Which shape has the most sides: a pentagon, a hexagon or an octagon?

Eight beats six beats five, and the names say so: the Greek numerals penta, hexa and octa with -gon, meaning angle. Once you know that, most of the family reads itself - a myriagon has ten thousand sides, and a chiliagon a thousand.

The hexagon is the one worth pausing on. It is the only one of the three that tiles a flat surface with no gaps, and of all the shapes that do tile - triangle, square, hexagon - it encloses the most area for the least perimeter. That is why bees build hexagonal cells: less wax for more honey. Basalt columns like the Giant's Causeway crack into hexagons for a related reason as the rock contracts.

The chiliagon has a job in philosophy. Descartes used it to separate imagining from understanding: you cannot picture a thousand-sided figure, and certainly cannot distinguish your mental image of one from a 999-sided figure, yet you can reason about its properties perfectly well. Understanding, he argued, is therefore not the same faculty as imagination.