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Math + Programming = Art: Iñigo Quilez wants you to know just how beautiful math can be


It seems to be a universal character trait of high school math teachers to preface every sketch of a sine curve with, Now I’m no artist… as if any group of sixteen-year-olds expects a blackboard masterpiece. For most of us, the linkage between art and math rarely extends beyond shoddy sketches of unit circles and, perhaps, a doodle or two in the margins of our notebooks. But just how wide is the gap between math and art? According to engineer Iñigo Quilez, it’s all but disappeared.

Quilez understands that math often gets a bad rap with the general public. He points out that most people will fall into one of two camps in regard to their relationship with math. The first are the folks who see math as being cold and mechanical, most likely the former high schoolers who found themselves at the kitchen table at 1 am, alone with their calculus homework, asking: When am I ever going to need to know this? “The second type of people,” Quilez tells ARTpublika Magazine, “are like me; a bit of an artist and a bit programmer. They’re nerds and like art.”

Quilez, a freelancer currently based in San Francisco, is trying to reach both groups to prove to the former that math is not, in fact, cold and mechanical; and to recruit the latter to help carry this message forward. “Just as you use words to make poetry or words to do the reporting of the morning,” he asserts, “you can use maths to do the reporting or you can use them to do something beautiful.”

For most of us, it’s not intuitively clear how typed equations become sweeping landscapes or cartoony portraits. Quilez’s YouTube channel, with its plethora of tutorials, is probably the best place to get a better understanding. In one video, he illustrates the process at a basic level by using one of the most universally accessible tools.

“We are here to paint [a] flower with mathematics by describing every shape and color through formulas,” his voiceover says as he primes a Google Sheets page to be a canvas. He goes on to enter formulas into the cells and columns, assigning colors to numerical values and using equations that are at least vaguely familiar to most people. He occasionally cuts to shots of his notebook, sketching out diagrams of the trigonometry he uses for those of us who have not held too tightly to these concepts over the years. The end result is a cutesy, pixelated flower, delivered with patient explanations and contagious enthusiasm.

The rest of Quilez’s work follows that same blueprint: consider which theorems and formulas will create the shapes he wants, enter formulas into a processing tool (Google Sheets was used in this example, but most creators will use more complicated tools called shaders), and tweak the resulting image.

Perhaps without realizing it, you’ve likely already encountered some of Quilez’s work. Think back to 2012, when Pixar’s Brave told the story of strong-willed Merida adventuring through the Scottish countryside and avoiding her royal duties. Those luscious landscapes through which our fire-haired heroine loved to gallop? That was all created using math.

“I would orchestrate the choreography of nature through the equations,” Quilez says, explaining that he’d start with just individual trees and rocks, then populate entire forests with those elements, before going through and making adjustments to suit the parameters of each scene. Quilez convinced Pixar to use math instead of hiring an army of technical artists to create Brave’s scenery. But the programmer didn’t learn to make art using math for the purpose of saving production companies, like Pixar, time and money.

“Intellectually I don't care if there is a use [for] it. Even if there was no use I would keep making it, I wouldn't mind,” he says.“It gives me total satisfaction when I find something new and think: I'm the first human ever who has been in this particular place of intellectual activity. No one has done this cosine plus logarithm equals a fern, or an oak tree. No one has been here before. And that's amazing, even if it doesn’t impact the world or it’s not a cure for cancer or anything like that.”

With the release of Brave, Quilez had achieved a major goal: To create something beautiful using math to be seen by millions of people all over the world. The downside? Most viewers didn’t realize it was math they were looking at. He asked Pixar to create a short video of him explaining how the vegetation in the film, called Wonder Moss, was made. “It didn't get that many views,” Quilez says, laughing, “so I have to continue my efforts, these messaging efforts.”

Over the years, those efforts have taken many shapes. Quilez co-created ShaderToy, an online platform akin to social media that allows other developers to share their works, and his YouTube channel covers everything from the principles of math-painting to two-hour-long recorded live coding sessions. Quilez shares what he knows freely, hoping to fill the resource gap for a relatively niche computer-art field.

As a teenager, he was curious about how computers work and make images, and by his early twenties, he was using his math and programming skills to create digital graphics. His education in electrical engineering provided the fluency in math he needed to create graphics but, beyond that, it was difficult to find resources to further develop the craft. And so began his journey of self-education.

“I started realizing that, Oh wow, this is amazing!, and no one taught me this. No teacher told me I could use cosines and logarithms to paint a tree, or a mountain.” He concludes: “I have to learn first more of this and then tell everyone about it.” In his pursuit of “telling everyone about it,” he found a community of like-minded creators in the demoscene, a computer-art subculture that features combinations of coding and hand-created art. The demoscene was most popular in Europe in the 90s and 2000s, and Quilez, who was born and raised in San Sebastián, Spain, found it to be a good space for the innovative works he was creating.

From abstract pieces to renderings of dreamy landscapes, Quilez’s work appears to be the fruits of a digital artist’s intense labors. But he hardly considers himself an artist. “I can craft pretty images sometimes — well, for some questionable definition of pretty, because probably what I do is not pretty to many people,” he says, surprisingly self-deprecating for arguably the greatest pioneer in his field.

Perhaps more surprisingly, Quilez doesn’t use reference images when creating his work, nor does he draw on film or literature for inspiration. Instead, he often starts with a challenge. “[I’ll] have an idea — wouldn't it be nice to plant trees on the ground in a hexagonal pattern? How do I teach about hexagons to the computer?” From there, he just starts coding, getting the basic shapes down before adjusting shadows and colors, often improvising as he goes along. “I land on something that is a mix of what I could do and what my taste told me I should do,” he says.

According to Quilez, there are only a few people besides himself that are using solely math and programming to create elaborate works of digital art. There aren’t many universities willing to fund research, and it’s hard to come by a production studio that’s pushing to further develop the techniques used in Brave. This, he says, is all the more reason to jump right in. “For those who want to do innovation, it's very easy to invent new things, because there are so few people exploring it,” he explains.

“If you are here for the innovation and not for the money — although there is money to be made — you're going to be Einstein, you're going to push the field further and invent new things. If people are looking for those kinds of rushes and getting that feeling, jumping into using maths to make pictures — there are lots of things to explore and it's easy to get that.”

Note* All images were provided by and are the property of Iñigo Quilez.

Feature Stories

VOL. 15 


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