The 4D Julia Set

[@dtimer319]

Dude I thought this would be from someone with hundreds of thousands of subs. This is a top tier video.

[@verymuchtom]

Thank you so much, that’s incredibly encouraging to hear! Who knows, maybe I’ll crack the one thousand at some point. I’m happy you enjoyed it :)

[@4m3y4]

@@verymuchtomthis video is very high quality, you deserve much more subscribers

[@robertosutrisno8604]

​@@verymuchtomroad to 10k subs 🎉

[@diggigamefr2156]

Real

[@AZALI00013]

this is insanely high quality what :0 I'm shocked this doesn't have more views !!

[@kaderen8461]

you know what you being here isnt that surprising

[@verymuchtom]

Thank you!!! It's insane seeing you here tho omg!! :D

[@Han-b5o3p]

whoa it's azali

[@kitansan]

Hi azali

[@A2music]

last place i'd expect to see you azali, so cool

[@stewartcopland7676]

unironically one of the best introductions to quaternions and why they're needed

[@theeyeofomnipotent]

How about visualizing julia set with a p-adic number system hahaha

[@agustinyavar525]

The algorithm has blessed me with this gem, as a programming student I'll enjoy trying this out myself, amazing vid

[@verymuchtom]

Happy coding! I can strongly recommand the blog by Inigo Quilez mentioned in the sources for guidance :)

[@tamarpeer261]

I'm all for class conscious mathematics

[@kwinammartinez7746]

i cant believe a 7th grader did all this while bored in math class!

[@burgerbobbelcher]

The embedded rant about video games, mmmm, so good.

[@verymuchtom]

Ha, thanks mate! Yeah there were some game companies worker layoffs whilst I was writing the script and I made the mistake of reading some beyond stupid takes online about it, so I needed to vent a bit :D

[@ancientfish6872]

Now's the time on Sprockets when we dance.

[@ernestolombardo5811]

"Jesus Christ what am I doing with my [effing] life?" I'll tell you what you're doing: 1) Something that you love. 2) Wading in the beach of a mathematical sea, towards future insights into areas of knowledge that we cannot yet even begin to glimpse, the way imaginary numbers were conceived a century and a half before it was realized they perfectly described subatomic particle spin. These mathematically rigorous visualizations are unique, their reliance on computational power and programming skills to be created make them something new in the human experience, which give them an almost mystical quality.

[@zzink]

Thanks for the video i had been doing a bunch of reading about this recently but this has everything i was looking for all in one place!

[@verymuchtom]

Glad I could help :) Yeah before making this video I was also looking for something like this and was surprised there wasn’t already some video on the topic 😅

[@somebodyhere3160]

wow this is is really high quality from such a small channel, loved the fact that you implemented your Julia Set visualization in Julia

[@verymuchtom]

thank you! implementing it in julia was one half using the language i know best and one half committing to the bit :D

[@onoff5815]

Information dense and full of good entertainment. A real joy to watch through!

[@strahinjasubotica]

Well i can say with almost utmost certainty i have never before watched a video with this much interest while understanding next to literally nothing. I am truly humbled at the amount of knowledge i do not poses and yet your enthusiasm and effort to explain something completely otherworldly to me made me want to watch and listen to what you had to say. From one student to another you have my applause and admiration. Keep up the amazing work!

[@endersteph]

golden video, seriously the quality is insane please have millions of views

[@dragoon347]

Brilliant video! This actually helps me so much with how to image n-dimensional vector spaces. Keep up the awesome work

[@HaiderAli-wm6mq]

This is fuking insane. definitely one of the top 10 KZbin videos imo. You break down complex topics and make them easily understandable and you dont assume that we know anything. Thank you for giving us this beautiful video.

