Euclidean tilings by convex regular polygons

Example

Consider tilings of the plane whose tiles are convex regular polygons. The tiling is edge-to-edge if two tiles that meet share either one full edge or one vertex. The interior angle of a regular $n$-gon is $(n-2)\pi/n$. A tiling by copies of a single regular $n$-gon is called regular. At each vertex of such a tiling some number $k \ge 3$ of tiles meet, so $k(n-2)\pi/n = 2\pi$, which gives $(n-2)(k-2) = 4$. The only solutions are $(n, k) = (3, 6), (4, 4), (6, 3)$. So there are exactly three regular tilings. They use equilateral triangles, six at each vertex; squares, four at each vertex; and regular hexagons, three at each vertex. In particular, regular pentagons do not tile the plane edge-to-edge, since $3 \cdot 3\pi/5 < 2\pi < 4 \cdot 3\pi/5$.