Why does plotting Collatz sequences in polar coordinates produce a cardioid and nephroid?

Solution 1:

Update from OP

Based on the discussion, the cardioid can be generated based on the envelope of lines drawn between points around a circle, with each line drawn between point line between points n and 2n (mod N), as described here. The nephroid is based on the same concept for points between n and 3n, as described here. The Collatz sequence contains both behaviors: some elements in sequence follow $2n\to n$ (the order of the two doesn't matter if we are connecting lines), and some elements follow $n\to3n+1$ (but the $+1$ component is negligible for the overall shape), resulting in the cardioid and nephroid.