Painting a Landscape with Maths

[@matthewkaemmerer7569]

Great quote at 13:47 "This is your little mathematical world, and you can do anything you want in it." It's like I'm watching Bob Ross, but with functions.

[@DavoGalavotti]

I was thinking about Bob Ross as well, when Iñigo said: “happy little clouds” 🥹

[@g-pr]

I‘m quite sure this phrases are a tribute to Bob. A „Hommage“ in German.

[@millercentretheatre]

And "happy little trees"!

[@g-pr]

@@millercentretheatre For sure. Also noticed :)

[@Smittel]

"Happy accidents" is also something i at least associate with BR but im not entirely sure

[@diegolg1189]

Just beautiful.

[@amyshaw893]

Its a small thing, but i really appreciate how you alternate between the old and new views when you change something small, it really helps me see the change you have made, as just showing it once really makes it hard for me to tell the difference sometimes

[@InigoQuilez]

Noted. I'll do more of that and better next time

[@ethancheung1676]

@@InigoQuilez I appreciate the comparison too. however, it is a bit hard to keep track of the graph and the image change at the same time. especially for the vignette effect, mid-green tone pull up, etc. sometimes a left/right split slider animation may be used too

[@InigoQuilez]

Yes, I totally agree, I'll try the slider idea next time.

[@rangersmyth]

@@InigoQuilez Hey Indigo, outstanding work! Wondering what program are you using? I would love to learn how to do this!

[@nephew_tom]

​@@rangersmyth Man... His name is Iñigo with Ñ: en.wikipedia.org/wiki/%C3%91 Not Indigo! That is a color! 😂 Anyway, I guess you do not know Spanish, so the mistake is understandable. But if you read the description, you will see the link to shadertoy website, where you can write shaders code, compile it and render images. Of course, you will need to understand how shaders work and all the math underneath to be able to build that image.

[@fudan1]

Just came from a class and my professor showed us one of your videos, and he said that you're basically his idol

[@RobattoJ]

My jaw was on the floor for half of this. I almost can't believe it's possible to render this much detail in real time. You have no idea how inspiring this is!

[@halian.vilela]

Talking about Inigo, my jaw is on the floor because I can't belive it's possible for someone to know how to apply THIS MUCH of math with such clarity. I guess it's the most impressive technical feat I've ever seen!!

[@agfd5659]

I thought this was rendered offline

[@Lumegrin]

@@agfd5659 Did he say it wasn't?

[@InigoQuilez]

It is realtime, yes. Just click the Shadertoy link in the description of the video, it will run on any browser and relatively up to date computer. It could be much faster, the ode is optimized for being mathematical, no fast. But still it should ran at 30 fps at least in a gaming PC.

[@Lumegrin]

@@InigoQuilez I'm excited to try this program; I have been looking for something just like this for a while. Until then, I'm going to need a lot more mathematical experience! 😅

[@randyprime]

this is fucking beautiful

[@marsl8603]

Honestly, I'm not suprised to see you here.

[@user-pn4dp4br7n]

Hopefully he doesn't try to recreate his game using only maths

[@wellox8856]

oi

[@Cayman_Clubsport]

Language

[@timothymakesgames]

Fancy seeing you here mate

[@konstantinkh]

Me: "Bounce light is going to be a nightmare with this terrain." Inigo the Wizard: "Super easy, barely an inconvenience."

[@daonlyowner6631]

How does he do the bounce lighting? Is he using raytracing or what?

[@konstantinkh]

@@daonlyowner6631 No. He's just taking reverse direction from sunlight with average terrain color and an ad hoc multiplier. On average, that's going to be in the ballpark. Though, the multiplier needs to be hand-picked for sun elevation and terrain shape, but since this is done for a set piece, that works!

[@daonlyowner6631]

@@konstantinkh Ah so no actual bounce lighting but just an approximation.

[@dsp4392]

Bouncing light is TIGHT!

[@dmitriytuchashvili8594]

didn't expect Pitch Meeting reference here

[@gordonfogwill4885]

40:21 The clouds shifting and the shadows following was absolutely delightful to watch, and made it seem like a time lapse. Great video! :)

[@g43s]

The quality in this videos is amazing, I love how he uses 3d animations to explain all the equations. it makes it so easy to understand what each thing is doing.

[@HyperMario64]

This is a masterclass. And completely deterministic, this could run in real-time. Oh wait, it does!

[@slickytail]

If by real-time you mean 3 FPS, sure.

