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.


Architectural Geometry

“In an architectural context, geometry is often used as an ordering device, that is, a way to link spaces to functions.” (Williams, K; Ostwald, M.J. 2017)

In the introduction to vol.19 of the Nexus Network Journal, Manifestations of Geometry in Architecture, Kim Williams and Michael J. Ostwald highlight that the geometry of architecture is not only a functional measure to ensure the most efficient and structurally sound building, but also to create a more appealing aesthetic.

“Once a structure is codified and built, geometry can add beauty to a design.” (Williams, K; Ostwald, M.J. 2017)

This idea rings true to me as a creative who has explored geometric graphic design for a long time without paying mind to the precise mathematical foundations that fortify such design. I rarely associate mathematics with beauty, not as much as with its creative cousins art and music, but in the precise geometry of buildings there can be found a great sense of rhythm, movement and intent.

Architectural design can range from the simple to the complex, just like with any other creative field. Examples of creative architecture include the Corpus Museum in the Netherlands which advertises itself as a “journey through the human body,” and the Spiraling Treetop Walkway in Denmark, created to give tourists a bird’s-eye view of the forest without disturbing the natural surroundings. Both embrace nature in their own unique way, and both attempt to represent the purpose (subject) of the building (Fig. 1).


Geometric architecture also encompasses the bizarre. For example, the Dancing House in Prague (Fig. 2, left). Designed by Frank Gehry and Vlad Milunič, this unusual building that leaps over the intersection symbolises the direction the Czechoslovak people have taken to move away from the totalitarian rigidity of the past.


The other two photos (Fig. 2, right) are buildings that do not conform to conventional architecture but rather deviate to create a new and unexpected environment and an entirely different experience. I thought about how this relates to contemporary design and my own work. I feel that we are often prone to working within pre-defined boundaries and expectations, but sometimes it is necessary to not just step outside the box but take a positive leap. The above architects were not afraid to break tradition; to them, it was extremely important to do so.

Architect Frank Lloyd Wright designed impressive buildings that worked alongside their surroundings rather than dominated them. Lloyd Wright himself described his buildings as “organic architecture” (Pfeiffer, B. 2004) and you can find examples of this in his work:


In Geometry And Architecture, Steven Firth writes:

“The uses of geometry in architecture tend to be instrumental, dovetailing with the process of building as a species of order. The modulation and arrangement of plans and the surfaces of elevations go hand in hand with this fiction, but also depend upon geometrical play.”

The implication here is that geometry exists to bring order and structure to a landscape or a people, but I cannot help but feel when looking at Lloyd Wright’s harmonious meshing of man-made and nature that there is a synergy, with the one blending seamlessly with the other. The water flows beneath the house and out over the rocks, much in the same way that people flow around and through the house, and in the way the creative ideas flow through the architect or designer.

The notion that geometry can influence a person’s perception of a building cropped up frequently in my reading. In Aesthetics of Simulated Soiling Patterns on Architecture, Carlota M. Grossi and Peter Brimblecombe explore the reactions in people to different soiling found on buildings, in an attempt to see if there was a preferred type of pattern. The idea is that if there is a less obnoxious soiling of architecture then building managers wouldn’t have to have their buildings cleaned so often. To me this is a study with only a tenuous connection to my own research, but what I found interesting was the results: people generally prefered a more uniform type of soiling–a pattern that they could make sense of–rather than a randomised formation.

The more I dig into geometry and shapes, the more it seems that we are drawn to pattern, repetition and order.

Finally, I looked at geometry at work in architecture from cultures other than my own. One fascinating article written by Carol Bier is Art and Mithāl: Reading Geometry as Visual Commentary. Bier explores the idea that the decoration on Islamic monuments is often viewed through a Western lens (Bier, C. 2008), but the pervasive use of pattern must be an expression of something more than simple ornamentation. The implication is that elaborate decoration somehow implies a lack of deeper meaning and significance. Bier also writes:

“It now appears that aperiodic patterns with five-fold symmetry related to that of quasicrystals may have been understood by craftsmen in Iran, if not mathematicians, hundreds of years before this phenomenon was discovered in the West.”

It might be naive of us to assume that geometric pattern relates to decoration or mathematics.

In the year 2000 I spent three months in India, and had the opportunity to see the Taj Mahal up close. The mausoleum itself is comprised of countless geometric shapes and precise lines and symmetry, but inset into the white marble are row upon row of black onyx script. The two details combined created an incredible sense of peace, flow, history and reverence.


Again, it fascinates me how geometry can be used in architecture to evoke different feelings from those viewing it. I do wonder how an opinion might shift if the story of a building is changed: if you transformed the Taj Mahal, built as an elaborate mausoleum by emperor Shah Jahan for one of his wives, into the stately home of a wealthy oil family, built to show off status and power. Suddenly the romantic spirit of the building is changed. The geometric components remain the same, the patterns are no less intricate and the work that went into it was no less laborious, but now it becomes a structure of arrogance and vanity.

This relates to my Practice 1 project, where I want to look at how small changes made to a geometric design can change people’s opinions and perceptions. My project relates to branding and logo development, but from my research into architectural geometry I can take away that geometry does not always have to be rigidly ordered to work; it can be combined with more fluid design (elements surrounding it) and they can work together harmoniously.


Bier, Carol (2008): Art and Mithāl: Reading Geometry as Visual Commentary. From Iranian Studies, Vol. 41, No. 4, Sciences, Crafts, and the Production of Knowledge: Iran and Eastern Islamic Lands.

Firth, Stephen (2010): Geometry and Architecture. Architectural Theory Review, Taylor & Francis Online.

Grossi, C. M.; Brimblecombe, P. (2004): Aesthetics of Simulated Soiling Patterns on Architecture. School of Environmental Sciences, University of East Anglia, Norwich.

Pfeiffer, Bruce Brooks (2004): Frank Lloyd Wright, 1867-1959: Building for Democracy. Taschen, Germany.

Williams, Kim; Ostwald, Michael J. (2017): Manifestations of Geometry in Architecture. The Nexus Network Journal, Architecture and Mathematics, vol. 19. Birkhäuser Publishers, Switzerland.