The Mandelbrot Set - Numberphile

Famously beautiful, the Mandelbrot Set is all about complex numbers. Featuring Dr Holly Krieger from MIT.
More links & stuff in full description below ↓↓↓
The next part is on Numberphile2 at: • Filled Julia Set
Animation courtesy of team fresh. Check out more at: hd-fractals.com --- Music: Alan Stewart. Support him at bit.ly/1sdwTHF
More videos with Holly Krieger: bit.ly/HollyKrieger
Since this was filmed, Holly has become a mathematics Lecturer at the University of Cambridge and the Corfield Fellow at Murray Edwards College.
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[@AlanKey86]

Allegedly, when Benoit B. Mandelbrot used to be asked what the "B" in his name stood for, he would reply: _The B? It stands for Benoit B. Mandelbrot!_ Legend.

[@IxDarkxNinjaxI]

AlanKey86 awesome comment lol

[@JS_SN_UQAU]

+AlanKey86 So his name is Benoit Benoit Benoit Benoit Benoit...Mandelbrot Mandelbrot Mandelbrot!

[@Em-gj2sg]

+Jacob Scholte No the Benoit would go on forever

[@Tossphate]

oi you stole my joke!!

[@Em-gj2sg]

Matrix29bear But if you tried to say it you would just be saying "Benoit" forever

[@hd_inmemoriam]

I want her handwriting as a font.

[@TacomaPaul]

At about 2:55 she says, "1+1=2". I got that ! The rest... ? Yikes. Fascinating stuff.

[@nosuchthing8]

It's just a wild formula. Some points stop bouncing around, others don't. You color each spot based on how long it takes to settle down.

[@chappie3642]

@@nosuchthing8 yeah, that's way too dumbed down. You didn't explain what those numbers are and how you get them, and why they bounce around, and how we know if they "bounce around". It's not a simple concept at all that of the Mandelbrot set

[@nosuchthing8]

@@chappie3642 yes, true. I coded it up as a child on a very primitive computer, taking all night to generate one image. It's the butterfly effect, small differences in initial conditions giving huge results down the line, and results that appear random.

[@error.418]

@@chappie3642 you added nothing and only tried to take away.

[@chappie3642]

@@error.418 what do you mean?

[@michaelbauers8800]

Those super deep zooms of the set never get old. I especially love the zooms that don't move...they just descend. And I think, it's pretty amazing how much complexity you get in such a small number space

[@Infinite_Omniverse]

Math can be very beautiful... The Mandelbrot set proves this.

[@VFM89]

well...it's not the set itself being beautiful...but its border indeed is.

[@TheMarcosutra]

but isn't the visualisation technically just the mathematical symbols we use to write the function...?

[@bornalidadhara8214]

Deanna

[@DoctorMaxMoebius]

So does Dr. Krieger

[@chappie3642]

@@TheMarcosutra what do you mean with simbols

[@lollertoaster]

That presenter is very good at explaining. I love how she reiterate on the things that can be more difficult for some people.

[@Majestic469]

Lol

[@uniqueusername_]

Nice pun

[@PC_Simo]

@@uniqueusername_ Exactly! 👌🏻🎯😅👍🏻

[@ker0356]

complex, not difficult

[@tomolonotron]

I hear words, but I'm not understanding them

[@Crazylom]

That's how i feel most of a time

[@aidabit7554]

If you turn on subtitles @5:05 to 5:06 you see "[evil giggle]" lol

[@Hududding]

best part hahaha

[@zakusa9891]

its a evil laugh

[@aryakomal]

Lol

[@yasyasmarangoz3577]

xD

[@yasyasmarangoz3577]

@@zakusa9891 an

[@billlson]

I think we need more Holly on Numberphile

[@axianerve]

You know I suck ass at any kind of math, but for some reason I love watching these vids.

[@RedStefan]

f(me)=I still don't get it.

[@cbr7170]

RedStefan This deserves way more likes haha nice one.

[@mightyemperorshaokahn7135]

So what you are saying should be this f(me)⇌I still don't get it mayby now you get it

[@kaustubha7371]

Lol

[@abdullahal-ahmati5030]

There is a function f(x) = x^2 + c. You calculate function on f(0), then on f(f(0)), then on f(f(f(0))), and so on forever. Either the result grows towards infinity, or the result remains in some small range. If it grows forever for a certain number c (which is a complex number, so it is a point on a plane), then it isn't part of the set, otherwise it is part of the set.

