*“Why is geometry often describes as “cold” and “dry?” One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line” *(Mandelbrot, 1982 p. 1)

These are the words of the polish-french mathematician Benoit Mandelbrot, who famously introduced the concept of fractals and its applications. He named the phenomenon fractal, derived from the latin word fraʹctus, meaning broken. In the introduction of his book The fractal geometry of nature (Mandelbrot, 1982) he states that most fractals tend to have fragmentation and statistical regularities or irregularities occurring at all scales.

**Fractal curves**

Fractals differ from Euclidian geometry and its straight lines and smooth curves by being fractioned and not having a tangent at any given point. Fractals have been used since the end of the 17th century. There are numerous fractal curves discovered and named by different mathematicians, for instance the Koch curve, the Minkowski curve and the Peano curve. (NE)

When discussing fractals there are a few aspects to consider, one of them is the fractal dimension. Bovill (1996) offers the following definition:

*“Fractal dimension (…) is a measure of the extent to which a structure exceeds its base dimension to fill the next dimension”* (Bovill, 1996 p.14).

The above mentioned curves, Koch, Minkowski and Peano, have different fractal dimensions ranging between 1 and 2. The Peano curve has a fractal dimension of 2, implying the curve completely fills a two-dimensional plane when iterated to infinity. The Minkowski curve (fractal dimension: 1.5) fills the plane to a higher degree compared to the Koch curve (fractal dimension 1.26), thus having a larger fractal dimension.

**Self similarity**

Self similarity is another important concept within the field of fractals. The fragmentation tends to recur identically at all scales. Although, when discussing natural occurring fractals, the concept of infinite iterations becomes less useful. Naturally there are physical limitations such as molecule size and finally atom size. Mathematical fractal structures on the other hand, reaches no such limit.

*“Fractals are objects with roughness at all scales or, for natural fractals, over at least several orders of magnitude of scales.”* (West, Deering, 1994, p.9)

## FRACTALS IN NATURE

In nature, energy efficiency is crucial, and high performing structures are created with simple material and the information itself being the key to success. In order to minimise the energy and material spent on this information, nature has conceived an impressive ratio between amount of information put into the system and the complexity of the outcome. Self similar structures require only one rule, which applies in all scales, and by being information efficient fractal structures are created. (Gruber 2011, p.97)

Natural occurring fractals can be found in the branching of a tree, the veins of a leaf, mountain ridges, rivers, vegetables and in the bronchial structure of lungs, to name a few.

In The self-made tapestry (Ball, 2001), the author analyses natural occurring fractals. When treating the topic of branches, he states that there are branching patterns with similar qualities occurring in both organic and physical systems, thus implying a universality in the geometry.

*“This ubiquity of branched formation in both the living and inorganic worlds begs the question of whether their formation can be ascribed some unifying features, in line with the idea of universality in pattern formation…” * (Ball, 2001 p. 110)

Ball writes about DLA (diffusion-limited aggregation), a theoretical model developed in order to describe the way in which particles of dust aggregates in air. For instance the DLA model could be used when describing branched metal crystals, formed in a process called electrodeposition. Ball connects the DLA model to Mandelbrot’s fractals by invoking self-similarity independent of scale. When returning to the topic of fractals, Ball states the DLA model useful when analysing bacterial growth, crack patterns and even as a tool to describe urban growth. The later, however, needs additional models and parameters in order to be accurately described and analysed. (Ball, 2001)

## FRACTALS IN ARCHITECTURE

An aspect of fractal architecture is how it affects humans from an environmental psychological point of view. In the article Fractal Architecture Could Be Good For You (Joye, 2007) the author presents numerous architectural examples where fractal geometry plays an important role, from Hindu temples, where the self repeating and self-similar components are supposed to reflect the idea that every part of cosmos contain all information about the whole cosmos, to gothic architecture, with a high degree of self similarity and complex detailing.

Joye discusses whether the positive effect nature and natural elements has on humans may in fact be derived from their fractal dimension, rather from the natural elements themselves. He writes that similar positive psychological and physiological responses can be triggered by elements not found in nature, such as in certain paintings and architectural elements.

Joye draws the conclusion that our love for certain fractal dimension can be derived from our biological urge to find suitable habitats. Environments with high fractal dimension offers too much coverage for possible danger, whereas environments with lower fractal dimension do not provide resources enough to prove sustaining, implying that mid-range fractal environments are suitable. This biophilic trait is the reason that architecture containing fractal geometry is beneficial for humans.

