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A note on these shapes for people who want to draw them from scratch... I found it quite difficult to draw a nice looking tiling using the basic monotile. It's

A note on these shapes for people who want to draw them from scratch... I found it quite difficult to draw a nice looking tiling using the basic monotile. It's quite hard to draw it entirely from the angles directly on the paper - not for a single tile, but extending any patch to a much larger one. Tracing and transferring the tile, on the other hand is easier to keep track of, but slight inaccuracies quickly propagate across the diagram. Since there is (by definition!) no larger-scale structure, there can be no guidelines to help correct this - unlike for repeated patterns. The way I found was best was just to print out a patch of tiles, and then trace that. I suppose that using a metatile approach might also work, although doing that by hand is a little daunting.

fanf2 2y ago
The Hat lives on a hexagonal grid which ought to help you correct inaccuracies.
- gilleain 2y ago
  Well the metatiles are hexagonal - does that literally mean using an underlying hex grid would work? I'll certainly try that, thanks!
  - fanf2 2y ago
    You can see the underlying grid in this image https://cs.uwaterloo.ca/~csk/hat/examples/hats.png https://cs.uwaterloo.ca/~csk/hat/examples/hats.png from this page https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/
bazzargh 2y ago
I came up with some L systems for drawing hats and spectres based on the metatiles eg https://trinket.io/python/274cc18bd5 https://trinket.io/python/274cc18bd5 https://trinket.io/python/108e28ab75 https://trinket.io/python/108e28ab75 (the second program has a bunch of variant versions) this cheats a little: it turns out you always want to draw the H8 metatile (which will overlap) instead of keeping track of the zoo of metatiles. And the flipped hat is never drawn; it appears as a hole in the plot. Similar things happen with the spectre. This was all done so I could make a program small enough to fit in a toot to generate patches using the bbc microbot: https://mastodon.me.uk/@bbcmicrobot/110295593252210445 https://mastodon.me.uk/@bbcmicrobot/110295593252210445 (also for hand drawing hats, use a hexagonal grid under your tracing paper, since it's just an assembly of kites)
- gilleain 2y ago
  Thanks, I'll try the L-system. I did try drawing on triangulated grid paper (dual to hex grid), which helped a little.
- fanf2 2y ago
  Very nice! Have you seen Simon Tatham’s articles on Penrose tiles, Hats, and Spectres? https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-tilings/ https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi... https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-spectre/ https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...

fanf2 2y ago

The Hat lives on a hexagonal grid which ought to help you correct inaccuracies.

gilleain 2y ago
Well the metatiles are hexagonal - does that literally mean using an underlying hex grid would work? I'll certainly try that, thanks!
- fanf2 2y ago
  You can see the underlying grid in this image https://cs.uwaterloo.ca/~csk/hat/examples/hats.png https://cs.uwaterloo.ca/~csk/hat/examples/hats.png from this page https://cs.uwaterloo.ca/~csk/hat/ https://cs.uwaterloo.ca/~csk/hat/

bazzargh 2y ago

I came up with some L systems for drawing hats and spectres based on the metatiles eg https://trinket.io/python/274cc18bd5 https://trinket.io/python/274cc18bd5 https://trinket.io/python/108e28ab75 https://trinket.io/python/108e28ab75 (the second program has a bunch of variant versions) this cheats a little: it turns out you always want to draw the H8 metatile (which will overlap) instead of keeping track of the zoo of metatiles. And the flipped hat is never drawn; it appears as a hole in the plot. Similar things happen with the spectre. This was all done so I could make a program small enough to fit in a toot to generate patches using the bbc microbot: https://mastodon.me.uk/@bbcmicrobot/110295593252210445 https://mastodon.me.uk/@bbcmicrobot/110295593252210445 (also for hand drawing hats, use a hexagonal grid under your tracing paper, since it's just an assembly of kites)

gilleain 2y ago
Thanks, I'll try the L-system. I did try drawing on triangulated grid paper (dual to hex grid), which helped a little.
fanf2 2y ago
Very nice! Have you seen Simon Tatham’s articles on Penrose tiles, Hats, and Spectres? https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-tilings/ https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi... https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperiodic-spectre/ https://www.chiark.greenend.org.uk/~sgtatham/quasiblog/aperi...