[@DailyPoptarts]

In an easier sense It’s a complex series of numbers that when geometrically plotted, and with different colors representing different iterations, you can essentially see a very esthetically pleasing mathematically correct picture.

[@tacchinotacchi]

Isn't there any video from this girl outside of Mandelbrot, julia set, and -7/4? Her voice is so relaxing

[@pampam7093]

[@soulcheckw2]

Maths ASMR :)

[@dalmacietis]

+Find 'N' Frag Well, there is a one-hour lecture on the dynamical Andre-Oort conjecture ;)

[@tugger]

soothing math

[@gwenmcardle3866]

i did a presentation on fractals last year and the mandelbrot set was my big finale. this video helped me a ton! i actually kind of understand it now, but my classmates didn't. im not the best teacher.

[@haidarsaab7588]

hi do u have the presentation?

[@SlyMaelstrom]

I'm too much of an iterate to understand this...

[@THEEditor-in-Chief]

I actually lol'd at this.

[@joemanses]

yeah, I can barely write in recursive

[@jamesmcginn6291]

Me too.

[@charlesklimko492]

Me, two.

[@denelson83]

Charles Klimko Me, π.

[@michaelk9080]

I love when Dr. Holly does anything with Numberphile. She doesn't sound like she's droneing, but rather is excited to teach and loves that which she is teaching. She's the type of teacher I bet more people wished they had had growing up when teaching Math or other subjects. I'll never understand the teachers that don't have passion for what they are teaching.

[@noradosmith]

I feel stupid for only just understanding this. And I feel doubly stupid for knowing they're trying to dumb it down for people lik me to understand. I like the idea though, of being on the cusp between 'blowing up' and not 'blowing up'. Pretty much summarised my brain watching this.

[@vccancerkill5047]

John Doe you ain't gotta lie to kick it we know you don't get.

[@rezilla1]

It's at a pretty normal difficulty level for numberphile IMO. After wikipedia-ing complex numbers and thinking about it for a bit, I now get this entire video. It feels really great to have finally learned this too.

[@hweigel528]

Someone asked, "why is two the bound after which everything blows up?", which is a very good question. The reason becomes more intuitive if you know a few important properties of complex numbers, namely that |u*v| = |u|*|v| for all complex numbers u and v, and that |u + v| >= |u| - |v| for all complex numbers u and v. Using these two properties, consider the magnitude of a given number going through this procedure. Given that z has magnitude |z|, f(z) = z^2 + c has magnitude |f(x)| = |z^2 + c| >= |z^2| - |c| = |z|^2 - |c|. Now we can consider a function based on some |c| >2. Clearly f(0) = 0^2 + c = c, and so |f(0)| = |c| > 2. Next, f(c) = c^2 + c = c*(c+1), and so |f(c)| = |c|*|c+1|, and since |c|>2, |c+1| >=|c|-|1|>1. Therefore |f(c)|=|c|*|c+1|>|c|. Now, assume that we have done this procedure enough times to reach some arbitrary number z, such that |z| > |c| > 2. (We already know that we reach a number with this property after two steps). |f(z)| = |z^2 + c| >= |z|^2 - |c| > |z|^2 - |z| = |z|*(|z|- 1). Since |z| > 2, |z| - 1 > 1, and therefore |f(z)| > |z|*(|z| - 1) > |z|. Since this is true FOR ALL |z| > |c|, we know that |z| < |f(z)| < |f(f(z))| < |f(f(f(z)))|

[@barrytone6581]

Thanks

[@juntjoonunya9216]

Um no

[@yogeshkapila1179]

This comment completes the missing explanation! Thanks :)

[@64rkerner]

Probably my favorite Numberphile ever. It's certainly amazing how such a simple function can lead to the most wonderful art... I've never been a fan of science fiction nor art for only just the sake. Rule fact here is so much more beautiful and amazing because it is absolutely so very honest to the core. Dr Krieger, I very much appreciate your patient explanation. Thanks!

[@will4432]

Who knew you could mathematically calculate an LSD trip?

[@juntjoonunya9216]

I'm going back on a "trip" now with a totally different pov and expectations.

[@halonothing1]

You can describe anything with math.

[@victorl.6128]

Try watching at .5 speed with head phones. Cool

[@alcirfigueroa3712]

Lol that's what I thought

[@tomauberwenig2116]

More like DMT, I think^^.

[@manuel8179]

I love how piano and harp music starts when zooming the Mandelbrot heart

[@YipYapYoup]

When I started reading the description, I thought "Famously beautiful" was describing the mathematician.

