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kitwallace

Fibonacci tiles

by kitwallace Mar 16, 2019
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It's quite easy to see the holes. Look at the notches in the outer edges. If you placed another tile anywhere, it would leave a hole there. You just needed to expand one more sequence.

See the added 5x5 tesselation - no holes

Yes, but you are using an expanded version of the tiles used in your first example. Try going larger with the exact same tile you used in the first example. By expanding the tile, you also expanded the failure threshold.

In both images, the tile is an order 3 Fibonacci tile - the snowflake version - there are several configurations depending on the rule and this version does tessellate without holes - I had misread the Wikipedia article. In fact I also missed one order, now added so this version is now order 4. I just found the paper https://arxiv.org/pdf/1212.1368.pdf and see what can be done with the other configurations, which do have holes.

Wow, its been 40 years since I have dealt with mathematics like that. Yes, there are configurations that tessellate without holes and others that do not.