The Julia Sets: How it Works, and Why it's Amazing!

hi I'm going to talk about the Julia sets and how they relate to the Mandelbrot set the Julia sets are a series of fractals discovered by Gaston Julia they are generated with the same recursive formula as the Mandelbrot set but how could that be bear with me here here is a list of iterations of our formula here is the complex plane for the Mandelbrot set you substitute all of these C's for a number on the complex plane such as this one then each result replaces Z in the next iteration if we see that the number keeps growing our chosen point is not considered a part of the set and it's colored depending on how fast it grows a number that stays bounded and doesn't grow on the other hand is considered a part of the set and it's point is colored black we repeatedly do this letting each and every point take a stab at being C and the shape of the Mandelbrot set is generated zooming into it can lead into infinite arrays of complexity pretty neat now the Julia steps are made exactly the same way but instead of substituting all C's with each number on the plane we substitute only Z sub zero but then what shall we do is to see no problem we can just choose a different number to be C such as this one so we calculate and find out that this number stays bounded so it is indeed in the set now to calculate a different point we keep C the way it is and only to the different number for Z sub zero this one is not in the set if we keep doing this letting each and every point take a stab at being Z sub zero we end up with this shape if we zoom in we find that it is a rather simple repeating pattern okay let's do this again only choosing a different value for C we get an entirely new shape so by choosing different values for C vast varieties of different shaped fractals can be made and every one of these fractals is considered to be a Julia set [Music] okay so aside from the formula how do the Julia sets relate to the Mandelbrot set as you may recall each point on the Mandelbrot set corresponds to a complex number or a C value so we can essentially take any point in the Mandelbrot set and use it to generate a Julia set what's remarkable is that each Julia set has visual themes similar to the corresponding area of the Mandelbrot set we can see here that gradually changing the C value can shift the Julia set creating an almost kaleidoscopic effect let's zoom into this part of the Mandelbrot set take this point and use it to generate a Julia set the Julia set looks like this and we can see some obvious similarities the very shape of the whole Julia set is depicted in the Mandelbrot set with striking accuracy we can see this in a variety of different locations what's even more interesting is when we zoom into the Mandelbrot set to find a mini copy of it since we're looking at something so tiny the C values involved only slightly different from one another this makes the Julia sets appear to be the same everywhere in this mini Mandelbrot but suppose we were to zoom in to this Julia set look a mini Julia set that does change with only a subtle difference in see what we have seen has made it quite apparent that the Mandelbrot set and the Julia sets are directly interrelated even though at first glance they may seem like separate entities for the mind open to metaphor this phenomenon can be seen as akin to the nature of reality maybe things are more interconnected than they appear to be smaller things being intertwined with what fundamentally exists as a larger whole [Music] [Music] you [Music] you

Channel: Jimi Sol

Views: 82,768

Rating: 4.9519143 out of 5

Keywords: Mandelbrot, Mandelbrot Set, Julia, Julia Set, Mathematics, trippy, fractals, fractal, fractal zoom, psychedelic, math, Mandelbrot Zoom, interconnection

Id: mg4bp7G0D3s

Length: 4min 13sec (253 seconds)

Published: Mon Apr 03 2017