Mathematical Tiling and Tessellation 
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Miki Imura
 April 13 at 1:07 AM Shared with Public group
Logarithmic Spiral Tilings are Periodic Tilings
You’ve probably seen tilings based on logarithmic spirals using similar tiles. I’ve also posted a few examples myself in the past. However, I haven’t explored them very deeply, mainly because they are essentially equivalent to periodic tilings with period 2π. To make this more intuitive, I created an animation to illustrate the idea.
What is happening here is essentially a change of coordinates. The animation interpolates between plotting points in polar coordinates (log r, θ) and plotting them in Cartesian coordinates (r cos θ, r sin θ).
Changing θ corresponds to changing the argument (i.e., rotation), and changing log r corresponds to scaling. As a result, a translation in the original coordinate system corresponds to a combination of rotation and scaling in the transformed one. Note that if we rotate shapes in the original coordinate system, they do not necessarily remain similar after the transformation.
In that sense, logarithmic spiral tilings are essentially equivalent to periodic tilings (p1). For this reason, I tend to view them less as objects for deep mathematical investigation and more as a natural playground for motif-based work, in the spirit of Escher.
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Mark Bean
That's really a helpful animation! I am reminded of a really cool video I watched recently. It's long but I got a lot out of it. https://www.youtube.com/watch?v=ldxFjLJ3rVY
Escher's most mathematically interesting piece
YOUTUBE.COM
Escher's most mathematically interesting piece

Escher's most mathematically interesting piece

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