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Had another look at the tiling and detected some regularity. It seems to be a distorted Cairo tiling. If a paver just repeats the red and green groups (or the blue and yellow groups), there will never be a deadlock.
So the following question to #TilingTuesday remains: Are there weaker rules leading to less regular tilings but still deadlock-safe?
#TilingTuesday
Finally I have added a beautiful triangular/hexagonal tiling to the polyhedron-folding tool.
It is @ngons's tiling from this toot: https://mathstodon.xyz/@ngons/114636901870731820
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The fact that @DaniLaura 's turtles consist of Cairo tiles inspired me to also provide a basic Cairo tiling.
Try it
- with the provided backgrounds,
- on a cube or on the flattened star (using the "bending" slider).
Last week's #TilingTuesday contribution https://mathstodon.xyz/@DaniLaura/114658815682032097 by @DaniLaura had two kinds of symmetry centers: "four-heads points" and "four-feet points". Rotating around one of these points by 90° leaves the tiling unchanged, except for the coloring.
Thus the turtle tiling has the kind of rotational symmetry needed for cubes in my polyhedron-folding tool. So I added (simplified variants of) that tiling.
Notice that at the corners only 3 rather than 4 turtles meet.
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There's a new version of the star/polyhedron folding tool at
https://hcschuetz.github.io/polyhedron-star/dist/.
Changes:
- More shapes (some contributed by @GerardWestendorp)
- Better shape-development support:
- Various problems are now reported in the UI.
- The README now contains an explanation of the triangulation requirement for the "autobend" feature.
Thanks to Gerard for his patience with the old version. I hope that the improvements based on his experience simplify shape creation for everyone.
There's a new version of the star-folding app (https://hcschuetz.github.io/polyhedron-star/dist/). The major new feature is that now a few "classic" tilings are available. (@owenbiesel, our conversation motivated me to finally implement that.)
I'm not completely happy with the only triangular tiling that I have found. If someone knows other tiles, please let me know.
