Where do soccer balls come from?

Geometry Level 2

The dodecahedron has 12 pentagonal faces. Therefore it has $\frac{12 \times 5}{2} = 30$ edges.

The icosahedron has 20 equilateral triangular faces. Therefore it has $\frac{20 \times 3}{2} = 30$ edges.

How many edges does a truncated icosahedron have?

180 60 90 120

1 solution

Clara Blackstone
Oct 16, 2015

The truncated icosahedron is like the offspring of a icosahedron and a dodecahedron. (It's also the standard pattern for a soccer ball.) One way to make a truncated icosahedron is to slice each of the 12 corners of an icosahedron off to reveal 12 pentagons each surrounded by 5 hexagons. Or you can slice each of the 20 corners off a dodecahedron to reveal 20 hexagons, each touching 3 pentagons.

Either way, you get a truncated icosahedron with 20 hexagonal faces and 12 pentagonal faces. $\frac{20 \times 6 + 12 \times 5}{2}\ = \fbox{90}$ is therefore the number of edges.

Why? Because the 20 hexagons have $6 \times 20 = 120$ edges among them. And the 12 pentaogns have $5 \times 12 = 60$ edges among them. Which naively gives 180 edges total. But each of those edges is shared by 2 polygons. Therefore, each was double-counted so in reality there are only 90 edges total.

The maths in the question is wrong. 12x5/2 is 30, not 60. Similarly 20x3/2 is 30, not 60.

Jeremy Child - 5 years, 3 months ago

Correct that 12x5/2 is 30. The second part is not 20x3/2 but 20x6/2 because the slicing means the flat surfaces are now hexagons (with 6 edges) not triangles (with 3 edges). So 20x6/2 works out to 60. Add that to the 30 edges from the pentagons and you get 90 edges in total.

Angela Sullivan - 5 years ago

Another way to do this would be to use Euler's formula. We know that the vertex configuration for the shape is (6,6,5). At each vertex, 3 edges meet so E = 3V/2 . At each vertex, one pentagon meets so F = V/5. And two hexagons so F = 2V/6 = V/3. Putting it into Euler's formula, V+F = E+2, we get V equal to 60, E = 90 and F = 32. Whats amazing is that this is true for any polyhedra !

Atharva Vankundre - 5 months, 3 weeks ago

Where do soccer balls come from?

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