[@AdriansNetlis]

@@slickytail Shadertoy's preview window is 800x450 at which it runs around 140 FPS on my PC. The drop is quite significant at higher resolutions however, on my 1440p monitor at full-screen it runs below what I consider smooth, I'd say it's about 30-40 FPS, not sure how to get the exact number there. But it's still certainly way more than 3 FPS :)

[@slickytail]

@@AdriansNetlis On my laptop it runs at 3 fps in fullscreen (1080p). Let's not even talk about phones! Just pointing out that the majority of users are not using hardware that will run this in real time.

[@tornado100able]

Well, Minecraft generate "square" words following some kind of mathematical formulas i guess

[@HyperMario64]

​@@tornado100able Absolutely, mostly with noise functions seeded on the basis of the chunk to generate. If you are curious, you can check this out: kzbin.info/www/bejne/hJubmKeVbpeWY68

[@qTnD42hR]

I remember bumping into Inigo literal decades ago when first getting into computer graphics. Literally 90% of everything I know in that area comes from his website. He's like the Knuth of CG, reading through his articles feels like perusing a sacred text, equal parts art and math, all of it genius. Very happy to see him here after all those years. Makes me itch to play in ShaderToy.

[@ruedigerraab4089]

Same with me. I saw his demo "Elevated" back then in 200 9live at the Breakpoint demo party and I was blown away. Little did I know that the creator of this masterpiece would also become a master of teaching others and sharing his knowledge.

[@kanoest]

This shows the beauty of mathematics that I never knew existed. The concepts and executions are peak maths understanding on high level

[@sam_phys-yz7fr]

Its extremely nice but just basic (but very clever use of) functions. Not high level math.

[@wargreymon2024]

This is fantastic, not only paint but created the world with mathematics alone and no real source feed into it. The maths involved is advanced but not too harsh, which is an eye-opener for people to learn math and programming. You are awesome.

[@w花b]

True, I was surprised to know all of these functions that I've seen in high school, that was surprise but a welcome one.

[@ankur67678]

I barely know anything about 3d graphics but still watched it till the end....What I liked is the underlying maths, which i understood to the some degree. can't thank you enough for sharing your knowledge...

[@davecgriffith]

The art here is actually this video. Fascinating subject and wonderfully presented. Thank you for this!

[@MatijaHan]

Thanks!

[@InigoQuilez]

oh wow, so generous, thanks a lot!

[@euphonic3466]

Now I understand why are we made to study these topics in class 11 and 12 in India and what's its application in real life. Also got to know today that this world is literally mathematically built.Thanx to you.This indeed is beautiful to know the details of our universe.

[@parsuli.]

As someone aspiring to work at Pixar once I graduate, and as someone who loves math, this video was an absolute gem. It was delightfully composed and the end result was breathtaking. Thank you for making this for all of us to see. I have no doubt it was a lot of work.

[@konya8248]

Man.. this is impressive it's like he's explaining how people back then are doing it, but with the current beauty of today's engines.

[@tizilogic]

You are definitely the Bob Ross of mathematical art!!

[@angeltodorov4577]

This is by far one of my favorite math videos. I did not understand all of the math but this was still so interesting. I never knew math could be used for this

[@_shroob770]

I once wrote a high-school paper about approximating the surface areas of hills using piecewise functions (the functions were only linear however, I could not deal with more than that back then), it's really awesome to see someone take the concept to such a degree, incredible video!

[@wiredlifter]

This has to be one of the most beautiful videos I have ever seen on the internet. You sir have made a MASTERPIECE of a video. I cannot describe how engaged I was through the whole thing. My jaw was always on the floor

[@MathIsGreatFun]

Stunning. I have never seen mathematics applied in such a beautiful and captivating way. You are the multivariable calculus da Vinci. Everything in this video was presented in a precise and masterful way. I would LOVE if my students could make something like this for a final class project. Thank you for sharing your work and for all the surely grueling hours it took to produce this excellent video.

[@nameno7032]

you teach code, math, and art at the same time, what a beautiful combination Hope you the best, and keep making these

[@Lucas72928]

You can really tell the amount of work that goes into making one of these videos. Great work!

[@emp9083]

Absolutely beautiful , and was engaging for the entirety of the 40 minutes. Thank you for sharing this and making videos of this calibre free to watch.

[@DaveChurchill]

As a computer science professor, these videos always blow me away. I wish I could create lectures of this quality!

[@christelleaugustin1695]

Just play this video

[@redlaserbm_phd]

Okay, this is cool! Didn't think much of the polynomial surface till you started adding scaled and rotated copies on top, that was brilliant!

[@chrisxd146]

It felt great to need out for over half an hour and see mathematics I use everyday be applied in creative ways. It also showcases just how powerful linear algebra and calculus are when used together.

