The many disguises of rhombus tilings

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.

2 thoughts on “The many disguises of rhombus tilings

  1. It’s kind of interesting, if bittersweet, that this blog has become so quiet since the creation of mO. (And similarly, the n-category Cafe since the creation of the nLab.)

  2. There are lots of reasons the nCafe is getting quieter. The formation of the nLab drew away the energy of Urs Schreiber and others who want their online writings to add up to a kind of reference work or research project. Now they’ve started a journal with refereed papers, Proceedings of the nLab. I’ve never gotten too involved in any of that; instead, I’ve gotten interested in topics that don’t fit comfortably on the nCafe. And David Corfield has gotten quite busy lately. It’s sort of sad, a bit like how rock groups break up as the members pursue their own projects. But I’d say a lot of the main goals of the nCafe have been accomplished now, or at least brought to the point where chatting about them on a blog is not the best thing to do.

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