The inspiration for this visualisation comes from a puzzle called "Rhombus Tiling" which I first discovered in Peter Winker's book "Mathematical Puzzles: A connoisseur's Collection". The puzzle goes like this:
Form n choose 2 different Rhombi from the pairs of non-parallel sides of a regular 2n-gon, then tile the 2n-gon with translations of the rhombi. Prove you use each different rhombus exactly once!
In simpler terms with diagrams:
- take an even sided shape (square, hexagon, octagon, etc)
- colour each edge differently
- form a rhombus from each pair of differently coloured edges
- completely fill the shape with these rhombi
- you'll find you've used each shape exactly once
This is true for any (even) number of edges, for example:
I can recommend spending some time trying to solve this for yourself, so won't give the answer away. I will include a hint or two below though.
Once I'd solve the puzzle, I was really interested in seeing what such a random rhombus tiling might look like for shapes larger than the best decagon I could draw by hand. Hence why I built this tool.
More details and a web version can be found at Observable:rhombus-tiling-visualiser.
![regular_large_tiling](blob:https://davidquick.static.observableusercontent.com/0314e808-948b-4f88-a5bd-846bd9912d43