Number Sequences in the Mandelbrot Set
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- เผยแพร่เมื่อ 30 ก.ค. 2024
- Welcome to part 4 of our little Mandelbrot Explained series. In this video we explore the bulbs around the main cardioid, and find that they contain number sequences such as the natural numbers, Fibonacci sequence, and the rational numbers. We then investigate them in terms of their Julia Sets to try and understand visually why they are there. Finally, we look at precisely where the bulbs are attached to the cardioid.
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In this video:
00:00 Introduction
00:30 Period of the bulbs
01:45 Signposts in the Mandelbrot Set
02:42 Number Sequences
04:24 Rational Numbers
07:06 Julia Sets of the bulbs
11:36 Location of the bulbs
13:15 Why signposts?
14:30 Building a Mandelbrot Set
In this series:
1 - • The Mandelbrot Set Exp... The Mandelbrot Set Explained
2 - • Julia Sets, and how th... Julia Sets, and how they relate to The Mandelbrot Set
3 - • How to Build a Julia Set How to Build a Julia Set
4 - • Number Sequences in th... Number Sequences in the Mandelbrot Set
Extra Visuals (No commentary):
• [Extra Visual] All orb... - All orbits of the Mandelbrot
• [Extra Visual] All Per... - Period 1 orbits of the Mandelbrot
• [Extra Visual] All Per... - Period 2 orbits of the Mandelbrot
• [Extra Visual] Buildin... - Building a Mandelbrot step-by-step
• [Extra Visual] Period ... - Period 2 orbits of a Julia Set
• [Extra Visual] Period ... - Period 3 orbits of a Julia Set
• [Extra Visual] Period ... - Period 4 orbits of a Julia Set
• [Extra Visual] Period ... - Period 5 orbits of a Julia Set
• [Extra Visual] Period ... - Period 6 orbits of a Julia Set
Mentioned in video:
• Fibonacci Numbers hidd... Numberphile video (Mandelbrot & Fibonacci)
• Mapping a polar grid o... Mapping a polar grid to the Mandelbrot
Jeez, I never knew that the period 2 bulb was the only perfect circle.
None of them are perfect circles; they're roughness is just so small.
@@lilmarionscorner maybe they mean if you approximate what its surface would be without all the attached bulbs
Did that, Still not convinced
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.
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.
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!
The thing about the bulbs matching with the internal angle of the cardioid is insane
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 can you be more specific about where C is? I didnt really get that part, since you just defined B and then brought C up out of nowhere
@@circumplex9552 C is defined as the complex number 0.25 + 0i, which is exactly at the cusp of the main cardioid.
@@denelson83 ah, ok. I'll see if I can figure something out
@@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.
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.
Thanks for the kind feedback!
THANK YOU for no music. ❤️
Thank you for your intuitive visual explanations of these beautiful patterns. Can't wait for your future videos!
Shalom and evening howdy how.
Very nicely done and thank you for sharing!
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.
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!
Incredible, thanks again!
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.
That is beyond imagination. Thank you.
Glad you liked it!
All this is incredible. Btw, does that mean all the "mini mandelbrots" are distorted as well?
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? 🤔
Dang, these vids are real good
Thanks!!
Also, can you make a video on why minibrots are distorted? I would really like to know.
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
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??
Ur a magician mate
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.
Thanks
Buenisimo
Am I a member of this guild as like I'm a mathemagician with a new-be magic wand?
The fibonacci numbers are showing up may be because of the intersecting point between the circle 1 and the curve x^
Square root of x as well !!!
how did you run that mandelbrot simulation at the end of the video?
what software do you use to visualize it like this?
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.
Ammonites? As in the people who cut eyes out of the Israelite people?
(It's a biblical joke).
Why are the sign post branches arbitrarily labelled 1, 2, 3....?
my theory is that the size of the circle needed to build the main cardioid is the same size as the period 2 circle.
Me gustan tanto los fractales que nunca se cual elejir
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.
Why is it a cardioid?
A shape like a heart. You can make it by rolling one circle around another. (The large area of the Mandelbrot set is a perfect cardioid)
@@TheMathemagiciansGuild Yes I know, but why the Mandelbrot set is shaped like that?
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
Period 1 ?
Period 1 is only within the main cardioid itself.
@@TheMathemagiciansGuild Period -1
?
Mandelbrot kinda thiCC doe.
Hi, Could i get your email? I have a paid project that i want to talk to you about
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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.
Nerds
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