## The Platonic Solids

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