I'm doing some research in visualizing arithmetic sets (resp. properties, resp. sequences of integers). I try to create patterns (in which I hope to observe some symmetries) by injectively mapping $\mathbb{N}$ on $\mathbb{Z}\times \mathbb{Z}$ in different ways and highlighting the numbers that have a given property (belong to a given set, are in a given sequence).
You can find a little tool with which I'm playing around here.
These are three pictures I found when highlighting square, triangular and pentagonal numbers on a spiral (like Ulam's):
and I want to ask these questions:
How can it be explained in simple terms that the tree of square numbers (originating at the center) has two obvious straight branches, the tree of triangle numbers has three rather easily detectable, and the tree of pentagonal numbers has five hard to detect curved branches? Will this go on and on for arbitrary polygons? Why does the tree of square numbers not have four branches?
This is taken from my answer to your related question. The following picture from it isn't exactly your three armed spiral, but I'll venture that if you expand your central picture you will see $17$ spirals.