Number Sequences in the Mandelbrot Set

[@vladimirarnost8020]

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!

[@EvilSandwich]

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.

[@denelson83]

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.

[@wallywutsizface6346]

The thing about the bulbs matching with the internal angle of the cardioid is insane

[@denelson83]

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?

[@circumplex9552]

@@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

[@denelson83]

@@circumplex9552 ​C is defined as the complex number 0.25 + 0i, which is exactly at the cusp of the main cardioid.

[@circumplex9552]

@@denelson83 ah, ok. I'll see if I can figure something out

[@circumplex9552]

@@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.

[@RobCozzens]

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.

[@asherwilkins465]

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.

[@TheMathemagiciansGuild]

Thanks for the kind feedback!

[@zfloyd1627]

Jeez, I never knew that the period 2 bulb was the only perfect circle.

[@lilmarionscorner]

None of them are perfect circles; they're roughness is just so small.

[@circumplex9552]

@@lilmarionscorner maybe they mean if you approximate what its surface would be without all the attached bulbs

[@lilmarionscorner]

Did that, Still not convinced

[@brian.westersauce]

What is attached to the (perfect) circle except for bulbs? It’s only bulbs, how could there be any additional separate “roughness” unattributable to specific extending structures?

[@MarshallBrandt]

Thank you for your intuitive visual explanations of these beautiful patterns. Can't wait for your future videos!

[@ИванВажинский-я1м]

THANK YOU for no music. ❤️

[@SmokeyDope]

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!

[@TheMathemagiciansGuild]

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.

[@mnada72]

That is beyond imagination. Thank you.

[@TheMathemagiciansGuild]

Glad you liked it!

[@SmokeyDope]

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!

[@ronancoyle2495]

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.

[@johnadriandodge]

Shalom and evening howdy how. Very nicely done and thank you for sharing!

[@NonTwinBrothers]

Dang, these vids are real good

[@TheMathemagiciansGuild]

Thanks!!

[@mkgamesartvisuals]

Incredible, thanks again!

[@zfloyd1627]

Also, can you make a video on why minibrots are distorted? I would really like to know.

[@alcyonecrucis]

Ur a magician mate

[@danielvieira8978]

my theory is that the size of the circle needed to build the main cardioid is the same size as the period 2 circle.

[@utilizator1701]

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.

[@ivymike3459]

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? 🤔

[@19Szabolcs91]

All this is incredible. Btw, does that mean all the "mini mandelbrots" are distorted as well?

[@denelson83]

They are not exactly distorted, but their bulbs are offset. Another conjecture is that the central component of every mini-Mandelbrot set is itself a perfect cardioid.

[@justjack2131]

how did you run that mandelbrot simulation at the end of the video?

[@Snowflake_tv]

Am I a member of this guild as like I'm a mathemagician with a new-be magic wand?

[@enricobianchi4499]

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??

[@denelson83]

Look up the concept of mediants to find out why.

[@galaxygur]

00:06:34 - The Mandelbrot set reveals an infinite number of fractions between 0 and 1, each with its own unique bulb. 00:12:01 - Only rational numbers can find a periodic equilibrium in the Julia set, forming the bulbs in the Mandelbrot set.

[@bachirblackers7299]

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 !!!

[@jeremysender]

what software do you use to visualize it like this?

[@abacussssss]

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

[@bachirblackers7299]

Thanks

[@lookinwardstothe2349]

Why are the sign post branches arbitrarily labelled 1, 2, 3....?

[@brothermaleuspraetor9505]

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.

[@zfloyd1627]

Ammonites? As in the people who cut eyes out of the Israelite people? (It's a biblical joke).

[@niurovi2studio884]

Buenisimo

[@joeeeee8738]

Why is it a cardioid?

[@TheMathemagiciansGuild]

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)

[@joeeeee8738]

@@TheMathemagiciansGuild Yes I know, but why the Mandelbrot set is shaped like that?

[@TheMathemagiciansGuild]

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

[@niurovi2studio884]

Me gustan tanto los fractales que nunca se cual elejir

[@billclinton4913]

Mandelbrot kinda thiCC doe.

[@Lord_Volkner]

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.

[@denelson83]

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, i.e., all values that _z_ takes on must have a magnitude of 2 or less.

[@frankconley7630]

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.

[@gabenugget114]

Period 1 ?

[@TheMathemagiciansGuild]

Period 1 is only within the main cardioid itself.

[@gabenugget114]

@@TheMathemagiciansGuild Period -1

[@gabenugget114]

?

[@shahanbutt9037]

Hi, Could i get your email? I have a paid project that i want to talk to you about

[@HappyRaven-neilrw]

ı

[@mocha_snipez5154]

Nerds

[@r3d570n3]

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