Simon Gregg has interesting musings about and mathart with right triangles: http://seekecho.blogspot.fr/2015/02/i-wrote-last-year-about-how-pythagoras.html
That got me playing around again with right triangles. The fact that any right triangle can be decomposed into two similar right triangles is why the Pythagorean Theorem is true, so it's worth a deep look. As I was playing with it, by making a tool to make the one size smaller & one size larger similar triangle, it made the spiral naturally.
The extra tools here make:
rttri - an adjustable right triangle
tworight - the smaller and larger similar triangles
subright - the black line subdivision of a right triangle into two similar triangles.