[@maymayman0]

You truly have one of my favorite channels on KZbin, and I watch way too much KZbin.... thank you for everything you do

[@firefly618]

Sometimes KZbin's algorithm hits you out of nowhere with an absolute gem of a video. This was amazing! Thank you for your teaching and for making the code available. I will study it and use it as a basis for my own mathematical paintings! In fact, I always loved constructing the functions that I needed out of small, fundamental pieces. If you ever want to do the same thing with sound, Csound is a very old tool to do just that. (But there are probably better tools nowadays?)

[@gbkEmilgbk]

Very nice tutorial! You are someone who has been spreading the word about ray-marching for many years - thanks to that I found out about this great technique (from your website ~7years ago) - thank you very much!

[@JonathanJumper]

This a whole other level man, max respects I'm an computer engineer myself, but this is pure math, and an art the way its presented

[@TannerCh]

This is one of the most incredible things I've ever seen.

[@BoyceBailey]

It's just a joy to watch this. It like watching the grown ups do something you're amazed by but when you're an adult.

[@alexmarunowski6442]

I watched this entire video with my mouth agape. Purely magical to watch.

[@swampflux]

You’re back! Look how far even the master of shaders has come by working with Pixar. In this video I feel you’ve not just done great mathematics teaching, but deeply connected with artistic expression. I was waiting for this one! Congratulations on finishing.

[@RhogerAnacleto]

I didn't know this was even possible. I didn't understand a thing and I am very impressed with everything And this is clearly the most magnificent mathematical painting I have ever seen! Good job!

[@lunafoxfire]

I'm only 2:30 into the video and my mind has already expanded 100x Thank you so much Inigo both for your amazing art and your amazing love of sharing knowledge!

[@hfs-lk5ip]

Holy moley, I have been messing around with 2d math art for 4 years as a high school student and just started on a homegrown orthographic projection. Looking at this is like stepping into the 1000+ years future of what i was playing around with. its surreal. It's crazy. its mind-breaking. so fucking cool. You've just shattered my reality.

[@nurse_plissken8433]

This presentation is absolutely wonderful, I've been having some trouble getting the general math concepts behind some of the code I see out there, and this helps a ton. also holy crap that 3/4/5 matrix trick is beautiful

[@lucianojadur]

One of those few videos in this whole site that everybody could describe as "perfect". Genio total.

[@iestynne]

This is a great educational legacy you are building here, to inspire the next generation of graphics programmers. Bravo!

[@vildauget]

First I was surprised that 40 minutes had passed. Then I was surprised that I had watched the whole video again. Thank you so much for this. Late to the party, but this video needs all the attention it can get.

[@niklas2430]

I love this video so much. It is really helping me understand some fundamental low level concepts much better but also conveys the design process so well without ever getting stuck on details.

[@christelleaugustin1695]

Don't you mean "high-level"? If you talk about "without getting stuck on details"

[@nobodynoone2500]

Brilliant. Maybe the best explination of where you can use math creatively I have seen in years.

[@claytonleonardcook]

This is one of the first videos I've ever seen like this. I'm a computer science student so I've seen people use really complex algorithms to achieve realism, but I love the simplicity of this. Hope to see more videos like this because I find them super interesting.

[@PolyRocketMatt]

I think it is beautiful how something many people don't like (mostly because they miss the engagement like this) can result in one of the most complex yet simplistic, elegant yet basic things there are. Thank you for providing these wonderful videos!

[@bytesandbikes]

appreciate the Bob Ross references!

[@benwesch1405]

This is incredible. Thanks for sharing so much knowledge in such a beautiful way!

[@isaackay5887]

This was honestly the most useful and insightful video I've watched in a very very long time! I love applying the math I've learned from my undergrad to something creative, so thank you!! Also, someone else in the comments mentioned "It's like I'm watching Bob Ross, but with functions" - I have to second this statement. Well, done, sir.

[@3kcozadurnylol]

I'm absolutely impressed by your work. It is the first video that I saw, but surely not the last one. The harmony between beauty and maths is like visiting an art gallery, theatre and dancing at once. Almost spiritual experience :D Great work!

[@Conobipe]

I'm totally in love with your art and explanations. It is trully inspiring to see how far we can go in generating art simply by tweaking numbers and formulas, even if we don't get the science behind it. Big thanks for your work IQ

[@HiHi-iu8gf]

waow the noise for the trees works way better than i'd have thought

[@plazmaguy13yago9]

Even tough i can barely understand some of the steps you do i find this incredible!

