The Greater Toronto Area has a boatload of people and very very few nearby rock climbing areas. So when a few local climbers developed The Turtle, a new crag on private land, there was certain to be heavy use and a high risk of a pissed-off land owner closing the crag. Developer Gus Alexandropoulos thought a couple signs around the crag would do nicely to encourage good behavior, so in the name of good crag karma I offered to make them.

A tiling a pattern of one or more geometric shapes that can be repeated to cover completely an arbitrariliy large 2D area. There are all sorts of set of shapes that can be periodically tiled - they can be quite beautiful and mathematically interesting.

There’s an active community of both serious mathematicians and regular laypeople that try to discover new and interesting tilings. You start with a constraint, say your set contains a square and a hexagon, and you try to work out what different patterns can be generated from that set. OK what if instead of squares you have rectangles? And what if the hexagon doesn’t have to be convex? And are you sure you’ve found all the tilings that come from that set? Can you prove it? And so on and so forth - it’s an endless treasure hunt. One hunting ground that has produced really cool patterns is that of pentagons. The wikipedia page on pentagonal tiling is a whopping 3500 words and has all sorts of strange and unexpected patterns. Poking around on this page led me to [this paper](/Hirshorn paper.pdf) that found all equilateral convex pentagons that tile the plane. As a little bonus, they discovered one really interesting pentagon, with 5 equal sides of equal lengths and angles of 60, 160, 80, 100, and 140 degrees, that can be used to produce at least two different tilings with a beautiful rotational symmetry. They dubbed it the ‘versatile.’ Rimshot.