Naturally Occurring Geometry

One fascinating resource I encountered in my research is the book The Fractal Geometry of Nature, by Benoit B. Mandelbrot. It has been important to me to write about naturally occurring geometry as it crops up in a lot of my reading and is something I am inspired by in my own work.


Nature is fragmented and multi-faceted. This is where Mandelbrot got the idea for the term fractal, from the Latin frangere, meaning “to break” (Mandelbrot, 1982). But the fragmentation of nature means that there are an infinite amount of possible geometric shapes and formulas, surpassing Euclidian theories that there is a finite number of geometric shapes available to us. In art and design, Mandelbrot suggests that we are more likely to accept abstract art that resembles fractals because it is subconsciously familiar to us.

M. C. Escher, whose famous tessellations were inspired by the Islamic tiles that he saw while on a trip to the Alhambra in Spain (Van Dusen, Scannell and Taylor. 2012), is often described as a fractal artist with a keen interest in the properties of patterns found in the natural world. His pattern Circle Limit III repeats at various sizes and scales without losing its geometric formation (Fig. 1) and is reminiscent of naturally occurring geometry in plants and flowers. Similarly, the Koch Snowflake, a fractal curve that was presented by Helge von Koch in 1904, is a scalable geometric pattern that can repeat indefinitely (Fig. 2).


Mandelbrot’s vision of geometry was unrestrained by the mathematical theory that preceded him, although this is not to say he does not respect those theories that came before; rather, he builds upon them and tries to bring them into better harmony. He believed that mathematicians shied away from the possibilities that nature presented, and attempts to meld science with philosophy. This blending of precision and boundless creativity is a pairing that mirrors the job of a designer working in the present day.

“One must also recognize that any attempt to illustrate geometry involves a basic fallacy. For example, a straight line is unbound and infinitely thin and smooth, while any illustration is unavoidably of finite length, of positive thickness, and rough edged. Nevertheless, a rough evocative drawing of a line is felt by many to be useful, and by some to be necessary, to develop intuition and help in the search for proof. And a rough drawing is a more adequate geometric model of a thread than the mathematical line itself.” (Mandelbrot, B. 1982. P22)

Here Mandelbrot expresses the difficulty in representing something as broad and infinite as a geometric line, but appreciates that we must find ways to root our understanding of geometry—in the form of drawings and art—and pay homage to the shapes present around us. As an artist and designer, this is interesting to consider: shapes are merely representations of things we might otherwise consider unfathomable. We are trying to make visual sense of something as vast and infinite as the universe.

Mandelbrot also points out that while many believe that mathematics, music, art and architecture seem to be related to one another, it is only a superficial connection, as—in the case of architecture—one building can follow strict Euclidian geometry, but another building can be more rich in fractal aspects (Mandelbrot, 1982). I see the similarities drawn between geometry and art, mathematics and music often, most recently when studying Swiss Style graphic designer Josef Müller-Brockmann’s poster for Beethoven’s symphony, where the geometric choices are believed to have represented the structure of the symphony. In my Practice 1 assignment I look at modern geometric Swiss Style art used to portray grunge band posters from the 1980s-1990s, although I struggle to draw the connections and similarities between something that is rigidly structured, ordered and precise with something that stood for distortion, being true to yourself and individuality.

Mandelbrot also writes: “It is widely held that minimal art is restricted to limited combinations of standard shapes: lines, circles, spirals, and the like. But such need not be the case. The fractals used in scientific models are also very simple (because science puts a premium on simplicity). And I agree that many may be viewed as a new form of minimal geometric art.” (Mandelbrot, B. 1982)


These snowflakes are from a winter-themed graphic design project I worked on last year for a local restaurant and shop. At the time I was not thinking in terms of geometry, but I was faced with the problem of deciding on the shapes and formations of the snowflakes, as there were so many possibilities. Even before I began reading about geometry I was aware of the challenge of representing natural complexity in art.


In Fractal Architecture: Organic Design Philosophy in Theory and Practice, James Harris explores the fractal geometry of nature as the inspiration for some forms of architecture where the natural “recursive mathematical derivation” of form creates a new structure that is removed from the more logical and rigid Euclidian geometry (Harris, J. 2012). Harris believes that philosophers and mathematicians created geometric theories to cope with the unfathomable randomness of the universe, which was beyond their comprehension (Harris, J. 2012). I believe this is a natural human behaviour—the need to make sense of the vastness of the universe and all the endless possibilities within. We search for and even yearn for our own kind of structure within the study of complex, naturally occurring structures.

