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.
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 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
@@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.
@@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 :)
@@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 Absolutely, mostly with noise functions seeded on the basis of the chunk to generate. If you are curious, you can check this out: ruclips.net/video/Nj8gt_92c-M/видео.html
@@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!
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!
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!!
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.
@@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! 😅
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.
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.
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.
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.
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.
Sometimes RUclips'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?)
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
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.
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...
@@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.
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
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!
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.
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!
I loved watching this video. A career dream of mine was to be doing similar things to this, bought all the (expensive) books. Unfortunately, life and the maths required meant it stayed a dream..
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.
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.
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.
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.
*Abstract* This video demonstrates how to create a landscape painting using mathematical formulas instead of traditional paint and brushes. The process involves defining colors and shapes through formulas, which are then combined into one large formula to generate the final image. The video covers various aspects of the painting process, including creating the terrain, lighting, atmosphere, composition, and adding details like grass and clouds. *Summary* *Basic Polynomial Surface (Noise)* * *0:52**:* The terrain is initially created using a linear function of x and z coordinates. * *1:11**:* Increasing the degree of the polynomial adds undulations to the surface. * *1:37**:* To avoid computationally expensive high-degree polynomials, the terrain is divided into tiles, each with a unique low-degree polynomial. * *2:29**:* Coefficients for the polynomials are randomized to create a natural-looking terrain. * *3:53**:* The resulting undulating and random function is called "noise" and is commonly used in visual effects. *Fractal Surface (FBM)* * *4:13**:* The noise function is used to add finer details to the terrain by repeatedly compressing, squashing, rotating, and adding it back to the original surface. * *5:46**:* This process resembles a Fourier series or discrete cosine transform. *Key Lighting* * *6:30**:* Lighting is introduced by considering the sun as the key light. * *6:43**:* Derivatives of the terrain function are used to determine surface normals and identify areas facing the sun. * *7:18**:* The dot product between the surface normal and the sun direction determines illumination. * *8:09**:* Shadows are added by checking for intersections between the line of sight to the sun and the terrain. * *8:48**:* Soft shadows are created using a smooth transition zone between light and shadow. *Atmospheric Coloring* * *9:16**:* To create a sense of scale, atmospheric perspective is simulated by mixing gray into the surface color based on distance. * *10:42**:* Different attenuation factors for red, green, and blue channels create a blue tint in the atmosphere. *Composition and Cliffs* * *11:00**:* A specific section of the terrain is chosen and framed for the composition. * *11:55**:* Cliffs are sculpted by raising areas above a certain height. *Light Direction* * *12:33**:* The sun's position is adjusted to create dramatic lighting and shadows. * *13:03**:* Different sun directions are explored to find the optimal composition. *Sky* * *13:55**:* A blue background is added and adjusted to create a darker, more saturated sky at the top. * *14:18**:* Clouds are created by evaluating the terrain function on a horizontal plane and applying a smooth step. * *15:23**:* Cloud details are adjusted to achieve the desired look. *Fill Lighting* * *15:45**:* Overall contrast and saturation are adjusted. * *16:17**:* Areas not directly illuminated by the sun receive skylight based on their orientation. *Grass* * *17:43**:* Green is mixed into the terrain color based on the surface normal to create a grassy layer. *Tree Sculpting* * *18:32**:* Some terrain undulations are filtered out to create areas suitable for tree growth. * *19:22**:* Trees are created by tiling the space and evaluating a sphere's signed distance function within each tile. * *20:42**:* Randomness is introduced to vary the positions and sizes of the trees. * *21:13**:* Spheres are stretched vertically to resemble trees. * *22:11**:* Trees are positioned on the ground, leading to some appearing under the cliffs. * *22:43**:* The ground between trees is darkened to simulate reduced light penetration. *Tree Lighting and Coloring* * *24:55**:* Noise is used to distort the tree silhouettes and make them more organic. * *26:00**:* The lighting on the trees is adjusted to reveal the shape of the underlying terrain. * *26:56**:* Highlights are added to the trees, capturing the silhouettes of the hills. *Color Touchups* * *27:41**:* A rainforest color palette is chosen, and variations are added to the trees. * *32:04**:* The brightness and lighting of specific areas are adjusted. *Highlights* * *29:48**:* Highlights are added to the cliffs and trees using formulas that approximate real-world light behavior. *Clouds* * *34:45**:* More realistic clouds are sculpted using a solid block and adding noise. * *35:41**:* The plotting algorithm is changed to accumulate cloud density and color for a fluffy appearance. * *36:07**:* Fractal noise is used to create detailed cloud silhouettes. * *36:30**:* The cloud density gradient is filtered to soften the lighting. * *37:20**:* Cloud size and lighting are adjusted. * *38:14**:* Skylight is added to the clouds. *Cloud Shadows* * *39:26**:* Shadows cast by the clouds on the terrain are computed using the cloud density function. * *40:17**:* Shadow softness and size are adjusted. *Conclusion* * *40:35**:* The final painting is complete, demonstrating the use of mathematics to create a landscape image. i used gemini 1.5 pro to summarize the transcript Token count 11,153 / 1,048,576
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.
