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I love fractals. Do you have a really obscure interest that for whatever reason lights a fire in you to learn more? Ever felt the fustration of not being able to have a meaningful discussion about said topic because 99% of people including your friends and family have no earthly clue what youre on about, nor do they care? To me, fractals are that topic. For whatever reason i just love them. they represent a new frontier in understanding reality. Every fractal is a rabbit hole of unexpected connections between aspects of math, science, and the mechanics of logic itself. They also appear everywhere you're not expecting them to. learning about this stuff is mind bending once you really understand whats being talked about. finally: they just look plain cool! even the simplest fractal contains infinite complexity (well the abstract ones anyways)
The Goal of this project is to slowly build my own little compendium of fractal knowledge based on my own understanding and constant education on the subject. It will by no means be comprehensive. Partially because i am not a real mathematician and many concepts are above my education, and also because the subject of fractals are as infinitely complex and rife with connections as the fractals themselves.
Defining what fractals are is a real pain in the ass for even legitimate mathematicians. Its sort of a "know it when you see it" kind of thing. Typically, fractals are geometric shapes which have self-similarity at multiple scales and often scale at a non integer dimension. If what i just said makes no sense, dont worry the specifics arent too important. Self similarity is a fancy way of saying that if you zoom in or out of the shape it still looks the same. Fractal dimension has to do with scaling, a topic which is best left for another time. 3blue1brown does a phenominal job explaining scaling and fractal dimensions, far better than i ever could here so i recommend you check out
this video for more information on that.
just one example of the kind of rabbit hole you can fall down, theres a quadratic equation that can be used to model population growth which is of great interest to biologist. The eqation is
Xn+1= R * Xn(1-Xn)
where R is rate of growth, Xn+1 is the next population generation, Xn is current population, (1-x) is how many died the current generation.
It has like 3 vairables yet accurately describes the fundemental mechanics of growth for every species to ever exist.
This same equation was also used as one of the first ways computers could deterministically generate psudo-random number sequences because at specific rates of growth the population became aperiodically unstable in inherently unpredicatble ways.
ALSO ALSO if you try to graph out this equation using a biforcation diagram, you end up with a fractal. this fractal is actually a one dimensional slice of a two dimensional fractal known as the mandelbrot set that lives in the complex plane. Its a one dimesnional slice because were only using real numbers for the equation. We started from biology and the nature of organism populations, to computer science and using a deterministic formula to produce unpredictable results, all the way to a purely abstract geometric fractal which exist in the complex plane. WHAT?! all these things are interconnected in such a bizarre yet beautiful and seemingly meaningful way that it takes months to really wrap your mind around it (if youre me of corse)
the study of fractals is called fractal geometry. Since the ancient Greeks, studies on the nature of shape and form, which we now call geometry, has had a basis in perfect abstract forms which only really exist in the dimension of thoughts and ideas. the circle, triangle, square, pentagon, all these shapes which lie at the basis of geometry were the cornerstone of the subject for many thousands of years. Simple, perfect, easily understood and studied.
A side note on the greek mathematicians "world of forms" idea
Many formulas, theories, and proofs center around these basic shapes and for good reason. Pi shows up almost anywhere you look if you dig deep enough. so does eulers number, phi, and many other fundemental constants. For whatever reason, the world of the abstract really does seem to have invisible connections with the real world. The logic which we used to derive abstract truths seems not so far off from the logic the universe itself operates with. which makes sense, we are products of that universal logic after, as is everything in existance.
Still, reality is not perfect. there are no circles or squares, ratios are close but not exact, and you cannot measure something with infinite precision. So how can you describe a cloud or any real shape mathematically with all its creases and undefined edges? Fractal geometry aims to do just that, using new concepts to describe the reality we live in. The idea of trying to mathematically capture reality on its own terms has lead to huge advancements in many fields both academic and practical. Today it is very much regarded as a legitimate study, showing promising results in places such as engineering, biology, cardiovascular health, and complex data analysis.
The term fractal was first coined by Benoit b. Mandelbrot, regarded as one of if not the founder of modern fractal geometry. The Mandelbrot set, arguably the cornerstone of fractal geometry, is his discovery (though works on the julia sets of which the mandelbrot set is based on has been studied since way before the invention of computers). I will be discussing the M-Set later on. His book "The Fractal Nature of Geometry" (OF WHICH I HAVE A REAL PHYSICAL COPY! WOOO!) was of huge influence to many as a gateway to the subject.
With the basics out of the way, lets talk about some fractals!