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## The Platonic Solids

The Platonic solids are sold shapes formed by identical regular polygons. There are only five Platonic solids

The tetrahedron, octahedron and icosahedron can all be made from 4, 8 and 12 equilateral triangles respectively.
On - the cuber - can be made from 6 squares.
The Dodecahedron can be made from 12 pentagons.
There can be no more such solids, because for
$n>5$
the interior angle is
$\frac{180(n-2)}{n}$
and 360 divides by this number
$\frac{360}{180(n-2)/n} = \frac{2n}{n-2}$
must be at least 3 or we have have just two shapes at a point and would not fold the plane up and fit another shape in the gap. If
$n=6$
then
$\frac{2n}{n-2}$
is exactly 3 and the 6 sided shapes are regular hexagons. We can tile the plane with regular hexagons, but can not fold the plane up while keeping the hexagons flat to make a solid.