[@algorithminc.8850]

Beautiful work - takes me back to the early/mid 80's. It was a fun time for fractals, dynamics simulation, computer graphics and the like. Alan Norton at IBM sent to me some of his really great papers. I think his work was later on the cover of Smithsonian. I could never find it, but there was a jazz/music video that included quaternions in the mid 80's (on VH1, I think). It was fun to build ray tracers and renderers for such back then, but difficult (given the level of computational power needed). Thanks for the video ... I look forward to scoping your channel. Subscribed. Over the next couple of months I'll see, if I can find some of my old work from back then - if any interest ... might spark a thought for you about strange attractors or the like. Cheers, Christopher

[@verymuchtom]

Christopher, thank you so much! I can only imagine the feeling of discovery you guys back then must have had when writing those renderers. I recon you must have been among the first people to do so that’s amazing! And what a great offer, I would love to see some of the projects from back then, I’ll probably have to learn some C/Assembly/Fortran to understand what’s going on, but so be it. :)

[@sive-natura]

a random video a decade and a half ago that I saw on a random website I had stumbled upon was that of a Quaternionic Julia set, it made me fall in love with mathematics, in about a weeks time I am defending my doctorates thesis at the French ENS ULM on Iwasawa theory, what a nostalgic video this was, thank you!

[@positivity3311]

this deserves so much more

[@yesinsky552]

this has to be seen by everyone, the whole video is perfection. The subject of the video is fascinating, the contents are educational and the video was produced in an captivating manner such that it is fun to watch if i had to grade this in right to grade this id give it whatever highest score there is.

[@herpsenderpsen]

Those have got to be the most beautiful screensavers i've ever seen

[@bennyloodts5497]

Magic stuff! Interesting and beautiful.

[@rlyehpolaris4456]

this is amazing man!! thanks for doing this video it was so interesting

[@verymuchtom]

Gladly dude! Happy that you enjoyed it :)

[@primenumberbuster404]

Nah, this channel deserves more support! W channel.

[@thejusdeau]

Absolutely amazing ! Love the math, the graphics and the beauty of it

[@stanky-cheese-factorio]

Fantastic Video. Dont get so see such beautiful math animations like these very often. Very cool watch and excellent explanation

[@loveless-savage]

I actually started working on a real-time rendering program for the Julia set in high school. I tried to do it in OpenGL like a maniac, but I ran into a buffer issue and gave up a couple years ago. So needless to say, 4D raymarching is pretty close to my heart. This is really clean work, great rendering and solid explanations. Easiest 'subscribe' ever

[@diggigamefr2156]

Animation of the fractals would be insane wallpapers

[@incription]

very nice to see someone go through the same journey I went through after stumbling across inigo quilez' articles. It took me a while to understand the distance field derivations but after you realise how it works your mind is finally opened up to a lot of 3d fractal ideas

[@FunctionallyLiteratePerson]

18:20 had me laughing (out of depression)

[@kruksog]

Hope you keep making stuff. This was awesome.

[@menaceinsane]

i am very stupid - but understood quiet a bit (especially the sneaky jokes hihi) - very nice (person I totally don't know!) the music fits very nicely too! excited for more!!

[@verymuchtom]

man thanks so much person i don’t know either!! that’s so crazy we should collaborate on a music video or something :)

[@nsgcommando2861]

Just discovered your channel and I can say I will certainly enjoy watching all your videos. Subscribed

[@Mysteries-revealed]

With infinite regression and symmetry based shapes, one can speculate that 4d and 5d are real

[@DaxSudo]

Most underrated channel so far this year.

[@lucamorgese7475]

Beautiful video, it must’ve taken ages to do all this work, you should be proud of yourself

[@synterr]

Very cool! Now all we need is to put the model into a slicer and 3D print it! :)

[@verymuchtom]

That would be incredibly cool, I can alredy see my future getting in trouble for sneaking into my facultiy building at night to secretly use their 3D printer :D

[@lis6502]

i should wake up early tomorrow, yet i was unable to leave the video. Seeing at the end you have 358 subscribers and video gaining 9 likes in 90 minutes was more of a surprise than the actual content. i don't get too much of mathematics, yet complex numbers are somewhat comprehensible for me. I've got lost in 3d projection methods but overall material was very interesting, carefully balancing on the edge of boring math, bit of history, stunning visuals and clear explanation of subject's matter. Well, seems you've gained 359th subscriber ;)

[@verymuchtom]

thanks mate, don’t forget my horrendous attempts at humor, they are another essential part :D

[@qwkutiub]

Loved the subject, mood, aesthetics… A pleasure to watch, and inspirational. Hated to type the references on google though 😝 Please, put them in the description :-)

[@alfredwillins]

Amazing work. Thank you!

