Can beauty be defined mathematically?

Divine Proportions

Throughout the history of art in the Western world, there is a number that keeps coming back, one that symbolizes perfect balance – a proportion considered to evoke a sense of perfect beauty. This number is phi, or the Golden Ratio. Also known as the Divine Proportion or the Golden Spiral, phi is an irrational number, a formula that contains itself into infinity. Art historians have pointed to the presence of the Golden Ratio in different artforms, such as architecture and painting, but also in natural environments, like the curve of a seashell.

The Golden Spiral is an infinite spiral, a self-similar smooth curve that extends infinitely both within and outside of itself. The spiral has become popularised for the sense of beauty that it evokes, for its seemingly perfect, infinite balance. While it revels in its simplicity, the Golden Spiral can be likened to another form of mathematical phenomena – patterns which are also both infinite and self-similar, but which present something different: infinite complexity. Welcome to the world of fractals.

Infinitely Complex: Fractals in Nature

Fractals are commonly understood to be found in nature: you can see them in the arrangement of sunflower seeds, in coastlines, in clouds. You can see them in leaves, in vines, in romanesco broccoli. Undeniably, the beauty of fractals can be found in many different places in nature. What we see, however, are not real fractals, but approximations. Fractals, by definition, are infinite. Simple mathematical formulas within which infinitely complex worlds are contained.

It was Benoit Mandelbrot, in an exploration of coastlines, who first started delving into these fractals as a phenomenon in themselves, ultimately arriving at the Mandelbrot set: a visual representation of fractal formulas based on the Julia set. Following Mandelbrot’s rendition, others emerged, such as the Mandelbulb: the first three-dimensional depiction of a fractal. 

Fractal Worldbuilding

Unsurprisingly, fractals have caught the attention of several mathematicians and artists. This can definitely be said of Julius Horsthuis – an Amsterdam-based artist exploring the intersection of mathematics and art – who has dedicated a large part of his career to studying fractals. Although he works with fractal-rendering software Mandelbulb 3D, he does not see himself as the creator of these shapes. Instead, Horsthuis considers himself to discover them, as a documentary filmmaker would. The fractals, he argues, already exist – what he does is the work of finding them and displaying them to the viewer for their own interpretation.

Does the artist-mathematician build worlds through the exploration of fractals? Or are these worlds instead built by fractals, simply waiting to be discovered? On July 18th, Nxt Museum will host “Art, Science and Other Worlds: Julius Horsthuis in Focus” – an artist talk and panel discussion with sci-fi writer and editor Rochita Loenen-Ruiz and science writer and journalist Margriet van der Heijden (Dr.). The panel discussion will centre around fractals, art, science, and worldbuilding.

Find out more about the event here!