Composing the first frame…
Math × Shader No.01 · Live plates
Complex analysis
Five plates on the geometry of complex functions, each one computed pixel by pixel in the browser as you watch: the colours of a function across the plane, the sphere whose motions give the Möbius transformations, the hyperbolic plane tiled by a kaleidoscope with one curved mirror, the Julia sets of z2+c with the Mandelbrot set that charts them, and the circles within circles that four inversions leave behind.
Plate I · 4 movements · 1:00 loop
Every point z is painted with the value f(z). The hue is its direction, argf: gold where f(z) is a positive number and ice-blue where it is negative. The rings are curves on which |f| is constant, each a fixed factor larger than the last, and the rays curves on which argf is constant, each a fixed angle round from the last; the two steps are matched, so between them the plane is cut into cells that are nearly square. The rings drift the way |f| grows and the rays the way argf grows. The hue itself stays put.
A function with a complex derivative, an analytic function, keeps angles wherever that derivative is not zero. The rings and rays are the level lines of the two parts of logf=log|f|+iargf, so wherever f is neither zero nor infinite they meet at right angles and, drawn at matching steps, are spaced alike (the Cauchy–Riemann equations): each small cell stays nearly square however hard the picture bends. Where the cells crowd together, the derivative of logf is large.
Walk once anticlockwise round a closed loop that meets no zero or pole, with nothing worse than poles inside, and count how many times the colours go round, turns the other way counting as negative: that is the number of zeros inside minus the number of poles, a double zero or pole counting twice. It is the argument principle, and the first plate shows it at a glance.
Add a single point, ∞, and the plane closes up into a sphere. Stereographic projection keeps angles and sends circles to circles or lines, so it is the natural way to see the whole plane at once: the origin at the bottom of the sphere, ∞ at the top.
In the Poincaré disk the straight lines of hyperbolic geometry are diameters and arcs of circles meeting the rim at right angles. Angles are drawn true and lengths are not: the rim is infinitely far from the centre, so the triangles of a tiling, all of one size, shrink on the page as they go out.
A Julia set of z2+c is connected exactly when the orbit of 0, the one point where the map folds the plane, stays bounded, and otherwise it is dust (Fatou and Julia). The Mandelbrot set is the set of those c, which is why it works as a chart of every connected Julia set at once.
Apply every map of a group to a point and look at where the images pile up: that is the limit set. For the tilings of the third plate it is the whole rim of the disk; for the four inversions of the last plate it is the Apollonian gasket, and when the mirrors shrink so that no two touch it is a Cantor set.
A Möbius transformation z↦(az+b)/(cz+d), other than the identity, fixes one or two points of the sphere. With two, it is elliptic, hyperbolic or loxodromic as the other points circle round them, stream from one to the other, or spiral from one to the other; with one, it is parabolic.
Every Möbius transformation arises in the way the second plate shows: project the plane onto a sphere, move the sphere rigidly, keeping its top above the plane, and project it back (Arnold and Rogness, “Möbius Transformations Revealed”, Notices of the AMS, 2008).