This particular pattern sparked my interest recently. Lets do some dissection here.
Basically, each circle has multiples of six amount of dots: 6, 12, 18, … This creates almost equidistant spread of the dots. The task that I faced was to figure out what is the closest dot to a given point.
But why six? Well, it doesn’t have to be six. You can use any whole number, but spread of the dots would be the most equal for value six. The easy way to fill why is to look at the first circle. In case of six divisions, we a getting the hexagon. It means that all seven (including the centre) points are equidistant from each other and sitting on the verticies of equilateral triangles. Any other n-gon wouldn’t give us an equidistant spread of the dots.
Lets say we have a point coordinates as \(x\) and \(y\). Lets move it into polar coordinate space: \(r_1=\sqrt{x^2+y^2}\) and \(\theta_1=\left|tan^{-1}\left(\frac{y}{x}\right)\right|\). If distance between the dots is \(a\), then
$$ r_2=\text{round}\left(\frac{r_1}{a}\right)\text{ and }\theta_2=\text{round}\left(\frac{jr_2\theta_1}{2\pi}\right), $$where \(j\) is the number of rays, six in general case.
Moving it back into cartesian coordinates gives:
$$ \left(ar_2\cos\left(\frac{\theta_2\pi}{3r_2}\right)\text{sign}(x),ar_2\sin\left(\frac{\theta_2\pi}{3r_2}\right)\text{sign}(y)\right), $$where “sign” is the following funciton:
$$ \text{sign}(x)=\begin{cases}x\geqslant0, & 1\\x<0, & -1\end{cases}. $$While we at the topic of circular shaped distribution of the dots, I have to mention the Vogel Spiral dots. The dots are located on a spiral (Fermat’s Spiral) here, which could be described with this equation in polar form:
$$ r=a\sqrt{\frac{\theta}{\pi\left(3-\sqrt{5}\right)}} $$To get nice spread of the dots we need to take \(\theta\), such that it would be a multiplier of \(\pi\left(3-\sqrt{5}\right)\). Any other step in the angles will give us different distribution, as you can see on the following pictures.