Question

A soccer ball has 12 pentagons and 20 hexagons, making a perfect ball shape; I'm told that any other combination of polygon that also form a ball shape MUST have 'at least' 12 pentagons, yet I have a photo of a ball shaped with close to 90 hexagons and possibly ONLY 6 pentagons. I will email the photo if you wish to work out the exact numbers ... then tell me if you can how it can work with only 6 pentagons -- please.

Answer 1

Answer on Question#39429 - Math - Analytic Geometry

A soccer ball has 12 pentagons and 20 hexagons, making a perfect ball shape; I'm told that any other combination of polygon that also form a ball shape MUST have 'at least' 12 pentagons, yet I have a photo of a ball shaped with close to 90 hexagons and possibly ONLY 6 pentagons. I will email the photo if you wish to work out the exact numbers ... then tell me if you can how it can work with only 6 pentagons?

Solution

Soccer ball graph is a planar, 3-regular and 3-connected graph, the faces of which are only pentagons and hexagons. Let the number of vertices, edges, pentagons, and hexagons of a soccer ball graph $G$ be denoted by $v, e, f_5$, and $f_6$, respectively. It is easy to see that

$$3v = 2e \quad \text{and} \quad 5f_5 + 6f_6 = 2e.$$

Euler's formula:

$$v + (f_5 + f_6) - e = 2 \quad \text{or} \quad 6v + 6f_5 + 6f_6 = 12 + 4e + 2e. \text{ From last formula } f_5 = 12. \text{ Thus the number of pentagons in a soccer ball graph is exactly 12.}$$

Answer

It can not work with only 6 pentagons.