[@onnio7998]

Holy shit! I've been curious about the Quaternion Julia for like 2 years now, but haven't actually tried to implement it. You explained the process well, and the results... Beautiful! In the future you could try experimenting with orbit traps as well. I remember some of Inigo Quilez' images being amazing.

[@YoutubSosetXui]

Thank you very much ❤Clear and interesting explanation and beautiful images.

[@raphaelgonzales3481]

It looks amazing ! Good Job !

[@beaverbuoy3011]

I love this

[@Palisade5810]

My earphones back in the day.

[@math6844]

This channel rules oh my god, i'm sharing this

[@ETBCOR]

What a great high quality vid. Happy to be sub #716! :D

[@maxpetschack3342]

Goddamn that rendering is cool

[@kristijanpete4473]

your videos are SO GOOD

[@verymuchtom]

Thank you!! :)

[@fossil98]

How do I buy stonks in a youtube channel?i

[@mategido]

That was fascinating

[@DelandaBaudLacanian]

"What is i times j"? *the beat drops*

[@mukinga]

julia (programming language) mentioned!!

[@verymuchtom]

It's blazing fast and comprehendible for fools like me :D Also love the creature in your profile picture

[@zcu8404]

wow, this is great!

[@elvisbrum1]

you deserve a million subscribers!

[@ninjajakee]

Youre slowly but surely becoming my replacement for Sebastian Lague! Also not too surprised that you know acerola haha

[@joshuabrown4952]

I think the algorithm thinks I'm smart enough to watch this... so the machines won't be taking over anytime soon...

[@TestAccountUndetailed-dd7vp]

ah yes i too also go into pure math when playing with calculators then proceed to explain fractals

[@kevinpatricsmith]

Phenomenal. Exzellent.

[@bobbilly8821]

love this

[@beaverbuoy3011]

Very nice!

[@WengineerKimProductions]

spectacular!

[@mrhassell]

Everything lives in 4 dimensions! The 4th dimension is time. You spanner.

[@verymuchtom]

There’s a difference between the 3+1 physical dimensions as described by general relativity and the dimensionality of some vector space. You have a bachelors in computer science, you should know this.

[@mrhassell]

@@verymuchtom while spatial space corresponds to intuitive three dimensions, vector spaces can have different dimensions depending on the number of independent basis vectors required. Dimensionality extends beyond physical experience and plays a role in various mathematical and scientific contexts including the julia set, which exists in 3+1 space already. Happy?

[@GrifGrey]

thanks mrhassell

[@modolief]

Bravo!!

[@drdca8263]

From the title I imagined this was going to be about non-coordinate-aligned slices of the 4D fractal given by all the different Julia sets (or whatever the appropriate term is) Like, the set {(z,c) in C^2 | iteration starting at z_0=z , with c being added each time, doesn’t blow up to infinity} and taking 3D slices of this which aren’t given by fixing one of the 4 coordinates, but instead by fixing a particular linear combination of the 4 coordinates.

[@verymuchtom]

Oh wow that's actually also a great idea! There's a video out there by Damnimp which is somewhat similar to that, in case you don't know that one yet :)

[@drdca8263]

@@verymuchtom Thanks! I was not aware. I’ll check it out

[@MachinaInfinitum]

Insane good job here ! Come have a look at what we do with other 4D Fractals :)

[@anantagarwal8279]

subscribers gonna go trough the roof soon great vedio!!

[@verymuchtom]

here’s to hoping🤞

[@4sk3wrecital]

loved this, your style is very similar to the youtuber named MAKiT

[@6Azamorn9]

This was insanely fucking cool!

