Could you do the internet favor and just post images of that mandelbrot set made out of a circle with degree marks to as many social media sites as you can? That is probably one of the best and most important visuals I got out of this entire fantastic series.
4:57 - I never knew you could get the numerator that way! You should really put together a fifth video in this series, dealing with other concepts connected to the Mandelbrot set, such as finding other number sequences in the bulbs like the powers of two, external rays, equipotential curves, Misiurewicz points, Siegel discs, how you can derive a bifurcation diagram from the set, the Buddhabrot variation, and even how you can calculate π just by using the set.
Here's a conjecture: Call the centre of the inner circle "O" and the point where the outer circle is tangent to the inner circle "A". When the outer circle is at 0°, the highlighted point on the outer circle-the point that traces out the boundary of the cardioid, which we will call "B"-is exactly at 0.25, which we will call "C". Now roll the outer circle around the inner circle through a certain angle. The conjecture states that line segments OA and BC are always parallel. Can you come up with a proof for this conjecture?
@@denelson83 worked on it for a while, and decided I may just give up on this one. what I did was I first sized up the circles to have a radius of 1 (this also meant C became 1 + 0i) to make things easier, then created the variable z to represent the measure of the angle of the outer circle, or more accurately angle AOC, in radians. all I needed to prove OA and BC were parallel was a proof that their slopes were equal. OA's slope ended up being (sin z)/(cos z), and for BC it was (2 sin z + sin (2z + pi))/(2 cos z + cos(2z + pi) - 1). this was a really good start, but no matter how I approached it, I couldn't simplify BC's slope adequately due to that pesky -1. (note, the -1 is because of C) Edit: that last bit can be simplified in some way, but not in a way that helps. (sin z/cos z) can become tan z, and then we can invert that tan function so the equation becomes "tan^-1 of the slope of BC = z" but thats essentially just a rewording of the conjecture.
My jaw has dropped when watching this video and I can't find it. It's probably somewhere in the complex plane, in a dark place behind one of the Mandelbrot bulbs. Absolutely mindblowing stuff. 🤯 Thank you!
oh man I got into fractals with the discovery of the fibbonocci sequence, and how I kind of discovered it myself butlater in life learning its incredibly implications in life and physics, and I've watched that numberphule video probably 10 times trying to best understand what she is getting at, but this is just what I needed. Thanks you good sir, and just know I lovetthese videos so much.
Thank you for these videos! I've been interested in the Mandelbrot set for years and these are some of the most informative videos for giving a taste of "why" it looks the way it does.
Hey Mr. Mathemagicians Guild, when you get back into making mandelbrot set videos, can you please talk about Mandelbrots of different exponents (Zn=Z^X+C), what happens as the exponent N approaches infinity, and what a mandelbrot set looks like with a imaginary/complex exponent? I tried making a imaginary mandelbrot using the fractal imaging software Xaos and posted my findings on my channel, however i think a proper hand coded imaging method is needed to properly view them in good quality. Finally i would love to hear your take on how the mandelbrot set and bifuration diagram/logistic map are connected. Thank you!
Youre awesome dude i wish this series got more views. I finished all 4 and keep coming back. Hope you make some more videos about the Mbrot, if not its understandable as im sure this takes a huge chunk of time to produce with animations and all. Thanks for your work!
Thank-you! Yes, I will add more occasionally. I will do a couple of videos of the Complex Analysis series first though. The Complex derivative is also useful for explaining some features of the fractals.
Wow! thanks for the clear insights. I was always happy to just wonder at the the sets, knowing natural sequences were echoed in them. This video has just bent my head enough to kinnnd of understand a little more.
Imagine a VR Mandelbrot in 3D where you can also see the values for each point you follow. Although math is conceptual, we clearly see the corresponding mirror in tangible nature such as crystal formation, plant formation, and interstellar body formations. Is this by accident or design? 🤔
Love this series! Any chance external rays will be covered at some point? The fact that _any interesting point at all_ is representable by a rational number seems even more incredible than the rational numbers of the bulbs. 11:53 In fact, there are _definitely_ no other perfect shapes! math.stackexchange.com/questions/1857237/perfect-circles-in-the-mandelbrot-set
6:22 in other words, the rational number sequence in Mandelbrot set is the form m/n, where n is a natural number greater than 1, and m is a number which it's congruent modulo n admits inverse in mod n.
I don't understand why the Fibonacci sequence emerges from the rational number properties. Additionally, it seems that the _numerator_ of those bulbs follows the sequence as well! How come??
You will see in the image during the course of this video, Sea Horses, Ferns, and flower heads, Ammonites (keep up, Google spell), among other things found in nature.
Ah sorry. Cardioids show up because a circle through the origin on the complex plane will get mapped to a cardioid under the z^2 function. A longer discussion can be found here iquilezles.org/www/articles/mset_1bulb/mset1bulb.htm
Something doesn't make sense here. Look at 1:01. We have a cycle of 3 points but one of those points is NOT in the Mandelbrot set. How is that possible. If one iteration is not in the set then it goes off to infinity. All iterations MUST be in the set. You can't get bigger than 2 and then shrink back down again.
In order for a point _c_ to be in the Mandelbrot set, the orbit of _z_ that it makes has to stay within the circle of radius 2 centred on 0, e.g., all values that _z_ takes on must have a magnitude of 2 or less.
Not explained very well. What are the little green balls. And why do they all move if you move the middle one. I will watch it again. I understand about complex plane and iterations and coloring counts. I also enjoyed the period one and 2 Mandelbrot videos. Don't respond. I will comment later on with a good question. Thanks.
