Reading an excellent research article on the effect of fatigue on the fractal nature of force output by my fellow academic and friend Mark Burnley, and seeing the large tracts of snow that covered the United Kingdom the last few weeks got me thinking of fractal theory, a concept which I have become increasingly interested in during the last decade or so of my research career. A broad definition of fractals are that they are rough or fragmented geometric shapes that can be split into parts, each of which is an approximate copy of the original whole. Examples of fractals are all around us, from snowflakes, to the branching of a tree or vines, to the coastline of a country, to our own lungs, intestines and blood vessels, to the patterns of weather and tides, all of which exhibit fractal geometry and dimensions. Fractals have become popularized in art and paintings, and many folk will be aware of fractals through curtains, floor mats or wall art that display fractal art in all its beauty.
While fractal theory, and its twin theories chaos theory and complex system theory have been studied for many years, the concept of fractals was popularized in the 1970’s by the pioneering work of Benoit Mandelbrot, who coined the word from the Latin term ‘fractus’, which describes a ‘broken stone’. He used this word to describe ‘fractured’ shapes and geometry that focused on broken, wrinkled and uneven shapes, which of course is what most shapes around us in nature are. In the ‘normal’ geometry we learn at school and most remember geometry as – Euclidean geometry – shapes are regular, for example triangles, rectangles and squares. Fractal geometry is a description and a way of measuring irregular shapes and qualities that have no clear way of being measured using classical Euclidian mathematical techniques, such as degree of roughness, ‘brokenness’ or irregularity of an object. Mandelbrot first worked in economics, where before his work economists believed that small daily changes in share and stock prices had nothing to do with large / long term changes. But he looked at the patterns of share fluctuations over first years, then months, then days, then hours, and found astonishingly that the variable curves for each time period of share price data he examined were almost perfectly matched. In other words, despite the share prices fluctuating seemingly randomly when viewed over a set period of time, the changes were ‘self-similar’ whatever the time duration examined was varied. This indicated that some organizational ‘pattern’ or principle was occurring with the stock prices that ‘cut across’ / was universal to whatever time frame of change which had occurred. In what he became most well-known for, Mandelbrot showed similar self-similar patterns occurred when examining the coastline of the United Kingdom. The pattern of the entire coastline is never a straight line and is always variable and constantly different when viewed on a map, but when he looked at a section of the coastline compared to the whole coastline, it had a similar looking ‘fractal’ pattern, and when he looked at an even smaller section of that sub-section, it looked the same / had the same ‘pattern’. So as he described, this is the key feature of fractal shapes, that they are ‘self-similar’ no matter how much one ‘zooms in’ on them. As one zooms in on a fractal shape, no new detail emerges and nothing changes pattern wise, with the same pattern repeating no matter how small an area one zooms in on. So if one looks at a snowflake or part of a human’s lung, then zoom in on a component of each of them, the sub-contents will have a similar pattern and organization that will occur similar to what was seen at the initial scale of viewing of it.
What is perhaps even more interesting than the concept of fractal geometry in physical and anatomy structures is that there appears to be fractal geometry to the patterns or traces of our daily activity and physiological function over time, for example our heart beat, walking patterns, or our brain’s neuronal function. If you graph all these parameters at a high enough data capture rate, they exhibit a variable pattern that seems random, but has a fractal, and therefore ordered structure to them that looks similar if you look at either the entire physiological activity data trace or a component of it. Working with Andre Bester, Ross Tucker and other folk in the lab I worked in at the University of Cape Town several years ago, we showed that the power output trace of cyclists performing a 20 km time trial had fractal characteristics, and Mark Burnley’s interesting data described above showed that when extending the knee in an isometric (non-moving) contraction against resistance, the natural variable fluctuations in muscle activity one has when doing such an ongoing muscle contraction also have a fractal dimension. So during all activities of daily living, while we are not aware of it, our bodies appear to perform activities and have physiological function that is very structured and is performed according to some geometric design principle which seems random and variable when viewed with the naked eye, but is fractal in nature.
From a ‘control of life’ perspective, the most astonishing thing about fractals to me is that when scientists, mathematicians and computer programmers began working in the field of fractal theory, they found they were able to generate intricate fractal shapes, and reproduce the fractal geometry found in nature and our bodies, on their computer screens, using fairly simple non-linear equations and computer coding programs. For those mathematical minded folk, the key component of the computer programs and equations that generate fractal design is that they have feedback components that are iterative – namely when the equation generates an outcome value and it is fed back into the equation in a repetitive (iterative) manner, the equations outcomes when graphed shows the beautiful fractal shapes that are all around us, in our bodies, and which are evident as the coastline of the United Kingdom and how it developed for example, or how stock prices fluctuate during a day, month or year. This concept of an iterative equation underpinning life form and function becomes a concern for geneticists, from the perspective that the ‘language’ of regulation for those working in the gene field is that life is controlled and regulated by individual or arrays of specific genes activated sequentially in time. However, when looking at life and its regulation from this fractal ‘nature’ perspective, there would need to be an equation type activity occurring which would ultimately regulate and ‘synchronize’ the activity of all components of our life and earth structures, and it is of course perhaps impossible to ‘find’ an equation in any gene structure or physical domain. So where these ‘equation’ based life processes come from and how they are controlled are some of life’s big questions – which come first, the equation controlling the ‘whole’ of life, or the specifics of life which then built up to a ‘whole’ that incorporates the equation. In either scenario, one can’t get away from the concept that an equation-related process underpins all current and past life activity, and it difficult to get one’s ‘head around’ this concept, using the current theories of how life began and is regulated. When I describe these concepts in academic lectures I give, with a half-smile I always end my lectures by saying that God, or whoever created us, must surely have been a mathematician. Each time one sees snow fall, or clouds develop in an otherwise blue sky, or feel the pulse of a patient to determine their heartbeat characteristics, one should and can only marvel not just at the beauty of life itself and its manifest detail all around us, but at the intricate processes and requirements which appear to ‘govern’ both its development and its very existence. It’s going to be fascinating to see how these concepts help us uncover and understand the ‘big concepts’ in life, which to this point in time have been beyond our capacity to understand, and which remain mysterious, even if they are beautiful mysteries at that!