Talk like a mathematician

Fractal – an image or object that infinitely repeats itself

Render – to create a realistic computer-generated image

Ray tracing – a rendering technique that calculates the paths of individual rays of light through a computer-generated scene

Ray marching – an adaptation of ray tracing in which each ray is advanced along its path in steps

Distance estimator – a mathematical function used in ray marching to estimate how far a ray is from a surface

Multicomplex numbers – a multidimensional extension of complex numbers (imaginary numbers –the square roots of negative numbers – are examples of complex numbers)

Commutative – a system in which the order you apply an operation does not change the result (e.g., a + b = b + a)

What do trees, ferns and broccoli have to do with mathematics? The answer: they represent the mathematical concept of fractals – objects that repeat themselves at smaller and smaller scales. A tree trunk splits into large branches which split into smaller branches which split into twigs, while each fern leaf and broccoli floret is a miniature version of the whole fern frond or broccoli head.

“Strictly speaking, these natural examples are only approximate fractals as they repeat over a limited range,” explains Assistant Professor Pierre-Olivier Parisé, a mathematician at Université du Québec à Trois-Rivières. “In a true mathematical fractal, the object appears within itself infinitely, with new details that keep appearing no matter how far you zoom in.”

Previous mathematicians have created techniques to visually render 3D fractals, and Pierre-Olivier and his collaborators have built on these, which has involved developing new mathematical methods. “This is, first and foremost, a piece of foundational mathematics,” he says, highlighting that one of the joys of being a mathematician is having the opportunity to simply push the boundaries of our knowledge. However, his technique can also be used for practical applications. “Fractals have always been good at drawing people into mathematics because you can appreciate their beauty without understanding them,” he says. “I hope these 3D images can serve the same purpose – showing that even something as forbidding as multicomplex analysis has a vivid tangible side that can simply be an object of art.”

What is ray tracing?

Ray tracing is a rendering technique used in computer graphics to produce realistic images (complete with accurate shadows, reflections and lighting) by simulating the physical behaviour of light. It calculates the paths that individual rays of light take as they travel through a scene to reach a viewer. However, as most light emitted from a source will scatter into the scene and never reach the viewer, tracing every ray from a light source would be a waste of computational power. Instead, the ray tracing technique starts from the viewer’s perspective and traces rays backwards into the scene, calculating the path of one ray for every pixel in the image.

“Each ray is followed until it hits the first surface in its path, and at that point the program must work out what colour the pixel should be by gathering information about the lighting where the ray landed,” explains Pierre-Olivier. To do this, the program casts a shadow ray to detect if the object is lit by the light source or if it is in the shadow of another object, plus a reflection ray to see what the surface reflects and a refraction ray to see how light bends through transparent objects. “A single frame in an animated film can require millions of rays which takes hours of rendering time but produces high quality images,” says Pierre-Olivier. “For example, Pixar first used ray tracing for Cars in 2006 to render realistic reflections, shadows and metallic surfaces.”

Why does ray tracing not work for fractal objects?

While graphic designers can use ray tracing to create realistic-looking objects in animated scenes, the technique does not work for rendering true fractal objects. “The real difficulty appears the moment we pass to a genuinely 3D fractal,” says Pierre-Olivier. “As a solid object sitting in space, to obtain a realistic image we must cast rays, find where each one meets the fractal object and light the surface it lands on.” But this is where the assumptions built into ordinary ray tracing break down. Because fractals have infinite detail at every scale, no matter how far you zoom in, the computed ray will never actually reach the object’s surface.

How does Pierre-Olivier render 3D fractals?

To solve the problem of rendering 3D fractals, graphic designers need to use an adapted version of ray tracing known as ray marching, where the ray is advanced along its path from the viewer into the scene in steps, until it is close enough to the fractal to be considered to have reached the surface. At this point, the program follows the usual ray tracing technique to determine what colour the pixel should be. The length of each step in the ray marching technique is defined by a distance estimator – a mathematical function that estimates how far the ray is from the fractal at each step. “Producing such an estimator is a genuine mathematical problem,” says Pierre- Olivier.

Inspired by a distance estimator found for complex numbers, Pierre-Olivier and his collaborators have developed a new mathematical method for calculating distance estimators that uses multicomplex numbers in a commutative number system that models higher dimensional space. “As a result, the properties of 3D fractals can be mathematically proved, rather than merely observed as an image on a screen,” he says. “This means they can be rigorously analysed. The key value of this technique is that it provides a method for exploring the mathematics behind fractals.”

Pierre-Olivier’s work sits at the intersection between mathematical multicomplex dynamics and practical computer graphics, and he hopes both mathematicians and graphic animators will benefit from his distance estimator method to share the incredible beauty of mathematics with the world.

Assistant Professor Pierre-Olivier Parisé

Department of Mathematics and Computer Science, Université du Québec à Trois-Rivières, Canada

Field of research: Mathematics

Research project: Developing a mathematical method to create and analyse 3D fractals

Funder: Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant RGPIN-2026-04740

Website: mathopo.ca

About mathematics

“Mathematics is one of the few subjects you can pursue with almost nothing but your own mind,” enthuses Pierre-Olivier. “Once you have learnt its language, you are free to wander wherever your curiosity leads.” For Pierre-Olivier, the hardest part of being a mathematician is not solving mathematical problems but finding the right problem to tackle in the first place. “I want to solve problems that are hard enough that cracking them will actually mean something,” he says. “Searching for a solution to a difficult mathematical problem feels like a quest! And the quest never ends, because every problem you solve opens new questions.”