** Complex Numbers**

The use of complex numbers comes close to being a necessity in the formulation of the laws of quantum mechanics. Yet complex numbers are far from natural or simple, and they cannot be suggested by physical observations. ~ Eugene Wigner

If reality is a mathematical construct, its mechanics are not quite hiding in plain sight. Complex numbers illustrate.

A complex number is itself 2-dimensional. Complex numbers are ubiquitous for characterizing dynamics in physics, chemistry, and biology.

Construed through complex numbers, fractals are patterns which exhibit self-similarity at different scales. The numbers involved are extra-dimensional: fractional complex numbers.

Fractals are dimensionally discordant. ~ Polish-born French American mathematician Benoît B. Mandelbrot

Many aspects of Nature are fractal. Fluid turbulence forms fractal patterns. The patterns that tree leaves and branches exhibit are fractal. Romanesco broccoli (pictured) are brazenly fractal.

Fractals occur in time as well as space. The brilliant, distinctive symphonies of Austrian composer Gustav Mahler were temporally fractal, with self-similar themes recurring throughout. These were not instances of exploratory variations, which are common in composition (e.g., Bach). Mahler instead ingeniously crafted an interwoven temporal fractal fabric.