What is the "Thurston Model" of the Mandelbrot set

Question

I found this image on the wikipedia for the Mandelbrot set: It's referred to as the "Thurston model of the Mandelbrot set," and no explanation is given. What is it?

Answer 1

A quite similar picture appears on page 168 of The Beauty of Fractals. I couldn't find the reference given in the caption. It's probably a credit to the author of the picture. For a brief explanation, see the relevant paragraph on page 167 given below.

page 168

page 167

Answer 2

Called the Duoady-Thurston Farey Rational model of the Mandelbrot set as the rational phase locking elements of the set. The Douady picture is in the Beauty of Fractals as stated above. The Poincare disk model given is an Hyperbolic 3 manifold from the R. Brooks and J. Matelski paper of 1978 that predates the Mandelbrot discovery of the set where they use a PSL(2,C) Kleinian group hyperbolic model for the Mandelbrot set. A field lines or curvature lines model also follows these sorts of rational curves. Since the area of the Plateau of the Mandelbrot set is finite, it has been conjecturesd that the Mandelbrot set is a finite hyperbolic 3 manifold of fixed volume.