[@user-xd7eq9ot5g]

Bro imagine commenting on a comment from nine years ago... that would be crazy.......

[@ralfoide]

Keeping watching this particular video over the years and it's still the best Mandelbrot set explanation I've seen to date. Dr. Krieger is remarkable, and the series of Numberphile videos on Mandelbrot with Dr. Krieger are all extremely clear and interesting. Would be nice to see Dr. Krieger return to lecture us on whether the Mandelbrot set is local connected and what it means if it is.

[@henrikwannheden7114]

This is perhaps the most enlightened description of what the Mandelbrot set is that I've ever heard, and I've been listening to explanations for at least 25 years. Very good!

[@numberphile]

The next part of this video has been posted over on Numberphile2 - kzbin.info/www/bejne/pXTOgmqNgJypq7s

[@BOOOZB]

+Numberphile Quoting Mandelbrot , about the inventor of fractals is a scam ! This man has only stolen the position of the creator of this part of geometric art . As any thief of intellectual property , he is obliged to erase any natural occurrence of the actual inventor's name . This true inventor was Helge Von Koch , a Swedish mathematician ( # 1906), and the first fractal known was " the fractal of Koch ". That's why Mandelbrot imposed the name " snow flake " to this first known fractal , whose genuine name was " the flake of Koch " . Doing so , he erased the name of this annoying guy that he was trying to rob . Later on, Mandelbrot exerted a real terror pressure over any mathematical publication ,and threatened anyone that dared talk about fractals without quoting him as the almighty father . Render to Cesar ... Mandelbrot invented the word " fractal" . But no more .

[@NicolasDiazWahl]

+Numberphile What is the song used at the beginning?

[@Demonfire39]

+Nicolas Diaz-Wahl "Trypophobic" by Alan Stewart.

[@rishabhbhardwaj2873]

Numberphile Hey numberphile here is a CHALLENGE solve this summation - sum arctan(m/n) from m=1 ,n =1 to m=10 ,n=10 .

[@Forgan_Mreeman]

shame on me for thinking I would understand this. I'll go back to cat videos :(

[@SomeRandomFellow]

+Haukenslush best profile pic ever

[@Imtheonlyoneinmymind]

Michael Bauers Thanks for taking the time to explain that Michael. I'm still a bit sketchy as to why you would do all this though :)

[@michaelbauers8800]

Because it's interesting

[@oz_jones]

+Internal Dialogue because some men want to see the world learn

[@Imtheonlyoneinmymind]

***** Haha!

[@TheJuan72]

why I didn't have a teacher like her ?

[@TheJuan72]

only at MIT ?

[@lucaitaliano5865]

idk

[@UthacalthingTymbrimi]

The first thing I ever downloaded from the Internet was a Mandelbrot Set generator, in 1992. I've been fascinated ever since.

[@ksphysicist]

This is by far one of my favorite mathematics videos on KZbin. Fantastic explanation, I will refer my students here when they want a good understandable explanation of the Mandelbrot set.

[@Ovni121]

What's the middle name of Benoit B. Mandelbrot ? A: Benoit B. Mandelbrot

[@sufikhoirunisanisa5672]

Yeah you're right. It's Benoit (Benoit (Benoit (Benoit B. Mandelbrot) Mandelbrot) Mandelbrot) Mandelbrot

[@Max-wy3qo]

can we take a moment to appreciate her writing?

[@rippspeck]

It must drive her British colleagues mad that she says "zee" instead of "zed", haha. Great video as usual.

[@sanjaytumati]

Thank you Dr. Krieger. This was so easy to follow. You made what was intimidating, friendly. You have a gift.

[@KarlFFF]

for once a club that accepts zeros (7:40)

[@The-Urban-Goose]

"Mandelbrot" just means "almond-bread" in German

[@SavageGreywolf]

it's Ashkenazi biscotti

[@netiosys4677]

almond bread is mandelbrød in danish too

[@minnarew]

jow didnt i realize this... (almon bread in danish is mandelbrød)

[@Cygnus0lor]

It was the mathematician's last name...

[@redcoat4ever]

And “Brot” is pronounced “Broat” and not “Brought”

[@reallyWyrd]

Yes, more, please! I've read and seen stuff on the Mandelbrot set numerous times, and I understand all about the iterative nature, yet still this video was better explaining it than any of the previous attempts.

[@djaimes5]

Happy Birthday, Benoit B. Mandelbrot!

[@emchartreuse]

Wow, that was such a great explanation, thank you! Algebra is my highest understanding of math and I was able to understand everything you said. I'm looking forward to seeing videos on the other sets.

