(3) Dionysian “Gasket”

Start with a solution to the Soddy equation, such as (9,4,25,76) or (2,2,-1,3) or (2,2,3,15) or (1, 4, 12, 1). Draw four circles with those curvatires on your plane, and call it a foursome. For each circle in each foursome, to two things: first of all, flip the other three cirles over that circle and add it to your plane. And second, draw the OTHER circle that is tangent the the other three circles.

This should be (so far unproven) a space filling set of non-crossing circles.

Simple 1-d approace

Given two kissing circles of radius 1/a and 1/b.
Flip b over a and the resulting cirle has radius

solve( (1/a + 2/b)*(1/a - 2/c) = 1/a^2 , c );
                                    2 a + b

And c = 2 a + b.

Given a circle of radius R and another of radius
r a distance d apart, flip r over r.

r' = R^2*r /(d^-r^2)

d' = R^2*d /(d^-r^2)