In the paper Is the Geometry of Nature Fractal? written for Science Magazine, David Avnir, Ofer Biham, Daniel Lidar and Ofer Malcai offer an alternative idea:

“Fractals are beautiful mathematical contructs [sic] characterized by a never-ending cascade of similar structural details that are revealed upon magnification on all scales. Over the past two decades, the notion has been intensively put forward that fractal geometry describes well the irregular face of nature.” (Avnir, O. et. al. 1998)

While it is difficult to believe that all geometry in nature holds a perfect symmetry and fractal behaviour, I do not think that theorists like Mandelbrot were trying to constrain nature within the definition “fractal”, but rather offer another view of something that cannot easily be understood or contained. Again, a way for us to wrap our minds around this deeply complex and mysterious world surrounding us.

As part of my Practice 1 assignment, I am looking at the multi-faceted nature of geometry and how low-poly art emulates the geometric shapes found in crystals and ice—neither of which are uniform in size and shape, and yet both of which form a structure with many faces that reveal different colours and reflections. Mandelbrot raises the question in the opening of his book “Why is geometry often described as cold and dry?” and I would like to explore geometry used to expose a depth and intricacy behind the “cold, dry” surface.


Avnir, David; Biham, Ofer; Lidar, Daniel; Malcai, Ofer (1998): Is the Geometry of Nature Fractal? Science, Jan 2nd – Vol. 279, Issue 5347, pp. 39-40.

Harris, James (2012): Fractal Architecture: Organic Design Philosophy in Theory and Practice. University of New Mexico Press.

Mandelbrot, Benoit B. (1982): The Fractal Geometry of Nature. W.H. Freeman and Company, New York.

Van Dusen, B; Scannell, B. C; Taylor, R. P: A Fractal Comparison of M.C. Escher’s and H. von Koch’s Tessellations. Published in Fractals Research, School of Education, University of Colorado, Boulder, Colorado, USA.


The Scutoid Has Landed

Just a short post about something that happened recently which relates to my chosen keyword geometric. A paper written by Pedro Gómez-Gálvez and Pablo Vicente-Munuera was published in Nature Communications in July 2018 introducing a new geometric shape that is linked to the metamorphosis of tissue cells during organ development. This shape is called a Scutoid*.


“As animals develop, tissue bending contributes to shape the organs into complex three-dimensional structures. However, the architecture and packing of curved epithelia remains largely unknown. Here we show by means of mathematical modelling that cells in bent epithelia can undergo intercalations along the apico-basal axis. This phenomenon forces cells to have different neighbours in their basal and apical surfaces. As a consequence, epithelial cells adopt a novel shape that we term “scutoid”.” (Gómez-Gálvez, P. et. al. 2018)

Nature has found a way to conserve energy and pack cells more efficiently, and we are only now uncovering this phenomenon. While the science of the scutoid falls outside of my main stream of research, I do find it fascinating that yet again geometry plays a major role in nature as much as it does in mathematics. It is this blending of science and the creativity of nature that arises time and time again. The more I read about geometry the more I feel that it links everything–from art and design to the fabric of living creatures and the universe itself.

Gómez-Gálvez and Vicente-Munuera also write:

“In addition to this fundamental aspect of morphogenesis, the ability to engineer tissues and organs in future critically relies on the ability to understand, and then control, the 3D organization of cells.” (Gómez-Gálvez, P. et. al. 2018)

Discovering this deeper level of our biology will no doubt have an impact on how effectively and efficiently we approach medical and scientific problems in the future. The new shape could also play a role in art and design. 


Pedro Gómez-Gálvez, Pablo Vicente-Munuera, Antonio Tagua, Cristina Forja, Ana M. Castro, Marta Letrán, Andrea Valencia-Expósito, Clara Grima, Marina Bermúdez-Gallardo, Óscar Serrano-Pérez-Higueras, Florencia Cavodeassi, Sol Sotillos, María D. Martín-Bermudo, Alberto Márquez, Javier Buceta, Luis M. Escudero (2018): Scutoids are a geometrical solution to three-dimensional packing of epithelia. ( Nature Communications, 2018.

* According to Science Daily (28th July 2018), the new shape was labelled “Scutoid” because of its resemblance to the scutellum, the posterior part of an insect thorax or midsection. (