## geodesic dome

The slide show below shows a geodesic dome. Details of how to construct this dome can be found in the book Spherical Models by Magnus Wenninger.

It is basically a (spherical) icosahedron with each face split into 16 triangles (a mixture of scalene, isosceles and equilateral) resulting in 320 triangles. There are 12 pentagonal vertices (in light purple), the rest are hexagonal - see below for the maths behind this! The pentagons are regular but the hexagons are not (otherwise you would just have the Buckyball).

It took around 20 hours to construct...

## why 12 pentagons?

Suppose there are P pentagons and H hexagons, so the number of faces F = P + H.

There are 5 edges on each P and 6 on each H, with each edge shared by two shapes, so the total number of edges E = ( 5*P + 6*H ) / 2

There are 5 vertices on each P and 6 on each H, with each vertex shared by three shapes, so the total vertices V = ( 5*P + 6*H ) / 3

Now, Euler's formula states V + F - E = 2. so if we substitute each of these for the expressions above we get P = 12. So there are always 12 pentagons!