Fractal Patterns: The Trademark of Nature to Create Beautiful, Healing and Engaging Spaces

Terrapin Bright Green has identified a need and growing interest for a peer-reviewed, design-oriented document addressing key learning points for working with fractals. Terrapin (Browning et al., 2014) classified fractals under the pattern of biophilic design ‘Complexity and Order’ [P10], as to indicate “rich sensory information that adheres to a spatial hierarchy similar to those encountered in nature” which engenders restoration from mental fatigue, stress recovery, enhanced creativity, relaxation, and excitement. [If you’re not familiar with the context of biophilic design find a brief historical overview at the end of this paper].

A Toolkit for Working with Fractals will be published soon on Terrapin’s website. If you’re interested, you can be notified when available by leaving your email address here.

By identifying the most appropriate data and resources, this toolkit hopes to advance the understanding and discussion of fractals for direct application from the design community and related sectors.

Significant research insights such as this typically take an average of 17 years before making it into industry practice.

This work is the result of a joint effort between Terrapin Bright Green, European Cooperation in Science and Technology’s COST RESTORE action: “REthinking Sustainability TOwards a Regenerative Economy”, Eurac Research, International Living Future Institute and many other partners, including industry professionals and academics. 

Fractals Design, Architecture and Art in Human History

Fractals have permeated cultures spanning across many centuries and continents, classical art and vernacular architecture from the column capitals of ancient Greece, Egyptian, Aztec, Inca civilizations, the art of Ancient Mayans, Islamic and Hindu temples, Angkor Wat in Cambodia, the Eiffel Tower in Paris, and the structures of Santiago Calatrava. Fractals are also evident in such well known works as those of Botticelli, Vincent van Gogh, and Jackson Pollock (Taylor, 2002; Taylor, Micolich, & Jonas, 1999; Taylor, Micolich, & Jonas, 2002; Taylor et al., 2007). Their visual properties were also explored by mathematicians when Benoit Mandelbrot published The Fractal Geometry of Nature (1982) in which he catalogued nature’s statistical fractals and discussed them using mathematical methods for their replication. Fractals constitute a central component of human daily experience of the environment (Taylor & Spehar, 2016). While extensive research has documented the negative effects of environments that do not have a complement of rich experiential aesthetic variety (Mehaffy & Salingaros, 2013), their role in art and design can continue to grow and diversify, creating architecture, interiors and products designed for human needs (Taylor & Spehar, 2016). Over the past two decades, interdisciplinary teams have confirmed that the aesthetic qualities of nature’s fractal patterns can induce striking effects on health (relevant studies will be published on the toolkit).

A close-up photo of the bright center of a star cluster:

Curious fact about fractals [1]: The pioneer research on stress-reducing fractals was funded by NASA with the aim of maintaining the health of astronauts during long missions away from Earth’s scenery (Taylor, 2006). How can we expose people to fractals in Earth-bound everyday stressful activities?

Fractals as a Natural Phenomenon

Nature is characterized by a particular type of statistical geometry, different from Euclidean geometry, called fractal geometry (Mandelbrot, 1982). Humans evolved in complex and sensory rich natural environments, where all of natural structure are fractals on a hierarchy of scales, from the large to the microscopic. At present, the majority of global population live in built urban environments characterized by minimalist/Euclidean architecture (e.g., straight lines, right angles, empty planes, rectangles, cubes, cylinders, etc.) resulting in un-nurturing and inadequately sustainable spaces. Abundant research in environmental psychology suggests that humans need fractal scales, rich patterning, spatial layering, interlocking geometries, which are typical of nature. In other words, humans like fractal features in their habitat because these features have survival value. Ultimately, a good habitat is where people can function at their optimal potential (Kellert & Calabrese, 2015). When nature’s trademark for complexity and order is applied to architecture and design this can result in nourishing, satisfying and sustainable spaces, products and materials.

Sheikh Lotfollah Mosque, Isfahan, Iran:

Curious fact about fractals [2]: Ornament might be what humans use as a kind of ‘glue’ that allows different parts of the environment to echo and connect to one another in a cognitive sense and even deeper functional purpose. It now appears that the removal of ornament and pattern has far-reaching consequences for the capacity of environmental structures from coherent, resilient wholes. Ornament is not a mere decoration but an important tool to form a complex fabric of coherent relationships within the human environment (Mehaffy & Salingaros, 2015). 