[@PaigeDWinter]

I'm a fractal artist. Thank you for this post!

[@TessaGallant]

Dr Holly Krieger, fantastic explanation, great teacher! The questioning back in forth in the Numberphile videos is a great learning tool. Thanks for posting.

[@oneofthesixbillion]

Thanks!, after a lifetime of loving the images that's the most I've understood them. I wish I could get an explanation with this much clarity of the IFS fractals that I'm also entranced and fascinated with.

[@gaius_enceladus]

Famously beautiful, Dr Holly Krieger from MIT. Featuring the Mandelbrot Set.

[@ChrisRedfield1]

Probably not, but a lot do.

[@prajwaldeepkamble6617]

Professor at Cambridge

[@sth128]

Where can I find the fractal animation used in this video? I need something to compliment my marij... I mean uh, I want to uh, study math.

[@KabochaOu]

I love the way she underlines her words leaving space for the descenders on the letters.

[@p0t4t0nastick]

This concept really is beautiful. Made me fall into a calm, peaceful sleep in the end. Simple and great depiction of the same !!! Thanks Holly

[@keineangabe8993]

I'm impressed. It's the first time in these videos i see "i" introduced the correct way as a number with the property of i^2 = -1 and not just the squareroot of -1 (which is incorrect)

[@ganifraterdogan1062]

Why is it incorrect?

[@Toroidal_Vortex]

@@ganifraterdogan1062 Since i is technically both plus and minus the square root of -1. That's my guess. So i = -sqrt(-1) and i = +sqrt(-1).

[@AnimMouse]

So i is ±√-1

[@Limosethe]

Zooming into the mandlebrot set will be like exploring the world I see on psychedelics, but on the internet

[@hd-fractals]

Excellent explanation of the Mandelbrot set :) I cant wait to see the next video!

[@elliottmcollins]

Awesome. I have always wondered this and this was such a satisfying explanation. What's still lost on me is why that set would have such crazy fractal patters.

[@LordMarcus]

Numberphile -- Question: say I take a single number line, a one dimensional continuum of numbers, such as the x-axis of a graph. If I use complex numbers, it stands to reason that on this number line is another axis perpendicular to the x-axis at 0 for the complex parts of x -- in effect, our one-dimensional number line is two-dimensional. Say then that I take this complex x-plane and add, perpendicular to it at 0, a y-axis, so now my graph is three dimensional -- a complex x-plane and a real y-axis. If I then extend the y-axis to include the complex numbers by adding yet another axis perpendicular to the y-axis AND the complex-x plane to represent the y-axis' complex part, I now have a four-dimensional system with only two variables, x and y. Can I do equations in four-dimensional space using this system?

[@DragonAurora]

My mind is officially blown....I always knew about Mandelbrot sets, but I never knew the logic behind them.

[@krakenmetzger]

For someone named Dr. Kreiger, this person seems remarkably sane and competent

[@chmd22]

Not sure what is cooler, the Mandelbrot set or that neat handwriting. Amazing!

[@soapboychris]

as much as i love the maths here, i lost it when she gave me that look at 7:21

[@juntjoonunya9216]

It's not often something so cute blows my mind away

[@Juan-dc6yf]

5:56 is better

[@oldschoolman1444]

Smart and beautiful, a truly rare combination ! =)

[@borderingonnothing]

Typical man-like behavior. Always focusing on women’s looks, even in regards to something completely unrelated.

[@senoreverything6366]

@@borderingonnothing I assume you're female then?

[@Octopossible]

All the examples used are on the x axis, the real axis. I'd love to see you work out a few iterations off the axis, in the imaginary domain. I dont understand that part. Really weird how primes show up so much, how does that work off the axis? Definitely one of the harder numberphile videos to grasp.

[@brunovaz]

Second that. I don't understand how imaginary numbers come into play there, and why they're relevant

[@juanausensi499]

@@brunovaz Let's say c is 1+2i. You start at zero and the result is 0+c, so 1+2i. You now plug the result into the function again, so you need to calculate (1+2i)^2+1+2i. Operate as usual, just remember that i^2=-1

[@friedyamms]

Excellent depiction of the concept. Very easy to follow. I came for a refresher on the topic and that's exactly what I got.

[@MarkWladika]

This was one of the best descriptions of the Mandelbrot set I've ever heard, Benoit would be proud, huzzah Dr Krieger.

[@Budgieman67]

Brady, you tease! Give us more Mandelbrot Now!

