The Mandelbrot Set Explained

Подписаться 7 тыс.

Просмотров 188 тыс.

50% 1

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...
/********************************************************************************
Discord: / discord
Twitter: / mathstown
Patreon: / mathstown (Support, Downloads & Usage Rights)
Website: www.maths.town
********************************************************************************/

Опубликовано:

16 июл 2024

Ссылка:

Скачать:

Готовим ссылку...

Добавить в:

Мой плейлист

Посмотреть позже

Комментарии : 218

@TheMathemagiciansGuild 2 года назад

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

@JacobRy 2 года назад

Hello, is this channel still going to post?

@landsurfer66 4 года назад

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

@TheMathemagiciansGuild 4 года назад

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

@gabenugget114 3 года назад

Well done!

@radrook7584 2 года назад

You actually understood that? I salute you!

@CalvinWiersum 3 года назад

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!

@vincenthabay5109 18 дней назад

this is hands down the most crystal clear explaination i've seen on the subject. When you master a subject and you are still able to enter a novice's shoes to teach him you reach the master Yoda level of pedagogy. thanks for this video

@x-75hurricane65 3 года назад

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 3 года назад

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

@mrsavedbygrace2569 4 года назад

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 3 года назад

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

@charleswiltshire 4 года назад

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

@VitalSine 4 года назад

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 4 года назад

Thanks for the great feedback!

@willclark7314 3 месяца назад

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.

@Peterdeskater100 2 года назад

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

@winkyyao1 3 года назад

The best demonstration of Mandelbrot Set on RU-vid. Many details!

@jfr644 3 года назад

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

@rednicstone3299 4 года назад

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

@TheMathemagiciansGuild 4 года назад

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

@stanervin6108 4 года назад

🏆

@johnr3936 4 года назад

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

@TheMathemagiciansGuild 4 года назад

Thanks John

@lythmashni2657 4 года назад

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

@superilu 4 года назад

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

@mkgamesartvisuals Год назад

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!!!

@andrewhall9492 3 года назад

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

@HathaYodel 5 месяцев назад

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.

@stitzelmichael 4 года назад

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

@TheAffeMaria Год назад

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

@bachirblackers7299 4 года назад

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

@lagduck2209 3 года назад

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!)

@blakef.8566 4 года назад

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 3 года назад

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

@johnmarchington3146 3 года назад

Absolutely fascinating. Thanks for a marvellous video.

@abhishek.chakraborty 4 года назад

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

@gl0bal7474 4 месяца назад

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

@bartolomejkozorog3387 4 года назад

Much underrated channel. Love it!

@rfo3225 7 месяцев назад

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.

@mnada72 Год назад

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

@mahirfr 3 года назад

This was the explanation I was searching for a year.

@danakrull6122 3 года назад

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. :-)

@stanervin6108 4 года назад

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

@ElBellacko1 3 года назад

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

@hamzahamxa5951 3 года назад

Thank you for this great production

@Marcel.66 3 года назад

This is explained great! Thank you

@girogiro-vh5pz 4 месяца назад

Amazing. Very nicely explained. Thanks!

@ArcticYetiz 4 года назад

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 !

@tcf70tyrannosapiensbonsai 4 года назад

Just great. Thank you!

@NonTwinBrothers 4 года назад

Awesome channel! Sent here from Maths Town

@serma3498 3 года назад

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

@ray017ray017 3 года назад

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

@DrakenFire 3 года назад

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

@justjay926 2 года назад

You explain so very well. Thankyou 👏👏👋

@Paulsinke Год назад

Thanks this is exactly what I needed

@gavinirwin2913 3 года назад

Finally something to hold my attention thank you

@mickwilson99 3 года назад

Brilliant pedagogy!

@tictacX1 4 месяца назад

Great video, thank you!

@MichaelZP1 4 года назад

Mandelbrot hero!, thank you.

@mikedaniels3009 2 года назад

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 2 года назад

Awesome video, thank you :)

@jk1422 3 года назад

Very good video! Thank you :)

@Stranger_In_The_Alps 3 года назад

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

@Snowflake_tv 4 года назад

thank you!

@rickyardo2944 9 месяцев назад

WOW! Thanks!

@sigriit7886 2 года назад

Very interesting and relaxing

@IamKavot 4 года назад

Good video. Thanks!

@ogb18gang62 2 года назад

Love this video

@TheEulex 2 года назад

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.

@danielduerst5067 4 года назад

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 4 года назад

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 .

@sawdustwoodchips 3 года назад

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

@naringrass 4 года назад

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.

@pvdguitars2951 Месяц назад

This must be my favorite video on fractals. I found a ‘weird’ butterfly effect for the Vesica Pisces surface area coefficient (=4/6Pi - 0.5xsqrt3). Approximately 1.22836969854889… It would be neat to see its behavior as c in the Mandelbrot iteration

@bergarteric5713 3 года назад

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 ..

@shable1436 3 года назад

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

@alexking1129 3 года назад

Fantastic video

@TheMathemagiciansGuild 3 года назад

Thank you

@GabrielsEpicLifeofGoals 4 года назад

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

@DrewNewmanEngineer 3 года назад

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.

@skybridg5705 Год назад

Start at 1 keep going. It never ends.

@Nick12_45 4 месяца назад

thx!

@BountyLPBontii 3 года назад

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

@jesseliverless9811 3 года назад

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

@BountyLPBontii 3 года назад

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

@DundG 3 года назад

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 3 года назад

@@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 3 года назад

@@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

@unbearifiedbear1885 3 года назад

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

@trumpetzmainia 4 года назад

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!

@hussainali9999 9 месяцев назад

Thanks

@RowynOfficial 4 года назад

Hi maths town!

@Dr.Pepper001 4 года назад

Clear as mud.

@NEMES1-S 3 года назад

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.

@fineepl 2 года назад

excelent

@Ivan_Nano Год назад

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.

@MiketheNerdRanger 3 года назад

That last one was friggin terrifying

@girogiro-vh5pz 4 месяца назад

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.

@aplimsollpunk2738 Год назад

🎵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...🎵

@FreeStyleCrhymeR 3 года назад

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.

@IqbalHamid Год назад

@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.

@EbbtideCheque 3 года назад

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. 😲🤩

@byugrad1024 7 месяцев назад

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.

@GZWA 3 года назад

feels like a Modern Vintage Gamer video

@EnPee91 Год назад

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

@leanderlle7602 Год назад

With which program can you recreate stuff like that?

@jeninaverse 5 месяцев назад

The poet and Mathematian Without Division.

@stanervin6108 4 года назад

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 4 года назад

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 4 года назад

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

@danielvieira8978 Год назад

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

@sorcerersofstone Год назад

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

@melvynbraithwaite8563 Год назад

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

@bitcoinzoomer9994 Год назад

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