Mathematical Tiling and Tessellation | **Spirals in Modulo Krinkle Tiling — A Postscript Observation**

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Miki Imura
 June 19 Shared with Public group
Spirals in Modulo Krinkle Tiling — A Postscript Observation
In the paper, I focused on the construction and validation of the tiling system. While I didn’t cover the spirals, I’ve naturally made a number of observations — here’s one I’d like to share.
The number of visible spirals is closely related to m⁻¹ — the inverse of m modulo k. In particular, in the offset-free case, we observe:
  • t·m⁻¹ counterclockwise spirals
  • t·(k − m⁻¹) clockwise spirals
If m⁻¹ is small, the counterclockwise spirals stand out more; if m⁻¹ is large, the clockwise ones do. The reason lies in how many edges adjacent tiles across wedges share: the more edges two tiles share, the more we tend to perceive a visual continuation — while tiles that share only a few edges appear less coherent.
The following image shows an example with (m, k, t) = (4, 11, 2). Here, m⁻¹ = 3, since 4·3 ≡ 1 (mod 11). This gives us 2·3 = 6 counterclockwise spirals and 16 clockwise spirals. Since 3 < 11/2, the counterclockwise spirals tend to stand out more clearly.
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David DeBrota
I love the name krinkle tiling
Victor Borun
a psychological thing. to show that as soon as an observer manages to find a way to make the sense of a scene they tend to block other tries.
in this case they never get to see the other spiral arms…
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May be an image of trumpet and chess
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Samo Dernovšek
could you draw out some of the other spirals, my brain can only see one 😃
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