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The mandelbrot set and logistic map
The rise of computers lead to breakthroughs in science and math never before possible. Before computers, all computations were done by working women whos job was to be human calculators. Crunching data for a living. This introduced the human limits of iterative processing, we simply aren't that great at being calculators, our abilities mentally visualize or algebraically calculate have very finite limits.
Computers on the other hand have no issue iterating equations hundreds of thousands of times within a couple seconda and graphing the output with more precision than any human hand could ever try to approach. Questions about precision and iteration within complex equations could finally be visually represented. Would you like to see the first examples of these shapes?
At some point it starts to seem spooky with it showing up in so many seemingly disconnected fields of STEM. Whats the deal? Why is the M-set so special to pop up like this?
Lets think about a simple shape which also shows up in STEM, the circle and its related mathematical costant pi. If you look close enough you can find pi pretty much anywhere. To quote wikipedia for a second
Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres.
In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and sciences having little to do with geometry of circles, such as number theory and statistics, as well as in almost all areas of physics.
The universe *really* likes circles for some reason. throwing a stone in a pond yields circular waves, rainfall on a puddle, craters on the moon, the sun and moon in the sky (and for some reason are both the same apparent size which for all intents and purposes is a cosmic miracle, they have no explainative reason for being the same size),
all gravitational bodies past a certain density like planets and stars and black holes are squeezed by that gravity into spherical shapes, flowers like sunflowers and dandylions like to grow into spherical and radial circular structures. The eyeball tends to be circular in most animals, most microcosmos life like cells and algae and fungal spores tends to be circular if not perfectly spherical.
Many spiritual symbols such as mandellas and the oroboros make use of circles and outward spiraling radial bands of circles to present ideas of eternal cycles and infinite levels of reality spiraling forever above eachother. The Jewish faith has the Kabbalah or 'tree of life' which depicts 7 perfect spherical containers meant to contain Gods light. The judao-christian faith makes heavy use of halos to depict divine substance, and also some variants had circular ringed seraphims filled with eyes.
There is no specific "Why" which connects all of these concepts together into one prevailing explaination of why circles exist the way they do physically or symbolically. They are simply a canon of reality we are forced to contend with and subconciousnly incorperate into our multifacited understandings of reality.