[@srb20012001]

I wonder if something like a 4th dimension could explain Quantum Entanglement.

[@AB-gf4ue]

KZbin was severely lacking til this video was uploaded

[@verymuchtom]

Incredible compliment, thank you some so much mate! :)

[@duytdl]

Amazing. Subbed

[@borisadimov]

❤

[@sirjersey]

Love the Video!

[@PravdaSeed]

🧞 Stop making Another masterpiece.

[@decare696]

36:50 would be such a good screensaver!!!

[@verymuchtom]

true!! i might need to look into how to make my own screensaver :D

[@1997oreosFTW]

“Based on the script written by: take a wild guess” 😅

[@Ρενκο]

The quality is phenomenal! This deserves 1M views, considering how much effort should've been put to produce such a good video. But I think that your speech pacing needs to be slowed a little bit, as well as a bit of intonation would be required to make your voice less monotone and robotic.

[@verymuchtom]

thanks for the advice :D especially at the beginning i rush a bit trough the script, mainly because of nerves. but i know that i’ll need to work on that. in terms of intonation that’s just my normal speaking/reading voice but i’ll might put some more excitement into it in the future.

[@Ρενκο]

@@verymuchtom yeah, to be honest it's already alright and good, but if you compare it to 3b1b, Vsauce and other already well established science KZbin creators, you'll notice that they always try to use emotion in their voices. To be frank, I also have a very monotonous voice, when I'm serious, read or just forget to use emotion - but my experience in public speaking, teaching and debates have made me realize the power of intonation and emotion in one's voice. It literally changes everything. Oh, though you can make this your charm, as junferno did - he cracks really fun jokes with a deadpun facial expression while also having a very monotonous voice. But, at the same time he doesn't rush as much as you do and let's some space in between, to make jokes a little more emphasized, as well as important information. Anyways, I hope you continue your career as a content creator, these kinda long videos are the best - this is how science videos should be imho. And anyhow, you're doing an amazing job, thanks for uploading this.

[@__christopher__]

Great video! Just one thing I noticed: You used "angels" in your code comments. You probably meant "angles".

[@thedoublek4816]

You only needed to mention the "p-q-formula" for me to know that you're most likely German :)

[@verymuchtom]

I hummed the dorfuchs song whilst typing out the equation in latex :D

[@positivity3311]

amazing!

[@landonkryger]

I love the 4D julia set too. My only complaint is that it these images have too much symmetry. I wonder if there's more interesting 3d slices you could take, or if you could vary the `c` point with the camera's uv coordinates. That might make it more similar to the 4d mandelbrot which is even more symmetric though.

[@verymuchtom]

Hi there :) the 4D julia set is, to my knowledge, point symmetrical around the origin, so you’ll inevitably end up with a pretty symmetrical image. However the idea with the UV coordinates could actually be a pretty cool solution for that, i might have to try that out at some point 🤔

[@whosmaya666]

just started watching and I'm already really enjoying it! seems like a great video. small note - at 2:20, you talk about how x²-1 doesn't depend on the sign of x, but the equation on screen (x²-1=x) is sign dependent, which confused me for a bit. imo (subjective! you can disagree) you could've been more explicit about why the sign doesn't matter here (the way I understand it, your point is that -x₁ is still taken to the same fixpoint x₁ despite not [necessarily] being a fixpoint itself)

[@verymuchtom]

Thanks for the feedback, I really appriciate that. Yes you are correct, I maybe rushed a bit through that part of the video. The solution isn't a fixpoint in any strict sense for exactly your line of reasoning. What I wanted to convey, and which maybe I could've explained better is that -x₁ also converges to a true fixpoint in a finite amount of steps, in this case just the one. This is opposed to the values in between where the convergence is asymptotic.

[@galoomba5559]

That part confused me too.

[@whosmaya666]

@@verymuchtom thank you:) I think that was basically the understanding I came to after thinking about it, so at least it wasn't too confusing❤️

[@ulyssesk7325]

how can a liumes diverge by 8% from its approiximation ?

