Tiling the whole plane

So far we have tiled a GIVEN region. What about tiling the entire plane — a surface with no edges, going on for ever in every direction? Can you think of a shape whose copies would do it?

Clearly, SQUARES can. Lay them in rows and columns and they go on for ever, with no gaps and no overlaps.

Are there other REGULAR polygons that can? Equilateral triangles can — and that is no surprise, since six of them fit exactly round a point, as we found when building the hexagon.

And a plane can be tiled using regular HEXAGONS as well. So among the regular polygons, at least these three work. (Ask yourself why a regular pentagon does not: what happens to the angles around a point?)

A plane can also be tiled using more than one shape, and using non-regular polygons. The great Dutch artist Escher, who lived from 1898 to 1972 and whose works explored mathematical themes such as tiling, came up with creative ways of tiling a plane with ANIMAL shapes — birds and fish and lizards that interlock perfectly.

This is not a finished subject. Mathematicians are still exploring various ways of tiling the plane, and one of the tilings shown in the book was found as recently as 2023. A topic that looks like decoration turns out to have open questions in it.

Tilings are all around you — in buildings and designs, and in nature too. The front face of bee hives and some wasp nests are tiled with HEXAGONAL cells, used to keep eggs, larvae and pupae safe and to store food. Because the region is tiled, no space is wasted at all. Scientists still wonder how bees and wasps manage to make them hexagonal.