“The Mandelbrot set broods in silent complexity at the center of a vast two-dimensional sheet of numbers called the complex plane. When a certain operation is applied repeatedly to the numbers, the ones outside the set flee to infinity. The numbers inside remain to drift or dance about. Close to the boundary minutely choreographed wanderings mark the onset of the instability. Here is an infinite regress of detail that astonishes us with its variety, its complexity and its strange beauty.” Kee Deweney, Scientific American, August 1985
The “Mandelbrot set” is one of the most studied and most recognizable mathematical fractals in the history of the fractal geometry. The extremely simple formula, z:=z2+c, gives no clues as to the vast complexity and unending beauty hidden within this simple iterative system. At first glance, a connection between the Mandelbrot set (M-Set) and black holes may seem improbable. What does fractal geometry have to do with black holes? Interestingly, recent research suggests that a relationship between black holes and fractal geometry does in fact exist. Using a mathematical duality between Einstein’s relativity and fluid dynamics, simulations show that fractal patterns can form on the horizons of feeding black holes {HolographicTurbulence}. This important point shows that relativity and fractal geometry may be intimately linked.
One of the main objections to this body of work is that it does not reference the many successes of relativity. Aside from comparing M-Set to the Schwarzschild black hole, this research doesn’t look anything like relativity. This is a completely different approach to cosmology that is founded on different principles. Whether relativity is successful (or not) does not affect this line of thinking. What if relativity was never invented? What if the discovery of fractal geometry (and M-Set) predated relativity? Would we still have intuited the existence of black holes? Can the geometry of M-Set tell us anything new about black holes that we didn’t know before? Can it make any predictions and if so, are they testable? This body of work is an attempt to address all these questions. If this “thought experiment” can give us insights into the inner workings of nature, then is it not philosophically worthy of further investigation.