An example that may support Joye’s theory is the indigenous fractal architecture found in certain African villages. Mathematician Ron Eglash discovered numerous examples of self-similar structures, which did not only arrange dwelling units, rooms etc. but also corresponded to the social and religious hierarchy. Furthermore Eglash emphasised the self-similarity, not similarity between occurring fractal structures, in order to not convey a generalizing view of African vernacular architecture.(Eglash 2007)

**Fractal biomimetics**

Comolevi Forest Canopy is an example of biomimetics where the fractal nature of a tree canopy was imitated in order to achieve the effective cooling system natural leaves and branches constitutes. The Comolevi Forest Canopy works by filtering the sun rays and shading surfaces below without blocking the air flow – just the same way a tree does. The Japanese company Lofsee Co., Ltd. aimed to solve the urban heat and in lack of natural vegetation they managed to invent a system mimicking nature without having to actually work with natural elements, thus not needing deal with the issues connected to vegetation in an architectural or urban environment, such as draining systems, or slab thickness etc. (Ask Nature)

Biomimetics can also occur at nano-scale, one approach is focusing on structural elements. The lotus-effect is a term now used for describing self-cleaning. The fractal structural patterns of leaves, but also insect wings and exoskeletons makes it hard for dirt particles to stick, furthermore the surfaces repel water by using wax excretions. (Gruber, 2011 p. 30-31)

When comparing the two examples above the former seems more realistic in a fairly short-term perspective, at least regarding production, since it is not depending on accuracy at nano-scale. When choosing what scale to execute fractal biomimetics, many aspects need to be taken into consideration, for instance what the purpose is, aesthetic, performing, or both, available technology, economy etc. When defined, a wide range of different fabrication methods might be suitable; it might be 3d-printing for mid-range scales, chemical procedures for nano-scales, or more traditional production methods for larger scales.

The fractal structures of the African villages examined by Ron Eglash shows that also vernacular production methods can create intricate mathematical structures, although there’s no direct connection to biomimetics.

## THOUGHTS

*“Just saying that a structure is fractal doesn’t bring you any closer to understanding how it forms. There is an unique fractal-forming process, nor a uniquely fractal kind of pattern. The fractal dimension can be a useful measure for classifying self-similar structures, but does not necessarily represent a magic key to deeper understanding”* (Ball, 2001, p. 117)

The universality and omnipresence of fractals can be regarded as an obstacle when thinking of possible applications – how can fractals be useful when they are so diverse? Even the title of Mandelbrot’s book, The fractal geometry of nature” (1982), suggests ubiquity.

If so, if fractal geometries can be found everywhere, how can they be incorporated into architecture? The key as well as the challenge of further architectural research within the field of fractals might be finding fractal systems that both appeals to our predisposition of biofilia, or love of certain fractal dimensions, as well as demand of performance. If such fractals can be found, we will be able to create high performing structures, which incorporates psychological and aesthetic, as well as environmental aspects into an integrated design.

### REFERENCES

Ask Nature, Comolevi Forest Canopy, accessed 2016-09-07, http://www.asknature.org/product/9c1c823f81e0e7218a3286d38595d0a2

Ball, Philip (2001[1999]). The self-made tapestry: pattern formation in nature. Oxford: Oxford University Press

Bovill, Carl (1996). Fractal geometry in architecture and design. Boston: Birkhäuser

Eglash, Ron. (2007). The fractals at the heart of African designs. TED Global, accessed 2016-09-07, [Video file]. Retrieved from https://www.ted.com/talks/ron_eglash_on_african_fractals

Gruber, Petra (2011). Biomimetics in architecture: architecture of life and buildings. Wien: Springer Verlag

Japan for Sustainability (2010) Kyoto University Develops Artificial Shade Structure Inspired by Natural Trees, accessed 2016-09-07, http://www.japanfs.org/en/news/archives/news_id030459.html

Mandelbrot, Benoît B. (1982). The fractal geometry of nature. Rev. ed. San Francisco: Freeman

Nationalencyklopedin, fraktal, accessed 2016-09-07, http://www.ne.se/uppslagsverk/encyklopedi/lång/fraktal

Pohl, Göran. & Nachtigall, Werner. (2015). Biomimetics for Architecture & Design Nature – Analogies – Technology. 1st ed. 2015.

The Mind Unleashed (2014) 30 Beautiful Photographs of Fractals in Nature, accessed 2016-09-07, http://themindunleashed.org/2014/10/30-beautiful-photographs-fractals-nature.html

West, Bruce J.; Deering, William (1994) Fractal physiology for physicists: Lévy statistics. *Physics Reports*, Volume 246, Issue 1-2, p. 1-100.

Yannick Joye (2007) Fractal Architecture Could Be Good for You. *Nexus Network Journal*, Vol. 9,No. 2, p. 159-382

Featured image

Troup Dresser. Unfurreling fern, accessed 2016-09-10, https://www.flickr.com/photos/23271361@N06/5793227295/