It’s easy to see that a plane can be tiled with squares or hexagons arranged in regular ranks, but in 1936 Heinz Voderberg showed that it can also be tiled in a spiral formation. Each tile in the figure above is the same nine-sided shape, but together they form two “arms” that bound one another. If both arms are extended infinitely, they’ll cover the whole plane.
In 1955 Michael Goldberg showed that spirals might be devised with any even number of arms, and in 2000 Daniel Stock and Brian Wichmann did the same for odd numbers, so it’s now possible to devise a shape that will tile the plane in a spiral with any specified number of arms.
(Daniel L. Stock and Brian A. Wichmann, “Odd Spiral Tilings,” Mathematics Magazine 73:5 [December 2000], 339-346.)