Cartoon Girl in one Formula

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Can you create the portrait of a cartoon girl reminiscent of the Frozen movie using only mathematics? Yes. This mind-blowing video by 3d graphics genius Iñigo Quilez shows exactly how, using only high-school mathematics, a keen eye and some patience.

You know you can make computer graphics with a mathematical formula such as the one for a sphere, or a box, or other building blocks. You can probably guess that there are formulae for colouring, lighting and shadows. You might also know about “noise” and how “randomness” can be made to produce something like a TV tuned to a dead channel.

In intermediate high-school mathematics you learn to combine formulae. You could compose the formula for a sphere with that of a thin cylinder and make a lollipop shape. What you might be surprised to learn is exactly how to use this compositional technique in an entirely artistic process to produce a compelling cartoon “selfie girl”. How do you make freckles? What about lips and eyebrows, a hoodie, braided hair and a frozen snowscape.

Not only does Iñigo Quilez masterfully create all these things, he shows you how it is done and even lightly animates the figure to complete the illusion that the girl is posing for a photo.

What is unique about this approach is that it shows the power of mathematical modelling when combined with an artist’s eye. Artists do not need to use a brush to create art any more than they need to use a computer mouse. While there may be a defensible tradition of computer art that is constructed by the composition of hand-drawn elements, what this video shows us is that if you can describe in the language of maths the curve that your hand would circumscribe, you can type it into a computer and see the result appear before your eyes like magic.

Algorithmic computer graphics is more usually relegated to the realms of fractals and the infinite lacy filigree of alien geometric spaces. But trippy graphics is no more essential to the medium than the naked bosom of a Venus is to oils.

What makes this form of art peculiar is that it is expressed in a language most of us insist on remaining illiterate in, despite living increasingly in a world defined by algorithms. The attitude that mathematical illiteracy is an excusable allergy for a modern, educated person feels to me like more than a missed opportunity. Consider that its mathematical notation is broadly representative of the impenetrably cryptic and is often aped to refer to the toxic effluvia of crazed genius. Nobody is expected to know it and yet we are all alarmingly numb to the message that our world is increasingly defined by algorithms.

So think of this as a reminder of the oft-cited beauty laying wait beneath the veil of our own ignorance in the expressive, magical realm of maths. Why not lift that veil every now and then?

Having said that i appreciate that while this composition uses what is declared to be high-school maths, it does move quickly to compose a large aggregation. Click through to the detail without preparation and you might be reminded that approaching the uncarved marble block with a renewed willingness to learn does not imply that the tools in your hands will reveal a Statue of David. Take heart and take the next step.


This is not the first formulaic painting Quilez has done. There are a series of how-to videos for several of his impressive creations on his youtube channel. Note he is also the creator of Shader Toy – a tool designed specifically for creating graphics using this approach.



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