[@vlahovivan]

This is the best video I have ever seen on KZbin

[@krpnz]

Thank you Inigo for this video, and for all of the detailed articles & demos on your website for many years.

[@male20yearsold]

everyone who's making a digital art is actually making "mathematically" art. The digital brushes, colours, canvas, etc. it is all math. what we saw in this video is very mathematically from the very base, real definition of "from a scratch" with no instant tools like digital brushes or etc. Appreciate it Inigo Quilez! Very Amazing and inspiring.

[@Anonuser11131]

Keep making these videos man, you have found an audience.

[@snippletrap]

Beautiful. I could watch the clouds roll over that landscape all day.

[@TriggerlyDev]

Me not understanding 90% of the video but enjoying the visuals

[@trevorleake2010]

I've been a fan of ShaderToy for so long. Thank you Inigo!

[@Demoscene0x0801]

If my mathematics teacher had told me 30 years ago what incredible things you can create with mathematics, I would have slept less in class. 🥺

[@christelleaugustin1695]

Exactly

[@pinklady7184]

I am 3D artist and animator, and I am learning mathematics for 3D realism plus physical simulations?

[@marcianoforst6311]

This is not simple math like what you learned in school

[@poutouellet]

@@marcianoforst6311 I'm a high school math teacher and I promise you that although most of the video went over my head, it was because of the speed and my wonderment and not the complexity of the maths. Everything done in this video is within the grasp of a good high school math student had they the interest and time.

[@marcianoforst6311]

@@poutouellet Calc 3 in high school?

[@gamesvrtech6666]

I find it fascinating how detailed the perception of some people is. I was already convinced of the scene but hey, let's add some little yellow here and another bump there... Very impressive! 👍🏻

[@lightninghell4]

Combining physics, math, and computer graphics to create a staggeringly beautiful painting. Amazing video man.

[@Emboar565]

Thank you for this video. I am a 1st maths student and I enjoyed every single second of the video. I loved all the work and quality on this video from the way to explain it to all the animations and equations. The result is espectacular and beutiful.

[@icaroamorim3123]

Wow I can't describe how amazing this is

[@brycepoteat1928]

I would watch a hundred of these videos that are a mix of art, math, and tranquility. Absolutely beautiful!

[@vladshcherbakov3112]

I always wondered about this piece, but when I looked at the code, I was just baffled by everything going on there. I'm studying Electrical Engineering, but when I'm bored laying out circuit boards this type of art is a fantastic way to do something different and at the same time learn something new :)

[@InigoQuilez]

I am actually an electrical engineer too (I did my Masters, I mean). This Paintimg with Maths stuff was my way to have a break from semiconductors, Maxwell and Shannon :)

[@SimonSchumann]

@@InigoQuilez I also want to thank you very much too, because I as well am approaching the end of my Master's Thesis in Electrical Engineering & Information Technology, and your incredible work with shaders inspired me for my thesis topic. It takes place in the medical domain, and is about simulation of a specific microscope. I create artificial images from blood cells modelled with SDFs, by doing sphere marching with some tweaks for lenses, depth of field etc.. This yields training data for machine learning, which in turn can then hopefully reason better about patient's blood conditions than it does currently. Also your video is absolutely incredible! I cannot imagine how much work that must have been

[@Mukesh_Rana._]

I am also a student of electrical engineering 👍

[@Ontropy]

The animations are top notch, makes it really really easy to understand the functions. Really good explanations as well. Thank you very much. Thanks for attaching the code too.

[@Invalid571]

This video is sooo beautiful, I watched it in it's entirety twice!

[@deathTurgenev]

I discovered this channel just a few days ago, and it is already one of my favorites

[@ハェフィシェフ]

Your animations, for instance of the normals at 26:13 are absolutely amazing

[@julienmartin2253]

I have been waiting for this video for 8 months ! Thank you Inigo

[@DJR3H]

If I ever have 1 year of time to spare - I'll try to understand all the math I just observed. Beautiful

[@InigoQuilez]

It is true that mastering anything does take time

[@ZougooKing]

This video prove that you have master maths. This is so awesome. I understood quite few things cause of my engineerings maths class but still so much more in this video. Good job man.

[@lovelybunny9190]

I am not into math or anything I didn't even search this but it got recommended and the title was too catchy. would be fun if they taught us such stuff at school. this looks so fun! it's beautiful

[@swyveu]

This just blew my mind. I had no idea... So much easier to comprehend mathematics like this. And the quality of this video is through the roof. Subscribed!!

[@thomaswatts6517]

this is the coolest thing I've ever seen, the fractal surface sum gadget is sick!!!

