\({z\to \frac{2z^2u}{|z|} + c}\)

\({ u\to u^2-|z|^2+d}\)

While equations in real numbers need not have real solutions, all polynomial equations have complex solutions. So unlike the reals, the complex numbers form a closed system.

Complex numbers were initially treated as a useful fiction, but as they became more and more useful it became clear that they could be understood graphically. Euler unified logarithms and trigonometric functions with his formula

\({ e^{it} = cos{t}+i sin{t}}\)

The invention of the Argand diagram, a geometrical interpretation of complex arithmetic, let the complex numbers be seen no longer as a convenient fiction but as the true realm of mathematical analysis. It was as if we had only being seeing a small slice of reality, and complex numbers let us see the whole picture.

Problems with real and natural numbers which seemed intractable opened easily with the tools of complex analysis.

Complex numbers play an essential role in the analysis of wave functions and quantum mechanics. They have become an integral part of our understanding of physical reality.

The insight gained from the additional axis of imaginary numbers prompted the question: were there further axes awaiting discovery, which would throw more light on the nature of number and space?

Complex numbers are closely related to polar coordinates, so one way to extend them is simply to add another imaginary component parameterized by an angle, giving what has been jokingly called triplex algebra. Of course this is not a real algebra, lacking inverses, but it does give rise to some stunning shapes, which extend the Mandelbrot set into 3 dimensions. The bulbous nature of these shapes has given them the name Mandelbulbs.

The 3rd order Mandelbulb is defined by the formula

\({z\to \frac{z^2(|z|^2-3u^2)}{|z|\sqrt(|z|^2+u^2)} + c}\)

\({ u\to \frac{u(3|z|^2-u^2)}{\sqrt(|z|^2+u^2)}+d}\)

where z is complex and u real.

In the Mandelbulb sets we see echoes of Mayan stellae, Hindu temples, Rococo stucco, seashells, diatoms, trees and vines.