Definition of Fractals

The term fractal comes from Latin and literally means broken and to break, as to indicate broken patterns. The French mathematician Benoit Mandelbrot (1924-2010) coined this term to describe a "never-ending pattern" ubiquitous in nature. The term was suggested in Mandelbrot's 1967 book "How Long is the Coast of Britain -– Statistical Self-Similarity and Fractional Dimension" and indicates the consecutive magnifications of self-similar patterns.

Fractals are self-similar patterns over a range of magnification scales, resulting in visual stimuli that are inherently complex and organized (Fairbanks & Taylor, 2011; Mandelbrot, 1983).

Fractal dimension (D) is the parameter that indicates fractal complexity or the scaling hierarchy between the patterns at different magnifications. This D value lies across a range from 1.1 to 1.9, and D= 1 and D= 2 indicate no fractal properties. For example, a smooth line (containing no fractal structure) has a D value of 1, while a completely filled area (containing no fractal structure) has a D value of 2.

Leonardo Da Vinci, Diagram of the growth of trees:

Curious fact about fractals [3]: Perhaps the first description of a fractal pattern in nature came from the great artist and scientist Leonardo da Vinci in the 15th century: “All the branches of a tree at every stage of its height when put together are equal in thickness to the trunk [below them]” (fractalfoundation.org). Leonardo speculated about a logical relationship between tree branches at different heights, based on their volumes. Leonardo da Vinci was a polymath and pioneer in integrated design. His great endeavours across art and science might be fully understood only now.

Fractals Can Be Divided in Two Types: Statistical and Exact

Given the prevalence and vast variety of fractal patterns across nature, art and science, for this project we chose to narrow down the focus of fractal patterns that can be accessible for design use.

Statistical Fractals repeat the statistical qualities of the pattern at different scales and have randomness into their construction. This disrupts the precise repetition so that only the pattern’s statistical qualities repeat (e.g., density, roughness, and complexity). They reveal the organic signature of nature’s scenery.

Exact Fractals repeat a pattern at increasingly fine scales and appear exactly the same at different magnifications. They exhibit the cleanliness of artificial mathematical shapes. They are commonly generated by computers that repeat the patterns exactly.

Why People Need Fractals, Fractal Fluency, Preference & The Aesthetic Pull of Fractals

Fractals have quantifiable health benefits, including reduced stress, improved cognitive functioning, enhanced creativity and problem solving, heightened appreciation for nature and positive emotional experience. In the contemporary indoor world where exposure to nature is being reduced dramatically, humans are progressively becoming deprived of these restorative effects. This causes an unhealthy build-up of stress, placelessness and sick building syndrome. Prolonged stress mobilization in humans produces a plethora of harmful consequences, such as increased blood pressure, energy depletion, release of stress hormones, decreased cognitive ability, reduced immune function, etc. The World Health Organization declared stress to be the “Health epidemic of the 21st Century” with associated illnesses ranging from depression to schizophrenia (Smith, 2012). Stress-related illnesses cost countries such as the US over $300 billion annually (Taylor & Spehar, 2016). In the UK, poor mental health costs employers up to ￡45 billion each year. However, for every ￡1 spent on supporting people’s mental health, employers get ￡5 back on their investment in reduced presenteeism, absenteeism and staff turnover (Franklin, 2020). As people increasingly find themselves surrounded by urban landscapes, they become disconnected from nature’s fractals and their stress-reduction qualities.

Taking this escalating concern as an interdisciplinary challenge, the design world can embrace the opportunity to create fractal designs based on the science of fractal aesthetics. Fractal patterns have the potential to radically improve the built environment, and this reversal has been shown to occur in only minutes, even seconds, as demonstrated in research (Smith et al., 2020).

Humans are fractal. Just like trees, human lungs, circulatory system, brain, skin, etc. are fractal structures. Throughout evolution, the prevalence of mid-complexity statistical fractal patterns (D= 1.3-1.5) in nature has led the nervous system to adapt and efficiently process them with small cognitive effort (Aks & Sprott, 1996; Taylor et al., 2011; Albright, 2015; Taylor & Spehar, 2016; Taylor et al., 2018). Analogous to a language fluency, fractal fluency is the human ability to detect and understand fractal patterns easily and accurately. This ‘effortless looking’ for mid-complexity statistical fractal patterns has been confirmed by behavioral experiments, coupled with qEEG and fMRI techniques (Taylor et al., 2018; Taylor & Spehar, 2016; Hagerhall et al., 2008, 2015). Repeating lines in colinear, curvilinear, parallel and radial patterns in design, facilitates visual perception by tapping into the highly organized neuronal system for representing contour orientations (Albright, 2015).

