Trigonometric Tapestry

Nesting one trigonometric function inside another, as in \sin(\cos z), gives a function of a complex variable with a surprisingly rich structure: zeros and poles arranged in rows, and essential singularities where an inner function has a pole. Adding several such functions together mixes their patterns into a single field.

Here f(z) = \sum_{n=0}^{5} e^{in\pi/3}\, g_n(z), where g_n runs through \sin(\cos z), \cos(\sin z), \tan(\cot z), \cot(\tan z), \sec(\csc z) and \csc(\sec z), and each term is turned by a sixth root of unity. Every pixel z = x + yi is colored by (1 - M)\,z + M f(z): hue shows the argument and lightness the modulus, so zeros and poles appear as points where all the colors meet. As M animates from 0 to 1, the plain color wheel of the identity map morphs into the full sum, which repeats with period 2\pi along the real axis. The string of tiny features on the real axis comes from \tan(\cot z) and its neighbors, which have an essential singularity at every multiple of \pi.

After a Desmos graph by DragonFire28.