[@crumblinggolem6327]

Very cool video. Unless I misunderstood, the final result is a "slice" of the actual 4d set, so in addition to moving the parameters and moving around the projection in 3d, could you not also move through the fourth direction (the one not directly visible) in order to see the different 3d slices of the 4d fractal from ana ("4d-left") to kata ("4d-right")?

[@verymuchtom]

Great question! Yes you could absolutley do that, I thought about including such an animation, but rendering them out out always takes a few hours and in my mind there were already enough animations, but for illustrative purposes that would've probably been useful. Maybe I'll upload something like that as a short in the future :)

[@robert_trirop]

Some anti-aliasing would be nice. Maybe you can try to do some random super-sampling (kind of like Blender Cycles) to get rid of some artifacts and get nicer looking details :)

[@muscovado09]

For the algorithm 👌👌

[@knarkknarkaren]

Name of tune starting at 25:40?

[@verymuchtom]

That's Cipher by LEMMiNO, I put half of his discography into the video :D

[@aliriza1688]

Does this relate in any way to twistors? The drawings appear to be similar.....

[@verymuchtom]

not to my knowledge no, but I also haven’t heard of them before your comment and I just skimmed the wikipedia article, so you know.. grain of salt 😅

[@Qwerasd]

The "bored in math class" framing is very Vi Hart

[@verymuchtom]

True, the inspiration was certainly there! I should have put a hexaflexagon in the animation :D

[@govindagarwal3310]

why does he only have only 2k subs

[@user-nd7rg5er5g]

26:09 I did pause and ponder, but I wound up drawing a mouse. :(

[@verymuchtom]

great success, what more can one ask for!! :)

[@ulyssesk7325]

okey but if it swing between 2 values as the same imput the calculater assumes 2 diffrent spaces

[@ulyssesk7325]

4 example a pure thermical spcale and z

[@ulyssesk7325]

no if you want ti imagien 5 dimensions you jsut imagine 2 cubs with diffrent fuctions

[@Finnnicus]

subscribed!

[@UnboundCinematography]

digger, watn video!

[@fedang]

Subscribed ❤

[@rockman1942]

Does it means 3d Julia set can only be made by “slice” from 4d Julia set? Is it possible we create a “n number system” to solve the infinite dimension of Julia set

[@verymuchtom]

Correct, the type of behaviour we want from a number system to create julia sets can’t be achieved in 3D, that’s why we have to use four. You can also further increase the number of dimensions but the needed number system, so called hypercomplex numbers, can only be formulated in 2^n dimensions so 2,4,8,16… dimensions. The higher the number of dimensions the more properties you loose: for quaterions (4D) you loose commutativity, for octonions (8D) you loose associativity, but theoretically you could keep going.

[@rockman1942]

Somehow, I feel sad about the fact that we are all behind higher dimensions, and we are just experiencing the projection from higher dimensions. For a long time, we have been trapped in Euclidean geometry mathematics based on what we have experienced, but in fact, Non-Euclidean geometry is a more accurate way to describe our universe, thanks to Albert Einstein.

[@lih3391]

Can you do the mandelbrot set with quaternions next?

[@verymuchtom]

Can't guarantee it'll be my next video, but it's definetly on my list. There's already some papers I put in my "to-read" folder on the topic, now it's a question of when I'll find the time, uni sadly has to have priority :D

[@lih3391]

@@verymuchtom no worries then, it's just that the julia set is very very similar to the mandelbrot set, where the +c instead of being constant is just the designated quaternion value for each voxel. In fact, when you zoom in close on a mandelbrot set at one point, it will look like the julia set for that point! It's kind of like the mandelbrot set is the grandaddy of all julia sets. If you read up on geometric algebra/clifford algrbras, you could probabaly take any power of the quaternions(bivectors) and make a fractal out of it by using x^n=e^(n*lnx)

[@rutalorp4777]

8:04 missed opportunity ;)

[@verymuchtom]

@@rutalorp4777 ahhh how didn’t i notice, that’s a golden opportunity for a horrible dad joke