Birds foraging for food across multiple size scales:

Curious fact about fractals [4]: The human eye adopts a fractal trajectory! This was found in studies of animals foraging for food in their natural terrains (Viswanathan et al., 1996) as their foraging motions are also fractal. For example, the short trajectories allow a bird to look for food in a small region and then to fly to neighbouring regions and then onto regions even further away, allowing efficient searches across multiple size scales. The eye adopts the same motion when ‘foraging’ for visual information.

Preference for mid-complexity fractals is universal whether fractal aesthetics is generated by nature, mathematics and art (Spehar et al., 2003), hence it is independent from the generation method. This ability to grasp nature’s complex ‘sense of order’ (Gombrich, 1984) brings the advantage of saving cognitive energy, allowing it to be allocated to the more complex and novel stimuli that human survival depends upon, such as understanding others’ emotions.

Clouds are an eample of mid-complexity fractals:

Curious fact about fractals [5]: Why are we all familiar with imaginary objects induced by clouds? Our pattern recognition processes are so enhanced by these fractal clouds that the visual system becomes “trigger happy” and we see patterns that aren’t actually there (Taylor & Spehar, 2016).

Healing Spaces

A positive visual preference or aesthetic pull of fractals over simple Euclidean patterns occurs among 95% of people (Taylor, 1998), and it’s due to the ease with which fractals can be processed. Designers, intuitively and by training, tend to create fractal patterns that are exact/geometrical rather than statistical/organic. Environments that have been found as non-healing do not have fractal scaling relationships and are typically austere or Euclidean. By contrast, these can be stressful environments and can induce anxiety and depressive behavior, and ultimately pathology in their users and residents (Salingaros 2012).

Natural forms are all co-generated together, interactively, as part of a mutually adapted evolutionary process (morphogenesis). Instead of a world of objects set apart from their contexts, with distinctive, attention-getting qualities (Gifford et al., 2002) we can build a contextual world of harmonious geometric relationships and connectedness, that are not mere aggregations of parts.

With this perspective… Current contemporary architecture has barely no fractals in it, so any increase in fractal complexity (D) would make a difference for human health. Today’s urban landscapes can be dominated by fractal patterns, as a powerful tool at our disposal to make architecture and design as wholeness - more than the sum of its parts - and capable of delivering a richer, better-adapted, healthier world.

Rita Trombin

Environmental Psychologist

The Context of Biophilic Design

1960S: The term biophilia was first used by psychologist Erich Fromm to describe a psychological orientation of being attracted to all that is alive and vital (Fromm, 1964).

1980s: The Biophilia Hypothesis was conceived by biologist Edward Wilson to suggest that there is an instinctive bond between human beings and other living systems that has been mapped in our brains for thousands of years (Wilson, 1984). Biophilia is our ‘Love of Life’. Most of human psychology and behaviour can be traced back to our ancient evolutionary connection with nature –indeed, the human species has grown up in nature.        Pioneering experiments demonstrated that exposure to natural scenery had striking, positive psychophysiological consequences for the observer. After major surgery, patients recover more rapidly and request fewer pain medications when exposed to the visual qualities of nature (Ulrich, 1981; Ulrich, 1984).

1990s: The Attention Restoration Theory (ART) asserts that people can concentrate better after spending time in nature (i.e., nature can sustain ‘effortless attention’). Natural environments are particularly rich in characteristics necessary for restorative experiences, also called ‘soft fascinations’ (Kaplan & Kaplan, 1989; Kaplan, 1995).

2000s: Humans attach aesthetic values from nature through its variability and coherence (Kellert, 2005). Biophilic design establishes a new framework for a satisfying experience of nature in the built environment (Kellert & Heerwagen, 2008; Browning et al., 2012, 2014).

Present: Biophilic design seeks to create good habitat for people as a biological organism in a modern built environment, advancing health, fitness and wellbeing.

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