Mathematical Tiling and Tessellation | **It looks like I’ve received a Group Expert badge in this group — thanks

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Mathematical Tiling and Tessellation 
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Miki Imura
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It looks like I’ve received a Group Expert badge in this group — thanks! Going forward, I’d like to share not only my original patterns, but also some general tricks and insights into tiling.
Let’s begin with a simple concept: edge modification.
As a simple observation, if we mark some (or all) edges of the prototiles so that marked edges match in the tiling, we’re free to replace those edges with any point-symmetric curve.
Furthermore, if we mark edges in such a way that their orientations also match, then we can replace them with any curve we like, as long as the resulting shapes are valid (e.g., not self-intersecting).
I think quite a few posts in this group can be understood with this idea in mind. Hope it helps you make better sense of them!
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Erlend Robaye
It's possible to make self-intersecting tiles:
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Timothey Lukinov
I wonder which other symmetries except of point (rotational symmetry) could be used. Apparently in 2d it is line (mirror symmetry) and their combination. Anything i missed?
Cody Ramseur
My mind wants there to be like a calculus for tiling that can bridge the gap between periodic tiling and periodic fractals. New math? Maybe there is no actual gap and this is just an imagined thing in my mind.
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