Painting a Landscape with Maths

Today we are painting a landscape using mathematics.
Support this channel: / inigoquilez
Buy this painting in a metal, canvas or photographic paper print: www.redbubble.com/shop/ap/398...
This is the link to the real-time rendering code (that you can edit yourself live) for the painting: www.shadertoy.com/view/4ttSWf
Music by Bent Stamnes ( / gloom303 .
Donate: www.paypal.com/paypalme/SMOOT...
Subscribe: / inigoquilez
Support: / inigoquilez
Twitter: / iquilezles
Facebook: / inigo.quilez
0:00 Intro
0:48 Basic Polynomial Surface (Noise)
4:12 Fractal Surface (FBM)
6:28 Key Lighting
9:15 Atmospheric Coloring
11:00 Composition and Cliffs
12:28 Light Direction
13:53 Sky
15:42 Fill Lighting
17:42 Grass
18:31 Tree Sculpting
23:53 A Story of Painting with Maths professionally
24:52 Tree Lighting and Coloring
29:48 Highlights
32:05 Color Touchups
34:48 Clouds
40:32 Conclusion
Related articles I wrote:
* Fractal Noise Summation, also known as FBM: iquilezles.org/articles/fbm
* Raymarching terrains: iquilezles.org/articles/terra...
* Rendering SDFs: iquilezles.org/articles/rayma...
* Derivatives of Noise: iquilezles.org/articles/moren...
* SDF of an Ellipsoid: iquilezles.org/articles/ellip...
* SDF Gradient: iquilezles.org/articles/norma...
* Soft Shadows: iquilezles.org/articles/rmsha...

[@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

[@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.

[@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

[@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

[@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

[@Lord_Drakostar]

@@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.

[@Lord_Drakostar]

@@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! 😅

[@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

[@kanoest]

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

[@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.

[@andremeIIo]

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.

[@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.

[@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?)

[@greymonwar9906]

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.

[@whannabi]

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

[@bytesandbikes]

appreciate the Bob Ross references!

[@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.

[@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!

[@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.

[@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

[@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.

[@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?

[@tizilogic]

You are definitely the Bob Ross of mathematical art!!

[@OkiSuwen]

"So do you use oil paint? Pencils? What do you use?" "NUMBERS."

[@marcinkrzeszowiec1538]

Bob Ross of tech art. Thank you. Perfect material to enjoy a meal in between fighting with ShaderGraph and Unity memory management

[@_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!

[@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.

[@johntom8863]

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

[@user-yb5cn3np5q]

29:50 "Now it's time to switch gears and do some computer graphics" Oh, so the last half an hour was just a warmup...

[@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

[@Lucas72928]

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

[@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.

[@joshua17111]

Love it!

[@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.

[@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 👍

[@mickobuenaventura4590]

Fascinating!

[@brunojani7968]

Beautiful!

[@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.

[@Ra11y]

This was absolutely incredible!

[@krisnrg]

Brilliant!!!!

[@ReBufff]

This is just amazing

[@plazmaguy13yago9]

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

[@krpnz]

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

[@qpdbwvwdbqp4730]

simply awesome art

[@dragonscalesvr]

Breathtaking.

[@AlanZucconi]

Thank you for making this, Íñigo! It was (un)expectedly relaxing! 😊

[@kostik-pi7me]

The quality of this video is top-notch at every aspect. Thank you!

[@panociagasta]

Wow! Simply amazing

[@khromemusic]

Absolutely brilliant!!!

[@CarstenMeyer]

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

[@kayaba_atributtion2156]

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

[@sapito515]

This is just incredible, thank you so much for putting this out there

[@user-ud9un9wz2b]

Amazing!!

[@thomasvanta]

Me encanta!!! Mil gracias :)

[@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

[@Limofeus]

This is just amazing...

[@SigurdBraathen]

This is amazing!

[@Invalid571]

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

[@icaroamorim3123]

Wow I can't describe how amazing this is

[@MrLuigiBean1]

This is INCREDIBLE!!!

[@iestynne]

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

[@thomaswatts6517]

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

[@shis10]

Amazing.

[@benedictsth]

So much love to detail!

[@rysea9855]

Wow, this is an amazing video. It's a very fun video to watch, and even though I probably won't try making this myself because I know nothing about programming, I still learned a lot. First 30+ minute video in a long time that I've been able to keep my attention to the whole way through, and that's saying something. Really well made

[@PauloLoma]

It's was like watching a happy mathematical movie. Thank you!

[@cnota69]

That's incredible

[@matteogiovannetti7071]

Truly astonishing.

[@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

[@unevenprankster]

The master of shaders teaches us about painting a new world pixel by pixel :D Incredible work!

[@panagiotisdeligiannis9610]

This is amazing! This is brilliant content!

[@diegolg1189]

Just beautiful.

[@lightninghell4]

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

[@user-xs9ey2rd5h]

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

[@Sid_music]

This is really special!

[@jaimebenhamu743]

this is beautiful

[@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.

[@somethingthatpops]

The combination of the terrain normals with the tree normals is brilliant and really shows the shape of the terrain and makes the image look so much clearer. I love these videos!

[@solcloud]

Super cool!

[@lavespa2758]

Mind blowing 🤯

[@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

[@ThePianist4Life]

I love the Bod Ross approach applied to the concepts of computer graphics

[@heidolf6002]

wow just so cool

[@MrMiguelfigueira]

This is great, thank you

[@gnorts_mr_alien]

Amazing. Years and years of research and experience in the dark arts... Finally out there, in such a digestible format. You are a legend Inigo. The field owes so so much to you. How generous.

[@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!

[@AJ-et3vf]

Awesome video! Thank you!

[@jamesbentonticer4706]

In the spirit of Bob Ross. I love it.

[@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.

[@rusticagenerica]

I love this concept

[@probablynot8154]

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

[@theomaia1460]

absolutely incredible work congrats

[@HBon111]

That's so cool! Inspiring!

[@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

[@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

[@vincentcleaver1925]

This is so gorgeous!

[@hellhand4786]

Outstanding work! thank you so much IQ

[@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!

[@lydianlights]

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!

[@fountaingoat6802]

Incredible, thank you

[@EquiliMario]

This is absolutely stunning.

[@Anonuser11131]

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