A bit of research around infamous polka dot pattern.
This pattern is very common. What I am trying to find out is how dark or light the pattern is for a given dot radius and distance between them. It is clear that dots are arranged in equilateral triangles. Technically it means that we have rows of circles width horizontal step \(a\) and vertical step \(\frac{a\sqrt{3}}{2}\). Our tessellation area is rectangle with width equal to \(a\) and height \(a\sqrt{3}\). See the diagram.
You can drag the orange dot to change the radius of the circle. As long as circles are not colliding, i.e. \(r<\frac{a}{2}\) the area taken by circles could be calculated as \(2πr^2\). As soon as they overlap we can see 6 “lens” shapes. Total area of those shapes could be calculated using this formula:
$$l=12\left(r^{2}\arccos\left(\frac{a}{2r}\right)-\frac{a}{2}\sqrt{r^2-\frac{a^2}{4}}\right)$$The maximum value of the radius is \(\frac{a\sqrt{3}}{3}\). After this value we can’t see any gaps between circles.
Darkness rate of the pattern is simply a relation of dark area to the whole area and we can calculate it like so:
$$R=\frac{2πr^2-l}{a^2\sqrt{3}}$$Of course if circles do not overlap, \(l=0\).