Logarithmic spiral tilings with similar shapes
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Mathematical Tiling and Tessellation 
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Miki Imura
 November 4 Shared with Public group
An Infinite Merry-Go-Round!
This time, I want to briefly talk about logarithmic spirals — or more generally, spiral tilings using similar shapes.
A logarithmic spiral is a curve defined by the equation r = a·exp(kθ) (you’ll find it on Wikipedia). That might sound a bit intimidating, but in simple terms: whenever the angle increases by a fixed amount, the radius gets multiplied by a fixed factor. This kind of multiplicative scaling makes logarithmic spirals a great fit for similar shapes.
So I’ve been thinking — how can we formulate tilings that follow this principle?
Let’s fix a complex number α such that |α| ∉ {0, 1}. This α determines both the rotation and scaling factor between adjacent tiles. Now define a relation z ~ z′⇔ ∃n ∈ ℤ s.t. z = αⁿ·z′. This is an equivalence relation on the complex numbers, so we can partition ℂ∖{0} into equivalence classes.
If we take one representative from each class and gather them into a set S, then all sets αⁿ·S (for n ∈ ℤ) are similar to S, mutually disjoint, and together they cover the entire complex plane, except for the origin. In other words: there’s no overlap, no gap — it’s a tiling!
There’s a lot of freedom in how to choose these representatives. But when designing actual tile shapes, it helps to start with a simple reference shape and deform it.
So here’s what I came up with. I called it a merry-go-round earlier in this post, but in fact it was inspired by Akabeko — a traditional Japanese toy. Hope you enjoy it!
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Matthew J. Scholtes
My first thought was “it’s turtles, all the way down”. 🐢 🤭
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Joseph Matthias Young
Your posts are the BEST!!!