The Mandelbrot Set Explained

We take a deep dive into the Mandelbrot Set, and try to understand what is happening under the hood. In particular we introduce the concept of orbits and their unusual behaviour. I hope by introducing such detail that you will find concepts in future videos easy to understand. This video contains many handcrafted visuals that took days code and animate, plus several hours to render them in 8k. Please subscribe!
Welcome to the channel! I'm new at making mathematical explainer videos, but I hope you get something from these videos. Feel free to leave some love or constructive feedback in the comments section. Questions are also most welcome.
If you loved this video then why not buy me a coffee: ko-fi.com/mathstown
In this video:
00:00 Introduction
1:51 Chaos & The Mandelbrot Set
2:40 The Equation
3:31 Looking at z squared
6:48 Mandelbrot, the real part.
15:24 Complex numbers
18:35 Full Mandelbrot Set
23:23 Mini-Mandelbrots
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
Deep Mandelbrot Zoom video (from introduction): • The Edge of Infinity -...
Panning around the Mandelbrot: • The Mandelbrot Set - F...
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Discord: / discord
Twitter: / mathstown
Patreon: / mathstown (Support, Downloads & Usage Rights)
Website: www.maths.town
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[@TheMathemagiciansGuild]

I am now on Discord: discord.gg/q4xsmSHV (under the Maths Town name)

[@JacobRy]

Hello, is this channel still going to post?

[@landsurfer66]

I believe that this is now the definitive video explanation for the Mandelbrot Set. Thanks for this great production!

[@TheMathemagiciansGuild]

Thanks for leaving such kind remarks. It's really appreciated.

[@gabenugget114]

Well done!

[@radrook7584]

You actually understood that? I salute you!

[@CalvinWiersum]

I've always been fascinated by this shape as a kid (hence the pfp), but I never fully understood its origin. Thanks for fulfilling by old wish!

[@x-75hurricane65]

I'm still none the wiser but I really appreciate your style of teaching...and I still think this stuff is absolutely beautiful! Cheers from NZ...

[@nicholasziglio]

It is rare to come across explanations as beautiful as this, absolutely wonderful work!

[@willclark7314]

I suck at math and can't tell you how much this made my day. You've completely opened my eyes and can't wait to see more. Subscribed.

[@mrsavedbygrace2569]

While I've been enjoying the Maths Town videos for about a year now, I never understood the math behind the Mandelbrot set. This video is giving me a better understanding of what's happening. I don't understand the math so much but I think I can understand the logic behind it. Great video and very helpful. Thanks

[@elibennett3034]

I know very very little about math.i was able to follow perfectly. Thank you for expanding my mind.

[@charleswiltshire]

This is the best Mandelbrot explainer video I've watched. Thanks for taking the time to create it.

[@VitalSine]

Fantastic video. I really enjoyed learning about the Mandelbrot set. I like how you paced your video, it was not too fast and not too slow, with well-timed pauses. I loved the animations, they really made the Mandelbrot Set come to life. I look forward to your next videos.

[@TheMathemagiciansGuild]

Thanks for the great feedback!

[@jfr644]

This really gives a short answer to the "how" question, but not to the "why". It seems to be quite philosophical though

[@bachirblackers7299]

Thanks . I think your favorite mini mendelbrot ( mn 25) is around the trancendental 1/e . Surprising .

[@rednicstone3299]

Verry nice and helpful video! I like the style of it and am looking forward for episode 2. Nice graphics btw

[@TheMathemagiciansGuild]

Thank you! And congrats on forever being comment number 1! :-)

[@stanervin6108]

🏆

[@johnr3936]

The best video ive seen on mandelbrot yet! Incredibly informative and concise

[@TheMathemagiciansGuild]

Thanks John

[@winkyyao1]

The best demonstration of Mandelbrot Set on KZbin. Many details!

[@mahirfr]

This was the explanation I was searching for a year.

[@Stranger_In_The_Alps]

The B in Benoit B. Mandelbrot stands for Benoit B. Mandelbrot 😎

[@Peterdeskater100]

By far the best explanation on KZbin I was able to find. Very very clear. Thank you.

[@andrewhall9492]

I really enjoyed this! Understanding makes fractals that much more enjoyable!