[@FlesHBoX]

You guys should do a computerphile on how this is actually plotted programatically on the computer!

[@alexthi]

It's really not complicated : for each pixel, it takes the corresponding complex number and iterates the z²+c thing several times, breaking out whenever the magnitude is greater than 2. If it gets to the end of the loop it's probably in the set, so it considers the pixel is. Drawing it with a gradient is a bit more complicated.

[@FlesHBoX]

alexthi94 Dangit, don't bring logic and what not into this, I want another video about it! :p

[@elliottmcollins]

I was just wondering that! One could set a computer to computing an incredibly fine grid, but given the crazy zooming effects they have, there must be some efficient way of doing this. And computerphile is too dumbed down at the moment. Some proper computer science would be great.

[@Azimuth1]

I saw these colourful images when I was a kid and just always assumed that what was behind it was some unbelievably complicated maths that I had no hope of understanding. Now having watched this video along with a little reading about complex numbers, I see that it's actually quite simple and I can now properly appreciate how interesting it is. Thanks!

[@kidkat279]

I'm so happy I watched this. It makes so much sense now. Thank you!

[@chrisofnottingham]

Famously beautiful indeed

[@dkamm65]

"A little messy?" The Mandelbrot Set of complex numbers is "a little messy!" This is chaos!

[@user-xd7eq9ot5g]

Literally!

[@jakel4901]

Thanks so much for all your videos Brady.

[@ZorkFox]

I promise this isn't the only thing I took away from your video, but Dr. Krieger's handwriting is lovely! It was a pleasure to watch.

[@ThioJoe]

Instantly thought of the song by Jonathan Coulton.

[@VejmR]

Oh

[@rahulzagade3778]

What thio joe with 2 comment reply's that sick and sad .....

[@rahulzagade3778]

And 54 likes is also interesting

[@spicypeanutbutteronion9943]

Even though I'm not currently in school, I learn something new every day.

[@MrTurbo_]

learning at school? HA, who does that these days...

[@BlissfulTortoise]

its a phase

[@user-xd7eq9ot5g]

@@MrTurbo_ ???

[@MrTurbo_]

@@user-xd7eq9ot5g Man, that comment is 7 years old, anyways, can still confirm i learned practically nothing useful in school, absolute waste of 15 years of my life, everything i use in my life these days is stuff i thought my self either at work or in my free time

[@roseclearwater9904]

This is the best video I’ve seen of this! I feel like I could actually understand this beautiful piece of math now THANK YOU 😭🙌🌟

[@zan5051]

I really enjoyed this video! I've been wanting a video explanation for what the Mandelbrot set is for a long time

[@mattv2099]

more digits than there are elementary particles in the universe?

[@lin4cba]

Very beautiful hand writing. ...and mathematician ;)

[@ishaim2]

Dr. Krieger, glad to see you're doing well for yourself. You once tutored me at UIC in remedial math courses and told me I had to be more "methodical", although it's a shot in the dark if you remember. I can write programs that multiply matrices, now. Cheers.

[@belas961]

Thank you so much for this! I have to do a project on fractals with emphasis on the Mandelbrot set and it was really confusing but this helped a looooot.

[@joshinils]

well, sure now i know what it represents, but how do i get it? for instance if i would not know how it looks like what do i need to do to find this particular structure?

[@Zhaggysfaction]

That was some trip... I wonder when I land...

[@jondo7680]

It's so beautiful, I'm not surprised to hear yet again that size matters.

[@Demonfire39]

The music is "Trypophobic" by Alan Stewart.

[@TheMarkoSeke]

Beautiful handwriting.

[@bobbyp21]

I love Dr. Holly Krieger. Yup, it's true.

[@helenamath]

Wonderful video that clearly outlines the creation of the Mandelbrot Set.

[@LawrenceDuffy477]

I'm a math nerd and I NEVER knew this. I thought I did. Thanks for being an awesome teacher. "Famously beautiful" YOU !!!

[@Intrafacial86]

Why must it be less than "2"? Why not "3" or "9834953479" or whatever? Do _you_ choose the boundary, or is that just a property of the set?

[@jamesbacon6825]

Intrafacial86 if you let c=-2, you will get z0=0, z1=-2, z2=2, z3=2... because 2^2-2=2. for c= any number x, in between 0 and -2, x^+x

[@guilemaigre14]

+James Bacon well ok, but why the circle of radius 2 ? or is that just a depiction of the "-2" and looks great and doesn't mean anyting... meaning that 2 is just the maximal norm accessible for it to work ?

