I tend to call the hypotenuses ‘spokes’ and each time the next spoke crosses the positive x axis (red line in figure below) I call that a new ‘winding’.

Now if we look at number of additional spokes in each new winding,
for x up to 10,000, we get the following data.

As we can see the number of additional spokes is either 18, 19, 20, 21 with 20 being the most and 18 being relatively rare. Going further to, for example, the first 100,655 windings the additional number of spokes is 18, 179 times, 19, 3612 times, 20, 4928 times and 21, 3612 times.

It is possible to plot this pattern in different ways:

First 100 Windings

First 1000 Windings

Pattern Art