For a while now, I thought I should write up a blog post on the many different combinatorial objects which are in bijection with rhombus tilings of centrally symmetric polygons: various constructions with reduced words, oriented matroids, projections of hypercubes, strongly separated sets and so forth. But I kept putting it off because I knew it would take a long time to write correctly, with all the motivation and lots of figures it deserved.
Yesterday I had a very nice conversation about rhombus tilings with Lionel Levine, and I decided it was time to consolidate my knowledge and fill in the gaps. So I sat down and dumped everything I could think of into a question on MO. Note that this is a question and even a community wiki one — if you know of more results to add, please head over there and add them!
Reflecting on the sociology of mathematics, it seems to me that we are seeing a growth in ways to do a quick and sloppy job publishing something. Fifteen years ago, this would have been a survey article that would have taken weeks for me to research and edit. Five years ago, this would have been a blog post written over several days. Now I’ve written something much less polished, but I was able to do it in an evening in between taking care of my baby. I’m not sure whether it’s good or bad, but it seems to have been the only way I could get this written at all.