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!
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
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!
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 :)
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 :)
@@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
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.
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!!
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!
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.
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
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.
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
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!
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!
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.
No you don't! There's tons of info online about linear algebra, programming and computer graphics! I recommend 3blue1brown on RUclips 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!
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.
@@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!
@@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.
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.
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
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 :) !
Wow. I'm so shocked it's 3am and I'm no longer falling asleep. I already know most of the maths you've applied, but it was so out of my field (physics) I barely recognized them. Thanks for this gem.
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.
I was thinking about Bob Ross as well, when Iñigo said: “happy little clouds” 🥹
I‘m quite sure this phrases are a tribute to Bob. A „Hommage“ in German.
And "happy little trees"!
@@millercentretheatre For sure. Also noticed :)
"Happy accidents" is also something i at least associate with BR but im not entirely sure
this is fucking beautiful
Honestly, I'm not suprised to see you here.
Hopefully he doesn't try to recreate his game using only maths
oi
Language
Fancy seeing you here mate
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
Noted. I'll do more of that and better next time
@@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
Yes, I totally agree, I'll try the slider idea next time.
@@InigoQuilez Hey Indigo, outstanding work! Wondering what program are you using? I would love to learn how to do this!
@@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.
This is a masterclass. And completely deterministic, this could run in real-time. Oh wait, it does!
If by real-time you mean 3 FPS, sure.
@@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 :)
@@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.
Well, Minecraft generate "square" words following some kind of mathematical formulas i guess
@@tornado100able Absolutely, mostly with noise functions seeded on the basis of the chunk to generate. If you are curious, you can check this out: ruclips.net/video/Nj8gt_92c-M/видео.html
Me: "Bounce light is going to be a nightmare with this terrain." Inigo the Wizard: "Super easy, barely an inconvenience."
How does he do the bounce lighting? Is he using raytracing or what?
@@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!
@@konstantinkh Ah so no actual bounce lighting but just an approximation.
Bouncing light is TIGHT!
didn't expect Pitch Meeting reference here
40:21 The clouds shifting and the shadows following was absolutely delightful to watch, and made it seem like a time lapse. Great video! :)
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!
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!!
I thought this was rendered offline
@@agfd5659 Did he say it wasn't?
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.
@@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! 😅
This shows the beauty of mathematics that I never knew existed. The concepts and executions are peak maths understanding on high level
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.
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.
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.
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.
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.
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.
True, I was surprised to know all of these functions that I've seen in high school, that was surprise but a welcome one.
Sometimes RUclips'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?)
The art here is actually this video. Fascinating subject and wonderfully presented. Thank you for this!
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
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.
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...
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. 🥺
Exactly
I am 3D artist and animator, and I am learning mathematics for 3D realism plus physical simulations?
This is not simple math like what you learned in school
@@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.
@@poutouellet
Calc 3 in high school?
If I ever have 1 year of time to spare - I'll try to understand all the math I just observed. Beautiful
It is true that mastering anything does take time
appreciate the Bob Ross references!
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
You are definitely the Bob Ross of mathematical art!!
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!
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.
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.
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!
I loved watching this video. A career dream of mine was to be doing similar things to this, bought all the (expensive) books. Unfortunately, life and the maths required meant it stayed a dream..
As a computer science professor, these videos always blow me away. I wish I could create lectures of this quality!
