Classification of Archimedean tilings

The first step is classifying all the vertex figures that can occur. These are arrangements of regular polygons around a vertex with interior angles adding up to $360^\circ$. The interior angle of a regular $n$-gon is $180 - 360/n$ degrees. By working out the resulting equation and considering the positive integers $\ge 3$ that can occur, we can show there are exactly 21 possible vertex figures.

The next step is to see how vertex figures would piece together. Usually just positioning a couple will show whether or not it is possible to find a tiling with that vertex figure. PostScript would be very helpful for experimentation on this. In this way, we will wind up with exactly all the Archimedean tilings.