with doyle spiral you can get double spirals that cover the plane! the trick is to use a Mobius transformation to all the circles of the doyle spiral.

Möbius transformation - Wikipedia

easier to understand is the book "Indra's pearl" (the vision of Felix Kelin) by David Mumford. it is like a course with recipes to model shapes. it goes in details on the mobius transform to understand the concept (in addition to the formulas...)

so, I prepared a user defined feature for Mobius transformation.

z -> az+b / cz+d

it means that I work in the complex plane. I can "multiply" point following complex algebra rules: "i" for example has coordinate (0,1)... and in general z has coordinate (x,y).

I had to prepare a few "user defined function" to support the calculation in the complex plane to make Mobius work: Z1XZ2 1/Z NormZ Zbar... most important is Z1XZ2

with this set of functions, I could "squeeze" the doyle spiral to get a double spiral

1. create doyle spiral

2. transform all circles with mobius transform (z+1/Z-1) -> points (1,0 and (-1,0) are the attractors of the circles

• a Mobius transform keeps the circles and angles -> these are most important properties
• and center of input circles are not transform to center of the output circles
• If you replace circles with squares, it will not become squares.

3. make a sphere with all the result circle

here below is the result rendered in CATIA Live Rendering (raytracing)

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