Just play this video
Wow, this is amazing.
Maths is beautiful 🤍
You can really tell the amount of work that goes into making one of these videos. Great work!
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.
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.
Just beautiful.
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.
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.
Combining physics, math, and computer graphics to create a staggeringly beautiful painting. Amazing video man.
*Abstract*
This video demonstrates how to create a landscape painting using mathematical formulas instead of traditional paint and brushes. The process involves defining colors and shapes through formulas, which are then combined into one large formula to generate the final image. The video covers various aspects of the painting process, including creating the terrain, lighting, atmosphere, composition, and adding details like grass and clouds.
*Summary*
*Basic Polynomial Surface (Noise)*
* *0:52**:* The terrain is initially created using a linear function of x and z coordinates.
* *1:11**:* Increasing the degree of the polynomial adds undulations to the surface.
* *1:37**:* To avoid computationally expensive high-degree polynomials, the terrain is divided into tiles, each with a unique low-degree polynomial.
* *2:29**:* Coefficients for the polynomials are randomized to create a natural-looking terrain.
* *3:53**:* The resulting undulating and random function is called "noise" and is commonly used in visual effects.
*Fractal Surface (FBM)*
* *4:13**:* The noise function is used to add finer details to the terrain by repeatedly compressing, squashing, rotating, and adding it back to the original surface.
* *5:46**:* This process resembles a Fourier series or discrete cosine transform.
*Key Lighting*
* *6:30**:* Lighting is introduced by considering the sun as the key light.
* *6:43**:* Derivatives of the terrain function are used to determine surface normals and identify areas facing the sun.
* *7:18**:* The dot product between the surface normal and the sun direction determines illumination.
* *8:09**:* Shadows are added by checking for intersections between the line of sight to the sun and the terrain.
* *8:48**:* Soft shadows are created using a smooth transition zone between light and shadow.
*Atmospheric Coloring*
* *9:16**:* To create a sense of scale, atmospheric perspective is simulated by mixing gray into the surface color based on distance.
* *10:42**:* Different attenuation factors for red, green, and blue channels create a blue tint in the atmosphere.
*Composition and Cliffs*
* *11:00**:* A specific section of the terrain is chosen and framed for the composition.
* *11:55**:* Cliffs are sculpted by raising areas above a certain height.
*Light Direction*
* *12:33**:* The sun's position is adjusted to create dramatic lighting and shadows.
* *13:03**:* Different sun directions are explored to find the optimal composition.
*Sky*
* *13:55**:* A blue background is added and adjusted to create a darker, more saturated sky at the top.
* *14:18**:* Clouds are created by evaluating the terrain function on a horizontal plane and applying a smooth step.
* *15:23**:* Cloud details are adjusted to achieve the desired look.
*Fill Lighting*
* *15:45**:* Overall contrast and saturation are adjusted.
* *16:17**:* Areas not directly illuminated by the sun receive skylight based on their orientation.
*Grass*
* *17:43**:* Green is mixed into the terrain color based on the surface normal to create a grassy layer.
*Tree Sculpting*
* *18:32**:* Some terrain undulations are filtered out to create areas suitable for tree growth.
* *19:22**:* Trees are created by tiling the space and evaluating a sphere's signed distance function within each tile.
* *20:42**:* Randomness is introduced to vary the positions and sizes of the trees.
* *21:13**:* Spheres are stretched vertically to resemble trees.
* *22:11**:* Trees are positioned on the ground, leading to some appearing under the cliffs.
* *22:43**:* The ground between trees is darkened to simulate reduced light penetration.
*Tree Lighting and Coloring*
* *24:55**:* Noise is used to distort the tree silhouettes and make them more organic.
* *26:00**:* The lighting on the trees is adjusted to reveal the shape of the underlying terrain.
* *26:56**:* Highlights are added to the trees, capturing the silhouettes of the hills.
*Color Touchups*
* *27:41**:* A rainforest color palette is chosen, and variations are added to the trees.
* *32:04**:* The brightness and lighting of specific areas are adjusted.
*Highlights*
* *29:48**:* Highlights are added to the cliffs and trees using formulas that approximate real-world light behavior.
*Clouds*
* *34:45**:* More realistic clouds are sculpted using a solid block and adding noise.