[@merimbungur7168]

This is the best thing youtube has recommended to me in years. I am simply amazed how well you apply mathematical concepts to get what you want. I made sure to subscribe and go down the rabbithole of your content

[@christelleaugustin1695]

i understand nothing but if i were smart i would understand it directly because of your visual clear explanations!

[@kellyhyland821]

This is the craziest thing I’ve ever seen. Holy shit dude. You are a wizard.

[@DenisIsidoro]

I never comment in videos, but I had to this time. This video is fantastic in so many ways: presentation, teaching, Math elegance, etc. Excellent work!

[@123lex123]

Honestly, this almost made me cry. The fact that this is all just math is so extremely beautiful!! I'm a computer science student and secretly in love with math. I think you just showed me a door I was looking for! Thank you so much!!

[@imnemo2327]

could you tell which branch of mathematics he is talking about in video.I didn't understood much!!!

[@lakshyarajsinghrathore1902]

@@imnemo2327 mainly algebra and calculus

[@nuqqets9570]

What's more impressive than just the maths is the understanding of art and lighting in nature

[@CarstenMeyer]

Another free lesson from the master himself! Great as always!

[@BreezyDefrag]

Holy wow! I new math was powerful, and in the back of my mind I know this is how my drawing programs work under the hood... but to see this scene built with ONLY math equations sort of kind of blew my mind... This was really cool.

[@oliverrevis4190]

This looks remarkably similar to the Blue Mountains in Australia. The Grose valley in particular

[@StevenHokins]

❤

[@rusticagenerica]

I love this concept

[@probablynot8154]

yes!! these are some of my most favourite videos on youtube.

[@aged_mozzarella]

The effort that goes into making a video like this is simply mind-blowing

[@lucabilotta]

Inigo your work is absolute stunning (I've seen among many, your "Elevated" 4kb winner of democomp) From an Italian fan, you are The Leonardo Da Vinci of mathematical painting and procedural gfx!

[@arindamn4880]

This is the I'd absolutely love to learn. Great video sir!🤩

[@harrybmichell]

What an incredible video! I'm taking a computer graphics class and this is just blowing my mind

[@Philoreason]

This is the best video I've seen explaining how landscape generation and rendering is done! good job! Thx!

[@voidbreak4756]

As a Desmos user, this has inspired me so much! I hope to try something like this when I'm older! I need to wait 5 years for university though, argh!

[@iwikal]

No you don't! There's tons of info online about linear algebra, programming and computer graphics! I recommend 3blue1brown on KZbin for the linear algebra stuff. As for the programming part, you'll have most luck searching for guides and web books about your particular programming language of choice. Mozilla Developer Network has a nice beginner's guide to WebGL for graphics in the browser, for which you will also need to know some basic JavaScript, but the internet is brimming with educational content about JavaScript. You'll find it if you look for it. There's other avenues though: you can learn Blender, or a game engine like Unity, Godot etc, or you can learn some other programming language and a native graphics framework. Just pick something that piques your interest and get started!

[@iwikal]

To back my claims up: I'm 25 and a uni freshman, but I've been doing computer graphics as a hobby since way, way before university. I'm mostly going to uni for my interest in deeper maths, the kind of maths you will never need as a programmer or technical artist.

[@voidbreak4756]

@@iwikal Oops! It seems like I've worded my comment wrong. I actually meant that I would try it, but I still have to wait 5 years for university. I don't know why I put the university part in lol, they don't matter. Anyways, some of people on the Desmos server and I are working on some code that would make Desmos work sort of like Shadertoy. Thanks for the inspiration, though!

[@iwikal]

@@voidbreak4756 have you tried shadertoy? I'm pretty sure you could make this exact scene in shadertoy. Just translate the expression to glsl and slap on a projection matrix! Super easy in theory haha, it'll just take a while. The framerate might not end up being the best though.

[@MScienceCat2851]

@@iwikal Snice im not good at math, what turials are best and what math was used in this video, its advanced

[@hallstein1]

This is absolutely amazing. There's so much wisdom and knowledge here to learn from it's mind-blowing. And the care made in the creation of the video is incredible too. Just wow!

[@kayaba_atributtion2156]

Wow this was beautiful to watch! You should make a Ted talk

[@W0otJul38]

The odds of this video happening now, 6 years after you uploaded this on shadertoy baffles me, 2 weeks ago I was trying to make a raymarching shader in unity to accomplish exactly this, I guess I'll pick the project back up! Thanks Inigo, you're a pioneer/inspiration for this kind of art :) !