Sébastien Labbé

In September 2025 in Montreal, Peter Selinger gave a talk at LaCIM in Montreal (where I am based this year) about the hex game. After lunch, I ask him to play against him. Since I needed to leave in about 30 minutes to give a colloquim at CRM at Université de Montréal, I thought playing againt Peter should not take too long as I would lose fast.

We ended up never even starting the hex game.

Peter was also short in time because he was also giving a second seminar the same day in the afternoon this time at McGill University.

Peter asked me what was my talk about. I said aperiodic tilings associated to the metallic mean, etc. Then, he showed me a picture he recently made. Right away, I recognized the partition of the internal space described by Baake, Gähler and Sadun for tilings by the meta-tiles T, H, P and F following the original terminology Smith, Myers, Kaplan and Goodman-Strauss. I was wondering since then if we could use their partition to describe the hat tilings the same way I did it for Jeandel-Rao tilings. So, I asked Peter how he got the picture. Then he told me exactly what I was hoping for. Peter's construction is marvelous!

With the semester Illustration as a mathematical research technique currently taking place this winter at Institut Henri Poincaré, Paris, I thought it would be a good occasion to share Peter's discovery. Together, we have been preparing a preprint presenting a construction of the hat tilings by a Markov partition following the approach I used in my work on Jeandel-Rao tilings. The preprint is not ready yet to be posted on arxiv, but we publicly share our draft today, March 26th, 2026.

An important aspect of the construction is the choice made by Peter for the placement of the anchors for each tile. Anchors are widely and commonly called control point in the community, but since Peter is an expert on Bezier curves, the terminology of control point did not make sense for him. And maybe he has a (control?) point.

The partition discovered by Peter can be seen as a extension of what happens with Jeandel Rao tilings that I now explain visually with a DIY laser cut puzzle on top of a partition.

Additional information and documents (slides, the partition, the grid, etc) will be posted here later. The talk at IHP on March 26 was recorded and will be available on their website.