Mandelbrot Set is a common term with Fractals, referring to that rather famous shape which Polish born French mathematician, Benoit Mandelbrot invented - or should one rather say discovered - in the early 1970s.

It looks initially a bit like a “jagged circle” and of course the main feature is: as you zoom-in to investigate the edge, new details appear, in principle an infinite number on its circumference, as well as an infinite number of further levels of depth!

The key point to understand is: you are usually seeing merely an utterly tiny segment somewhere deep within the Mandelbrot set, just a billionth of it or less, and yet, as infinite as all of those are - they still all belong to the Mandelbrot set.

One would think that “infinite” number of variations ought to be enough for a lifetime or two of exploring - but the fact is: self-similarity is one of the inherent properties of it, and so after a while you begin to recognize the various shapes…

There is a vocabulary of forms, probably many hundreds at least, but not the kind of infinity one intuitively associates with that word. Rather more like “no two strawberries look exactly alike” - so there are infinite variations of them - but it would lose a bit of its excitement after a while…

But - as infinite as all that is… it is merely one of many other possible universes like that!

In fact it gets more amazing still: for every single one of the infinite number of points along the edge of the Mandelbrot set, there is a matching othershape - and that collection of all of the infinite setsof them is called the Julia set.

By the way, just like the MSet being named after its discoverer Benoit Mandelbrot, the Julia set is bearing the name of the mathematician who first published their existence - and he did so over 50 years before Mandelbrot: Gaston Julia.(Independently Pierre Fatou worked in this domain)

On a noteworthy aside: Julia wrote his thesis right during WWI, where he lost his nose at age 23 - and had to live with various leather straps across his face - but still lived another six decades, with six children as well. Amazing work by an amazing man… on many levels.

Tip: Just to point out one direction: as you zoom into the classic MSet, you do realize after a while that there are many small copies of the set shapefound inside itself, at deeper zoom levels.

You guessed it - they are infinite in number really, but already after zooming in a mere 10x or so, one can find thousands of them, quite fast… (for instance following the long tip, they line up)

These so called “Mini Mandels” are copies of the full set, rotated, scaled - some are exact detailed twins, other with softer edges, some morphing towards circles and sometimes skewed at angles in amusing distortions…

Here is one example of where such things are hiding… and a fun skewed off angle one as well…

You can see in the lower left the zoom beginning, into the sharp needle shape and how the final big white MiniMandel differs from the main one:

Tip: Now the insight…: if you do know that certain spots on the Mandelbrot set generate interesting Julia shapes - say the fourfold symmetries at the tip of the tail end - you then realize: the same is true for the smaller Mini Mandels - they too produce these results in what is often termed “nested Julias”.

But as the Mini Mandels are surrounded by other areas, and not identical but rather morphed cousins of the main set - so are the nested Julias interesting deviations and mutations…… and great fun to explore!