[@superilu]

I knew the basics of fractals and Mandelbrot, but this video takes it a step further. Thanks, I really enjoyed that!

[@HathaYodel]

We thank you for the care and thought you put into creating this excellent and succinct exposition of all the main aspects that tease and puzzle so many people who enjoy exploring Mandelbrot Sets and yearn to understand WHY and HOW they behave like this. The visual display of period orbits is particularly illuminating.

[@lythmashni2657]

This is definitely the best Mandelbrot video out there. please make more :)

[@lagduck2209]

Amazing, this was so visual and intuitive without going too obvious or too abstract! (though slightly infuriating was the moments like 4:53 where upper scale and lower scale does not match (or 1 does not go to 1), but that's just little minimal thing that is ever could be imperfect, in otherwise perfect video!)

[@mkgamesartvisuals]

Awesome work, thank you so much for this video! Edit: Wow! I was really amazed by the fact that you've done a video about this topic but as I'm watching it completely I just have to say that the content is really well visualized and explained!!!

[@gl0bal7474]

thank you for such a clear precise explanation. Im looking forward to watching more of your videos

[@stitzelmichael]

this is exactly what i was looking for, thank yiou for the great explanation, kooking forward to the next one

[@johnmarchington3146]

Absolutely fascinating. Thanks for a marvellous video.

[@rfo3225]

Just came across this. A second viewing was required before it clicked in my brain. Thanks for an excellent presentation. I feel like I actually understand this well enough to probe further.

[@TheAffeMaria]

Best video on the topic thank you from the bottom of my heart for making it understandable for an absolute math novice. Such beauty!!!

[@girogiro-vh5pz]

Amazing. Very nicely explained. Thanks!

[@hamzahamxa5951]

Thank you for this great production

[@mnada72]

Thank you, that was a great demonstration for the subject.

[@bartolomejkozorog3387]

Much underrated channel. Love it!

[@abhishek.chakraborty]

Thanks for making this 🙏 Looking forward to be regular subscriber if I see a fairly periodic stream of mathematics-related insightful videos 👍

[@Marcel.66]

This is explained great! Thank you

[@ElBellacko1]

this is the best video on mandebrot set, explanation, thanks.

[@NonTwinBrothers]

Awesome channel! Sent here from Maths Town

[@tcf70tyrannosapiensbonsai]

Just great. Thank you!

[@blakef.8566]

This is amazing. I've never seen a visualization of this like you've done around 9:40. Thank you so much for making this video.

[@mariedamlarsen]

I absolutely love this! Thanks for the really visual explanation! I've pressed the subscribe channel, by the way 😉

[@tictacX1]

Great video, thank you!

[@mickwilson99]

Brilliant pedagogy!

[@stanervin6108]

I'm in! Subscriber #92! Woot Woot! Top 100 list! 💯

[@Paulsinke]

Thanks this is exactly what I needed

[@justjay926]

You explain so very well. Thankyou 👏👏👋

[@danakrull6122]

Thank you for this introduction which will help me share the beauty of mathematics with my sons in our homeschool classes. I am not gifted in math but am fascinated by these concepts (and images, of course). I hope that this will help me spark a love of numbers in our boys. :-)

[@sigriit7886]

Very interesting and relaxing

[@GabrielsEpicLifeofGoals]

The amount of numbers it maps to is the amount of branches the bulb has, and it doublss every smaller bulb.

[@gavinirwin2913]

Finally something to hold my attention thank you

[@naringrass]

this set maps the perceivable reality, I don't know why nor how, I will find out but also will probably pass away before I finish.

[@serma3498]

Es fascinante este mundo maravilloso en que vivimos ,las matemáticas esta en todos lados ! Maravilloso vídeo ,felicitaciones y gracias por divulgarlo

[@ray017ray017]

You've made a great job to make this interesting... Or I'm just interested and I don't know why

[@MichaelZP1]

Mandelbrot hero!, thank you.

[@DrakenFire]

This video and the one from Numberphile really explained the Mandelbrot set. Now i get it thanks.

[@ArcticYetiz]

I am completely foreign to this subject, but that was reaaally interesting. I can imagine all the work required to do this video, so thank you !

