Julia Set on the Riemann Sphere: 3D Globe with Stereographic Projection and Orbit Hopping

Setting up the gallery...

Presets

Live readouts

Graphs

Orbit of the probe point: |z_n| against n (log scale); a bounded orbit stays under the escape radius

Orbit of the probe point in the complex plane (blue), fixed points alpha (green) and beta (red)

Histogram of escape iteration counts over a 4 by 3 window (log scale); the spike at the right edge is the filled Julia set

Box counting of the Julia boundary: log10 N(boxes) against log10 boxes per side; the slope is the fractal dimension

Fixed-point multiplier |2 alpha| against Re c (Im c fixed): alpha attracts where it is below 1, inside the main cardioid

Numerical checks

running...

How to use

Use the sliders or presets to change c; click the globe to drop the probe point and watch its orbit hop along the sphere. Drag orbits the camera, wheel zooms; the north pole is the point at infinity. Try: (1) the Douady rabbit preset, then switch colouring to Equipotential bands; (2) Animate c around a circle with r = 0.7885 and watch the set break from connected to dust when c leaves the Mandelbrot set; (3) the Cauliflower preset to see the parabolic point (multiplier 1 on the last graph); (4) raise Power d to 5 and count the symmetry; (5) drop a probe on the unit circle at c = 0 and see |z_n| stay at 1. Space pauses the animation, R resets the view.

The gallery, plinth, brass meridian ring, cradle, plaque and lamps were modelled in Blender (baked PBR textures); the fractal itself is computed per pixel on the GPU.

The science

The iteration. For a complex number c the quadratic map is f(z) = z^2 + c (generally z^d + c). The filled Julia set K_c is the set of starting points whose orbit z, f(z), f(f(z)), ... stays bounded; the Julia set J_c is the boundary of K_c, the set where the orbit is chaotic and arbitrarily close starting points end up far apart. Orbits that leave the disc |z| > R with R = max(2, ...) can never return, because there |f(z)| >= |z|^d - |c| > |z|. That is why the picture colours each pixel by the iteration count at which its orbit crosses R.

Smooth colouring. Plain iteration counts give hard bands. The page uses the continuous count nu = n + 1 - ln( ln|z_n| / ln R ) / ln d which is continuous across the bands because ln|z| grows by the factor d at every step.

Green function (equipotentials). The potential G_c(z) = lim d^-n ln|z_n|, G_c(f(z)) = d G_c(z), G_c = 0 exactly on K_c measures how fast an orbit escapes. The "Equipotential bands" mode shades log2 G: each band is mapped by f onto the next band outwards, so the picture is a ruler of the dynamics. The page checks the functional equation numerically.

Fixed points and multipliers. For d = 2 the fixed points solve z^2 - z + c = 0: z = ( 1 +- sqrt(1 - 4c) ) / 2, lambda = f'(z) = 2 z A fixed point attracts if |lambda| < 1, repels if |lambda| > 1. The point alpha = (1 - sqrt(1-4c))/2 is attracting exactly when c lies in the main cardioid of the Mandelbrot set (|1 - sqrt(1-4c)| < 1); at the cusp c = 1/4 the multiplier is 1 (parabolic, the cauliflower), and for c on the boundary of the cardioid with an irrational rotation number the fixed point is the centre of a Siegel disc. Each attracting cycle captures the critical orbit (of 0), which is why the Mandelbrot set - the set of c whose critical orbit stays bounded - sorts all Julia sets: J_c is connected exactly when c is in the Mandelbrot set, otherwise it is a Cantor dust. The closed forms for the cardioid and the period-2 bulb, q (q + x - 1/4) < y^2/4 with q = (x-1/4)^2 + y^2 and (x+1)^2 + y^2 < 1/16, are checked against brute-force iteration.

Dimension. The box-counting dimension is the slope of ln N(eps) against ln(1/eps). For c = 0 the Julia set is the unit circle (dimension 1). For small |c| Ruelle's expansion gives the Hausdorff dimension D = 1 + |c|^2 / (4 ln 2) + ...; for large boundaries (dendrites, rabbits) the box dimension from a finite pixel mask is only a rough estimate.

Special cases used as checks. c = 0: the filled set is the unit disc, area pi. c = -2: with z = 2 cos(theta) the map is the angle doubling theta -> 2 theta (Chebyshev conjugacy), so z_n = 2 cos(2^n theta) and the Julia set is the interval [-2, 2].

Honest simplifications. (1) Escape-time images use finite precision (double); beyond zoom x1e13 pixels merge. (2) The maximum iteration count cuts off orbits that need more steps than allowed (near the boundary), so "inside" is a statement about maxIt steps. (3) Connectedness of a Julia set for d > 2 uses the orbit of z = 0, which is only one of the d-1 critical points of z^d + c (all are 0 for this family, so it is exact here). (4) The Ruelle estimate holds only for small |c|.

Checks run in the page. (a) area of the c = 0 filled set against pi; (b) the Chebyshev conjugacy at c = -2 after 18 doublings; (c) the closed-form fixed point residual and the measured contraction rate against the multiplier; (d) the cardioid and bulb formulas against brute force; (e) the box-count slope of the unit circle; (f) the Green-function equation G(f(z)) = 2 G(z) and G = ln|z| at c = 0; (g) stereographic projection round trip and chordal distance.