Cosh Mandelbrot

Cosh Mandelbrot leaves the familiar polynomial step behind. Instead of repeating z² + c, it repeats z → cosh(z) + c. Squaring is replaced by hyperbolic cosine's exponential growth along the real axis and 2π repetition along the imaginary axis.

Complex cosine and hyperbolic cosine are close relatives: each repeats along one direction and grows without limit along another. For cosh(z), the imaginary-axis period is 2π, so a detail band returns above and below itself. When |Re(z)| is large, its exponential terms race outward. Most starting points escape quickly; only slim bands hold an orbit — keeping its successive values bounded for many steps.

The parameter plane no longer resembles the Mandelbrot set’s cardioid and bulbs. It stacks fans, gilt plumes, and vertical echoes instead. Its line between escape and non-escape is drawn by the transcendental iteration itself; no simple algebraic escape radius is available.

In 1988, Clifford A. Pickover, a computer scientist and author at IBM's Thomas J. Watson Research Center, published the short paper "Chaotic behavior of the transcendental mapping (Z → cosh(Z) + μ)" in The Visual Computer. He presented images of chaotic behavior produced by this transcendental function, alongside maps such as z → sin(z) + c and z → eᶻ + c.

Pickover’s work brought the cosh-based map into the fractal-rendering community, where it became a standard transcendental escape-time example. Earlier, mathematicians including Robert Devaney and Michał Krych had developed the broader study of transcendental complex dynamics—exponential, trigonometric, and hyperbolic families—and analyzed the dynamics of exp(z) in the 1980s.

First notice the vertical repetition. Because cosh(z) repeats in the imaginary direction every 2π, fine plumes, layered fans, and gilt filaments return at regular intervals. The result is striped and tiered, far from the radial branching of polynomial Mandelbrot sets.

In the canonical rendering, plumes and fans often carry warm tones. The finest detail concentrates in narrow horizontal bands, where an orbit lingers near the boundary between escape and non-escape. Above and below, escape accelerates and the image becomes smoother.