[@jk1422]

Very good video! Thank you :)

[@mikedaniels3009]

I love the calm voice and the peaceful ambiance of this presentation. So happy you subtracted the music from the presentation. I have a question though: my enigma remains Benoît's paradigmA. WHY is the equation "supposed" to stay small? WHAT was Benoît's big idea to even come up with the equation ? Much obliged in advance for bearing with a mathematical oaf.

[@Teranova]

Awesome video, thank you :)

[@IamKavot]

Good video. Thanks!

[@rickyardo2944]

WOW! Thanks!

[@ogb18gang62]

Love this video

[@sawdustwoodchips]

Very clear and concise explanation!! if I may enquire, what software are you using to show the orbits?

[@Snowflake_tv]

thank you!

[@TheEulex]

I see 35 "thumbs downs." This is an awesome video and I can't understand what pinhead would ever click the wrong thumb icon. Thanks for posting.

[@skybridg5705]

Start at 1 keep going. It never ends.

[@DrewNewmanEngineer]

Fantastic explainer video! What is the application that was used to create the visualization starting at 35 seconds and ending at 55 seconds? Thanks in advance.

[@alexking1129]

Fantastic video

[@TheMathemagiciansGuild]

Thank you

[@shable1436]

Nice vid, you can't break it down much more easier than that.

[@danielduerst5067]

Thanks man for the explanation, and mostly on the mini brots!! That's amazing!! Can you go into more detail of how to find the most mimi brots in a fly over path around the shore lines of main Mandelbrot?? Simply put... Where are the best mini brot infestations??

[@bachirblackers7299]

Yes man ! We need to know more minis coz i think the mini next to the cusp is related to 1/e correct me if i were wrong . Also we need to know how transcendental numbers behave .

[@Nick12_45]

thx!

[@RowynOfficial]

Hi maths town!

[@bergarteric5713]

Mister thanks for your explications !!! realy thanks . now i understand more this fantastic and indcredible thing of Fractale and the genious of Master Mandelbroot . Sorry for my English !!! Eric from France ..

[@BountyLPBontii]

The Mandelbrot Set isn't chaotic, it IS chaos!

[@jesseliverless9811]

Actually it's what's outside the Mandelbrot set that's chaos? Since it's unstable, whereas the M.S. is stable...

[@BountyLPBontii]

@@jesseliverless9811 Sure you can look at the unstable area just around the border, but you can also look inside thru the buddhabrot set!

[@DundG]

Chaos is a general concept of not comprehensible complexity. The Mandelbrot set is a mathematical equation following this concept (so it can't be the concept by default) , and only partly as some iterations are shown to converge to a single point.

[@BountyLPBontii]

@@DundG Chaos is literally inside the Mandelbrot, since its everything that follows a bifurcation. You should watch Veritaseums Videos regarding that to learn more. Our whole reality is just the 10^99999th iteration at some point in the chaos of the original bifurcation. Everything we know is just a emergent property of the specific area of the mandelbrot our reality exists in. Im talking about us just being the next iteration after the multiverse with black holes creating another iteration yet again.

[@DundG]

@@BountyLPBontii yeah Chaotic behavior is part of the set but so is order in its convergent and non Chaotic oszilating solutions. And that about the multiverse is something we have no proof of. It's literary just an imagination of the beyond, based on our incomplete knowledge, just as people believed the earth to be flat and has an edge because the sun and moon evidently rise up and down... People can look for clues but unless proven it stays a diversion made to entertain the curios mind and is not science

[@hussainali9999]

Thanks

[@trumpetzmainia]

Pausing at 10:41, I was confused about the orbit that seemed to stay at [2] for about 10 iterations before blasting off. Correct me if I'm wrong, but have you chosen a number ever so slightly close to [-2] to start with, such that it would eventually diverge? [-2] is contained within the set, but I can see why you wouldn't want that hanging out on the screen when explaining the periodic nature of the converging values... Great stuff! Thanks!

[@Dr.Pepper001]

Clear as mud.

