"There is always another way to say the same thing that doesn’t look at all like the way you said it before."
- RICHARD FEYNMAN
The "Mandelbrot set" is one of the most recognizable mathematical fractals. The formula, z := z^2 + c, gives no clues as to the vast complexity and unending beauty hidden within this simple iterative system. At first glance, a connection between M-Set and black holes may seem improbable. For example, M-Set, as depicted in Figure 1, doesn’t look anything like Figure 2, i.e., the Schwarzschild black hole. What does fractal geometry have to do with black holes? 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.
In THE OM PARTICLE theory, the Mandelbrot Set is modelled as a quasi-black hole. Not the black holes of relativity, but the black holes that nature makes.
If we code for the Mandelbrot set, the algorithm begins an iteration loop. Iterating the function gives different trajectories which all has different behaviours. These trajectories are mainly divided into three different parts: domain of convergence, domain of divergence, domain separator.
The black hole, generalizes as the domain of convergence, belongs to the domain of counter-space. The photon sphere, generalized as the domain of divergence, belongs to the domain of space. The event horizon is thought of as a domain separator and separates the domains counter-space and space. In, The OM Particle theory, domain separators are always fractal in nature.
In THE OM PARTICLE theory, domain of convergence (inner black region) of The Mandelbrot Set (or M-Set for short) is analogous to a black hole. The domain of divergence (outer greyscale region) is analogous to the lesser known photon sphere of a black hole. The boundary that exactly separates the two domains is analogous to the event horizon. This is an event horizon in the truest sense as it exactly separates the converging domain (the contracting part) from the diverging domain (the expanding part) of the fractal geometry. Benoit Mandelbrot referred to this as “S” for separator. The event horizon is the most interesting part of M-Set as that is where all the beautiful fractal patterns are “stored”.
In the M-Set quasi-black hole model, black holes, event horizons and photon spheres are three components of a single (sacred) geometry, which I affectionately refer to as “The OM Particle”. The diverging domain is the expanding side of the event horizon. The converging domain is the contracting side of the event horizon. In this model, expansion and contraction conspire to the creation of the event horizon, where all the interesting fractal patterns emerge. In a similar manner, the universe itself is thought to have an expanding part (space: dark energy), a contracting part (counterspace: dark matter) and an emergent part (visible matter). The emergent part is the event horizon and the event horizon is the visible universe.
In the standard model of cosmology, space and time are combined together into a 4-dimensional "space-time" manifold where time is treated as another spatial dimension. In order to understand M-Set as a quasi-black hole, we need to understand this time. Time is an emergent property of change brought about by an iterative process. In this manner, time is analogous to iteration, and vice versa. It is the unending, unknowable uniqueness of each moment that gives us the sensation of time and the arrow of time. Without unending, unknowable, irreversible change, there would be no sensation of time and no arrow of time. That said, the only math that can mimic this kind of unending, unknowable, unrepeatable, entropic change over time is iteration, especially as it relates to chaos theory and fractal geometry. This, of course, includes M-Set.
This shows how beautifully the fractal geometry mimics the nature. Benoit Mandelbrot wrote a book “The Fractal Geometry of Nature” on this. Fractal geometry uses simple rules and a simple to generate an endless set of figures that strikingly resembles familiar cosmological objects like shown above. But of course, the very existence of a quasi-black hole is still in doubt!