I've got a hobby of making fractal patterns on Desmos, and a few years ago I saw this pattern and wanted to try and make it: Fractal of nested Circles
Basically, you start with two circles that are sitting on some vertical line and touching each other. There is a gap underneath them so you add the biggest circle that will fit in that space. This creates two more gaps so you add more circles and repeat the process recursively. In working on this, I realized there was a relationship between the radii of the circles. If your first two have radii $R_1$ and $R_2$, then the radius of the circle tucked under them will look like: $$\frac{1}{\sqrt{R_3}} = \frac{1}{\sqrt{R_1}} + \frac{1}{\sqrt{R_2}}$$
Or, if we define $S = \frac{1}{\sqrt{R}}$, this is just: $$S_3 = S_1 + S_2$$
Now, since we're doing this recursively, you might think this is just the Fibonacci Sequence, but it's not quite. Take a look the fractal again. We don't just take the "last two" in the sequence to get the next one. If circle $C_3$ is tucked under circles $C_1$ and $C_2$ ($S_3=S_1+S_2$), one new circle get's tucked under $C_2$ and $C_3$ ($S_{4,2}=S_2+S_3$), but there's also a new circle tucked under $C_1$ and $C_2$ ($S_{4,1}=S_1+S_3$). And so this recursive sum works the same way. Instead of leaving numbers behind, you're constantly coming back to old numbers to make new ones. It works more like a branching pattern than a linear sequence.
This numeric pattern is really what I'm asking about. Has anyone seen a pattern like this somewhere in math? I wrote some code in Mathematica to visualize the pattern starting with the points (1,0) and (0,1). This is the plot that came out if it: Branching Recursive Sum Plot
Notice how the points seem to be arranged in radial lines pointing out from the origin. Maybe this is similar to how the ratio of elements in the Fibonacci Sequence approaches the golden ratio. I've been fascinated by this pattern for a while now and would love to hear if anyone knows anything about it.
