Mathematics and the art of fractals

Mathematics and the Art of Fractals by Luca Pinto

Today, I have the pleasure of sharing a post authored by Engineer Luca Pinto, who graciously volunteered (perhaps without much choice, being my partner).

Engaging scholarly reading.

At a certain point, humanity was no longer content with the representation of the ideal world, as envisioned by Plato and refined by thinkers like Newton and Kant. The world of ideas, the “noumenon,” and the “linear” systems seemed inadequate. A quest began for a new path—one that represented reality more tangibly. In the early 1800s, thanks to Gauss, complex numbers found their place in mathematics. Represented as A + iB, these numbers encompass a “real” part (A) and an “imaginary” part (iB, where “i” stands for “imaginary”). This “imaginary” part was devised to encapsulate the impossible: a number that, when multiplied by itself, equals -1! …Try that with all numbers, from the simplest to the renowned Greek Pi, and you won’t succeed!

In 1975, a brilliant IBM employee of Polish origin, Mandelbrot, employed complex numbers to explain how “dynamic” systems change over time. Mandelbrot introduced “fractals.” The term “fractal” signifies “divided by,” as fractals lack a “full” size. This implies that understanding reality requires a “break” in perception, utilizing an “imaginary” element (complex numbers!) and choosing a “fractional” dimension—a dimension existing between a line (size n = 1), a square (size n = 2), and a cube (size n = 3).

The existence of fractals testifies to the intersection of mathematics and art: they reside in the ability to “imagine” new solutions, akin to how an artist delves into themselves and creates their work. Fractals exist in the capacity to “travel between dimensions,” similar to an artist who “opens” a window on a canvas, shattering preconceived constraints.

Fractal equations possess an undeniably simple elegance. Mandelbrot’s entire equation, when inscribed on a blackboard, is far removed from the portrayal of mathematics in movies—those chalkboards filled with complex formulas and curious symbols. The beauty of fractals transcends any false portrayal, akin to Japanese sumi-e or Fontana’s artworks.

Yet, fractals are of immeasurable complexity. Zooming into a graphic representation of a fractal reveals the same form repeatedly. Much like DNA, a fractal repeats itself irrespective of how it’s dissected. Dilation is a fundamental trait of fractals: it remains unchanged and whole, regardless of magnification. Fractals encompass everything within themselves; they cannot be fragmented for analysis. The entirety exists in every part. Fractals, like art, contain a kernel of life that resists dissection. Fragmenting a piece of art, analyzing its details, won’t unveil its essence: similar to a fractal, it’s to be apprehended as an intricate, complete whole.

Fractals aren’t a product of imagination: they’re a living reality, much like art. They exist in Romanesco cabbage—what my companion fittingly dubs “fractal hell” (… and imagine the response from the grocer when we visit the market!), in the alveoli structure of lungs, in snowflake formations, in the motion of particles in a glass of water, and in the dance of flames through the air. Here lies another connection between fractals and art: both narratives speak of life and the tangible world of reality.


Hi, I’m Manuela, urban potter and Italian artist who’s passionate about exploring the expressive potential of clay. With a love for Wabi Sabi aesthetics and the beauty of imperfection, I create bespoke ceramics, paintings, and fine art photography. My studio, nestled in the charming city of Milan, is a magical box where creativity runs wild and beauty is abundant. Whether you’re looking to bring a touch of magic into your everyday life, or simply to explore your creative side through art therapy I’m here to inspire and encourage you on your journey. I bring art into the world and celebrate the beauty in every moment. This blog is my Artistic Salad, filled with creativity, beauty, joie de vivre, passion and tools for creative minds.

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