* *35:41**:* The plotting algorithm is changed to accumulate cloud density and color for a fluffy appearance.
* *36:07**:* Fractal noise is used to create detailed cloud silhouettes.
* *36:30**:* The cloud density gradient is filtered to soften the lighting.
* *37:20**:* Cloud size and lighting are adjusted.
* *38:14**:* Skylight is added to the clouds.
*Cloud Shadows*
* *39:26**:* Shadows cast by the clouds on the terrain are computed using the cloud density function.
* *40:17**:* Shadow softness and size are adjusted.
*Conclusion*
* *40:35**:* The final painting is complete, demonstrating the use of mathematics to create a landscape image.
i used gemini 1.5 pro to summarize the transcript
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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
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.
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!
Ultimate video in youtube 😱
This is a great educational legacy you are building here, to inspire the next generation of graphics programmers. Bravo!
There’s no way the universe isn’t simulated yet. This guy could be from a far future generation, teaching us the basics of what is coming.
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
I can't believe it. Oh my GOD!!! It's awesome.
Wow I can't describe how amazing this is
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!
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 :)
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 :)
@@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
I am also a student of electrical engineering 👍
you teach code, math, and art at the same time, what a beautiful combination
Hope you the best, and keep making these
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.
Don't you mean "high-level"? If you talk about "without getting stuck on details"
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!!
could you tell which branch of mathematics he is talking about in video.I didn't understood much!!!
@@imnemo2327 mainly algebra and calculus
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!
This is one of the most incredible things I've ever seen.
Even tough i can barely understand some of the steps you do i find this incredible!
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.
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
Thankyou. A watch worth ones time.
Keep making these videos man, you have found an audience.
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.
Thank you Inigo for this video, and for all of the detailed articles & demos on your website for many years.
Wow. Just wow. I had no idea this was possible in a shader.
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
Amazing! Thanks for sharing.
This video is sooo beautiful, I watched it in it's entirety twice!
I watched this entire video with my mouth agape. Purely magical to watch.
i understand nothing but if i were smart i would understand it directly because of your visual clear explanations!
Wooaw it's truly amazing "Shadertoy". Thanks for everything amigo.
I love that website.
this is the coolest thing I've ever seen, the fractal surface sum gadget is sick!!!
Brilliant. Maybe the best explination of where you can use math creatively I have seen in years.
Another free lesson from the master himself! Great as always!
This channel is a gold mine.
Your animations, for instance of the normals at 26:13 are absolutely amazing
This is taking the "You'll use this equation in the future" in a new level.
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!
Wow I never think about it.
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!
This man is total crazy. Take a bow🙇♀️🙇♀️
"So do you use oil paint? Pencils? What do you use?"
"NUMBERS."
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.
I love this concept
yes!! these are some of my most favourite videos on youtube.
my jaw is on the floor the whole time. really renews my excitement for math again.
Wow this was beautiful to watch! You should make a Ted talk
Wtf did I stumble across. This shit is amazing.
Finally found the answer for why we need maths.
What an incredible video! I'm taking a computer graphics class and this is just blowing my mind
Stumbled upon your website a few years and pleasantly surprised to stumble upon this video today. Subscribed!
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!
No you don't! There's tons of info online about linear algebra, programming and computer graphics! I recommend 3blue1brown on RUclips 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!
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.
@@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!
@@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.
@@iwikal Snice im not good at math, what turials are best and what math was used in this video, its advanced
One of those few videos in this whole site that everybody could describe as "perfect".
Genio total.
This was absolutely incredible!
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.
This looks remarkably similar to the Blue Mountains in Australia. The Grose valley in particular
You truly have one of my favorite channels on RUclips, and I watch way too much RUclips.... thank you for everything you do
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
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 :) !
I'am simple man, when I see new video from raymarching god himself, I like the video. You should do the same
I would watch a hundred of these videos that are a mix of art, math, and tranquility. Absolutely beautiful!
So this is why , people tell maths is beautiful
Wow. I'm so shocked it's 3am and I'm no longer falling asleep.
I already know most of the maths you've applied, but it was so out of my field (physics) I barely recognized them.
Thanks for this gem.
This is a really well made video, so good to follow along! Just discovered your channel.
This is insane, i wish i'll be able to get to such a level one day, thanks for the video !