Domain coloring is a popular way to visualize functions that map the complex plane to the complex plane (equivalently, the Riemann sphere to the Riemann sphere). The range space (the plane or sphere the function maps to) is given a patterned coloring and those colors are mapped back to the domain space (the plane or sphere the function maps from) by evaluating the function at every pixel in the domain. This of course involves evaluating the function a million times or so, but computers make it easy.
For the image above, the function (-64*((1/z)^3+1)^3/(((1/z)^3-8)^3*(1/z)^3)), a Belyi function for the tetrahedron, is interpreted as a Riemann-sphere to Riemann-sphere mapping. The range Riemann sphere was decorated with a map of the globe oriented in a particular way: South Pole at 0, North Pole at ∞ (on the Riemann sphere, ∞ is the point antipodal to 0,) and Null Island (shorthand for the point off the coast of Africa at 0° latitude, 0° longitude) is at 1 (on the Riemann sphere, 1 is halfway around from 0 to ∞.) The picture below shows the range sphere in the same projection used above.
Imagine the two polar projections to be hinged at 1, then imagine closing them together like a face-down book to form a two-disk map of the world.
Always keep in mind that domain coloring produces a view of the domain. In other words, the Belyi function begins with the hollow, non-physical, decorated sphere in the top picture, and stretches and folds it, in perfect registration, to cover the globe in the bottom picture. Pretty remarkable!
The dessin, in this case a tetrahedron, is found by tracing in the domain (top picture) all the pre-images of the line segment [0,1] in the range (bottom picture). This geographic domain coloring makes: vertices correspond to Antarctica's, faces correspond to Arctic Oceans, and edges to Antarctica-to-Antarctica sea voyages that pass between two copies of Africa.
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