The Einstein hat does repeat periodically… on a cylinder!?

Months ago you may have seen the new shape that does not periodically tile (repeat) like most other shapes, in the plane. However, that is assuming an infinite (x,y) plane… What if instead of that geometry, we try building out a periodic cylindrical tiling?

To see this pattern for yourself you can go to the Ein-stein hat tile generator they have at https://cs.uwaterloo.ca/~csk/hat/app.html and generate some hats – here is the one I used (which you can right click and save):

Print this out on a standard sheet of paper and cut around the edges to experiment with some non-planar geometries: Bend and rework the pattern into something that matches the pattern above it until you get it to a cylindrical pattern like so:

What do you think – does this count as a periodic tiling? Have you found any other shapes or cases where the pattern repeats itself? Let’s hear it in the comments:

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