Thread by @FogleBird

Each pattern is similar to the one before it.

How does it work? Start with an infinite grid of filled circles. In the first image, their diameters are 0.9, so they don't touch.

In the second image, their diameters are 1.25, so they overlap with their neighbors. Overlapping regions switch color. (The even / odd fill rule.)

As the diameter increases further, there are more and more interactions between circles and interesting patterns appear.

Now it's time to REALLY nerd out. This plot shows the # of white pixels as a function of the circle diameters.

Anything interesting about those local minima / maxima?

I naively assumed the first peak was at exactly 1, right as the circles touch. After that point, we start getting more black, but we ALSO get more white as the circles grow. It's actually more like 1.0818, shown here.

After that is a local minima around 1.6288...

Those are suspiciously close to 2/5 * e and 3/5 * e, but I'm probably making things up. It'd be interesting to come up with analytical solutions to these values though...

So here are the first 9 local maxima/minima. Not sure if there's anything special about these, but there ya go!

To wrap up, here's an interactive JS / SVG version that I just now hacked up. Drag the slider at the bottom to change the diameter of the circles.

michaelfogleman.com/static/circles/

For the dedicated nerds, here's a CSV with the data up to diameter 10 in steps of 0.001. Columns are diameter, coverage (as a percentage).

michaelfogleman.com/static/circles…

So you can think about it as what percentage of the plane is covered by an even number of circles, and what percentage is covered by an odd number.

coverageEven = 1 - coverageOdd

I think these both approach 0.5 as the diameter goes to infinity, but not sure. Someone do a proof.

See what happens when you just run with something? All I did was start rendering some damn circles and look where I ended up. Certainly didn't see this end result when I started.

Spotted in a restaurant.