[@MiketheNerdRanger]

That last one was friggin terrifying

[@fineepl]

excelent

[@unbearifiedbear1885]

I have a fluorescent "Thunder Egg" crystal/agate about the size of a cricket ball and the formation in the middle looks _exactly_ like the Mandelbrot Cardioid.. its *incredible* Also have a raw octohedral diamond which is fractal layers of triangles on triangles on triangles, with negative spaces which are the same, triangles in triangles.. gives an amazing insight into how these objects actually form, the visual expression of the physics and mathematics which precipitate them Fractals, Mandelbrots, Paisley, Moroccan style rug patterns.. first time I did psychedelics and closed my eyes, it all made sense! I genuinely believe this is the language of creation

[@CavalloDiSpade]

what software are you using to render these?

[@leanderlle7602]

With which program can you recreate stuff like that?

[@jeninaverse]

The poet and Mathematian Without Division.

[@girogiro-vh5pz]

Are there any tools I can use to help visualise what's going on? In particular, I am interested in playing around with seeing a tiny change in C that causes a chaotic change in the result.

[@eduardogaldino8510]

Amazing nature!

[@aplimsollpunk2738]

🎵Just take a point called z in the complex plane Let Z1 be z squared plus C And Z2 is z1 squared plus C And Z3 is Z2 squared plus C And so on...🎵

[@EbbtideCheque]

Watched it through and all I learned was that I'm dumb as a rock. I have no idea what any of it meant. Very pretty though. 😲🤩

[@sorcerersofstone]

at the 17.54 can you explain how you came to 12? It has been a long time since I used algebra.

[@quitsevensix]

The opening advert was for beer 🤣🤣

[@IqbalHamid]

@04:11 Have you got your dx's and dy's mixed up? Surely it should be the other way around such that a = dy and b = dx in the diagram.

[@danielvieira8978]

fact: the number of branches of the bulbs indicate its period, just like the julia does

[@bitcoinzoomer9994]

Math always has a counterpart in reality. What in reality could a projection of finite and infinite space represent? My guess is that this is a map of the multiverse

[@RGJubilee]

How do I get my computer to do the Mandelbrot set?

[@GZWA]

feels like a Modern Vintage Gamer video

[@tworsfeline2302]

Where'd you get the color from? Some of those are between black & purple.

[@byugrad1024]

Is the area of the mandelbrot set known (does it approach a limiting value) or is it undefined? I would think it needs to be bounded by the area of the circle with diameter 4.

[@FreeStyleCrhymeR]

Dude you have great teaching skills, great computer simulations, both good combined and kept it simple. But you need to put some suitable sound or music to the background because the silence between your statements made my sleepy actually.

[@stanervin6108]

What about the use of imaginary number values other than roots of negative numbers? Or mathematical constants (pi, etc.)?¿ Anything interesting happen with, say, octal or hexadecimal? Thanks for your amazing knowledge and content you bless us with!

[@TheMathemagiciansGuild]

It's more that the roots of negative numbers was a gap in algebra. It turns out to be really useful to do this, in particular they simplify any problems involving waves, such as electrical engineering and quantum mechanics. Other constants don't really fill this void. As to octal or hexadecimal. They don't change the result at all. (This video was calculated in binary, and the colours specified in hexadecimal notation. Think of these notations as simply a different way of writing numbers down, much like Roman numerals. It turns out Roman numerals are very hard to multiply and divide, so we now use decimal notation. Hexadecimal may have been a better choice though, as it is more natural when working with computers.

[@stanervin6108]

@@TheMathemagiciansGuild Thank you for the quick, informative response to my questions.

[@melvynbraithwaite8563]

The Vertical line represents H-Chaos/Conformity This is what our zcreator can create in his Universe MBraithwaite Yorkshire zViking

[@NEMES1-S]

Started ok, because I had already watched several videos attempting to explain this. By 5.25 minutes, I was completely lost again. One obvious sentence say’s….’When you do algebra’……… hmm, yes well algebra was never a strong point in my secondary school maths, same as geometry. You people who understand this subject automatically assume the viewer has a good understanding of the basic terms. I so want to understand this, so I will persevere; a quality I did not possess in my teens.

[@Ivan_Nano]

YES!!! Thanks for making this video! It answered all my questions. I read the Wikipedia article many times but I needed this to finish my understanding of it. The Mandelbrot set makes me feel this depth within myself that I can’t explain. It feels like god shows itself more clearly in it.

[@EnPee91]

Wow, I never knew that all parts of the Mandlebrot set are connected.