The icosahedron – truncation

When you truncate the icosahedron (in other words, you divide each edge of each triangle in thirds, and then cut off each of the tips of the icosahedron to the 1/3 mark), you get a polyhedron called the truncated icosahedron (commonly known as the Buckyball). It has 12 pentagons, each surrounded by hexagons.

 
This, of course, is the shape of a soccer ball. It's also the 3-D shape you would get if you were to take a 2-D honeycomb of 32 hexagons, slice a pie-shaped piece out of twelve of the hexagons, and then fold them up to make pentagons.

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