[@xario2007]

+Guillaume Lemaigre It doesn't matter which bound you choose, as long as it's at least 2: If it gets bigger than 2, it will get bigger as any number.

[@denelson83]

+James Bacon That makes -2 a "Misiurewicz point", a point where the values of z after a certain iteration are precisely periodic, but that period does not include the c value itself.

[@rubenmejia9020]

Oh my god, she is beautiful.

[@lawrenceworrell591]

@@stage8790 It's true, though.

[@theonlycaulfield]

@@stage8790Notice that comment was made five years ago. Whoever made the comment likely would not have even known the ridiculous term "simp" had you called him that five years ago.

[@waynewalls5033]

@@stage8790 virgin

[@emmacovey6169]

incredible job explaining complex math in a simple and comprehensible way!

[@nxs7226]

Super interesting . Please, please, please more videos about fractals!

[@kanabalize]

she has both the brain and the beauty...

[@beard6329]

Lol no beauty

[@alcapope]

very interesting video, but i miss the videos brady used to do on numberphile, about primes and conjectures (specifically james grimes and matt parker; pi, grahams number, abc conjecture, enigma machine etc). yes they were simple, short videos, but they were fun and didn't make my head hurt! any chance for some more 'fun' 'easily accessible' videos mixed in with these newer, somewhat more advanced topics?

[@UCvow2TUIH0d2Ax2vik9ILzg]

Every topic becomes easily accessible when you put some effort into understanding it. People have been asking for more complicated topics from him for a very long time now. I don't mind less complicated topics too, though.

[@siddhantshekhar]

Is it just me or the scratching sound of the marker on the brown paper gives you goosebumps as well (and not the good kind, the kind you get when someone runs nails over a blackboard)...

[@SamMcinturff]

Understanding what it is makes the set even more beautiful.

[@EviIDuck]

where is part 2? I've been waiting for a month!

[@numberphile]

EviIDuck here you go (I put it on Numberphile2): kzbin.info/www/bejne/pXTOgmqNgJypq7s

[@NicolasDiazWahl]

+Numberphile What's the song? used at the beginning

[@AdrianSchray]

So Beautiful.... The Mandelbrot Set looks awesome too XD

[@anonkiddo]

I guess that's why kids from MIT end up doing great stuff like this. Beautiful fractals taught by beautiful frectels.

[@missjennbo]

Best explanation I have ever seen! Thank you very much!

[@Necroskull388]

The first comment I see is going to be about the woman's appearance... Yep, the first comment I saw was about the woman's appearance.

[@renardmigrant]

Dagda Mor was it your own comment? Because yours is the first appearance related comment I've seen.

[@RapiDEraZeR]

Martin Gardner okay,i scrolled down and it's actually true LOL. so now i am free to say that my blood flow went from my head to my pants just listening

[@MysteryMan852]

Your reply is about the woman's appearance.

[@jhyland87]

Understandably so.

[@twolostsoulsswimminginafis4795]

Kinda sad

[@karlboud88]

Fascinating :) Can anyone provide a screensaver or a program displaying the Mandelbrot set? I never thought learning math would be fun, I mean I always liked it but I can spend all day watching these videos! If they had a new video every day I'd have a phd or a degree in mathematics! :p

[@mamoonblue]

.

[@stevefrandsen7897]

Good instruction and music too. Well done. Thank you.

[@1001themaster]

How do they all have such beautiful handwriting!?!

[@TheSentientCloud]

SHADDUP, I KNOW I'M BEAUTIFUL.

[@HowToBasic]

She looks like Jenna Marbles

[@YZOBEL5000]

is how to basic smart ???

[@DraithVicious]

+HowToBasic Wow! I don't know what's crazier right now. The fact that I just came here to get a better understanding of what a Mandelbrot Set was only to have my brain bombarded with numbers far beyond the scope of my comprehension or that HowToBasic was here. I have to know what brought you here. Please tell me!

[@DraithVicious]

+MErCH Right? My mind has been blown twice from this one video. First by the numbers and now that somehow HowToBAsic ended up on this video. He doesn't strike me as the type of person who would be watching this video for any reason. I need to leave the internet and recuperate for a bit. My brain hurts.

[@faizanm1563]

HowToBasic dafuq are you doing here????

[@thegamingcat9212]

Like actually why are you here

[@johnhoens]

You make a nice presentation. I'll try to watch your other math lessons. Thanks!

[@MrDemented669]

it's a visual expression of chaos and is amazing