges.

enter image description here

So, in a sense, you were right both times, but just thinking about different polyhedra!

It's interesting to note that these are both examples of Goldberg Polyhedra, polyhedra made from only pentagons and hexagons -- although the faces are not necessarily regular (and in the chamfered dodecahedron, they are not).


What's wrong is that your diagram does not have the usual pattern of pentagons and hexagons that a soccer ball usually has. The seams of a soccer ball are given by projecting centrally the edges of the truncated icosahedron onto a sphere:

enter image description here

In particular, no vertex on a truncated icosahedron is shared by three hexagons, but that is not the case for the polyhedron in the diagram, which (per pjs36's answer) is called a chamfered dodecahedron. (A truncated icosahedron is also the shape of the molecular structure of buckminsterfullerene a.k.a. buckyballs.)


The problem is in the picture. In the picture, there are five hexagons adjacent to each pentagon. Moreover, each hexagon is adjacent to exactly two pentagons. This gives that $$ \frac{12\cdot5}{2}=30 $$ hexagons are needed, not $20$. Therefore, the given figure is does not match the given data.

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Combinatorics

Polyhedra

Algebra Precalculus

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