Reflections on Fractals

Our book Holonomics: Business Where People and Planet Matter explores many aspects of living and non-living systems, from sub-atomic particles right up to the entire earth. It also examines perspectives on evolution which fall outside of either Darwinian or neo-Darwinian explanations. These Darwinian theories explain evolution in terms of tiny random variations in genes but for scientists like Brian Goodwin and Stuart Kauffman they only have limited applicability. While they may well be able to account for variations as a result of adaptation of a species, they can not explain the spontaneous origin of species. This is a controversial position to take in science, and is by no means widely accepted. But what complexity theory does offer us is a potential explanation about how the forms of life actually do arise from patterns of interaction in a complex system.

A key aspect of systems thinking is the move away from quantification, the measurement of things. As we move into an analysis of complex systems, our conceptualisation of parts begins to break down, and so quantification becomes impossible. In their book Turbulent Mirror, Steve Briggs and F. David Peat describe this movement in scientific measurement as follows:

In the old quantitative mathematics the measurement of a system focuses on plotting how the quantity of one part of the system affects the quantities of other parts. By contrast, in qualitative measurement, the plots show the movements as a whole. In qualitative mode, scientists don’t ask, How much of this part affects that part? Instead they ask, What does the whole look like as it moves and changes? How does the whole system compare to another?

The first part of Holonomics describes the intrinsic dimension of organic systems, as explained by Henri Bortoft. A key insight is that in many dynamical systems, there is an intrinsic dimension, or some aspect or quality of that system which is not open to quantitative measurement. It would not until the 1970s and 1980s that computational power would be both fast enough and economical enough would enable the creation of mathematical systems to model this way of understanding wholeness. These systems, or mathematical models were fractals.

In the 1970s Benoit Mandelbrot revolutionised the way we think about irregular forms in nature, whose rich detail had previously been ignored by conventional geometry. For Mandlebrot, Euclidean geometry was “dull”. Instead, he saw irregularity as not just noise in Euclidean forms, rather it was within this noise that nature’s creative forces are revealed.

Photo: Simon Robinson

Mandlebrot first asked his famous question “How long is the coast of Britain?” in 1967. This may seem a strange question at first. According to Ordnance Survey, the coastline length around mainland Great Britain is 11 072.76 miles. In giving the figure to two decimal places, they are saying that they are accurate to within 10 metres. But shorelines continually evolve, and also this figure will have been derived based on the average high tidal mark. The figure was derived using a map with the scale of 1:10,000. But what if we were to use a less detailed map, for example a typical road atlas? The length of the coastline would shrink. This is because you would see far less craggy details of the coastline, and so if you were to measure the coast with a piece of string, it would be much smoother than on on the more detailed map.

Photo: Simon Robinson

Now imagine rather than being a person who is going to walk along the coast, you are an ant. Whereas the person will take a straight single stride, the ant will not necessarily be going in a straight line compared to the human, since they will be going in and out of rocks that are smaller than a human foot and sorter than the length of a human stride. The ant will ultimately travel a greater distance than the human.

Mandlebrot’s genius lay in being able to develop mathematics which could measure this type of irregularity. In fact, he showed that phenomena such as a craggy coastline could demonstrate the property of self-similarity, in that if you zoomed in to a small detail of a coast, it would show the same features that can be found in the whole shape. It was in fact Mandlebrot who coined the phrase fractal and his new geometry could in fact be used in fields outside of geography such as finance and economics.

When examining just how irregular a surface is, it is possible to state mathematically as a fractal dimension. We are normally used to three dimensional space, but a fractal dimension will fall between the values of 1 and 2. The coast of Britain has a fractal dimension of 1.26, where a dimension of 1.0 represents a smooth surface. This may seem like a strange concept, but the mathematics has proved useful in many differing applications, such as describing the strength of a metal, or the roughness of a surface for example.

Having looked briefly at these few examples, I now want to explore what I feel is one of the artistically and experientially most amazing fractals of all, the Mandlebrot set. This was popularised by A.K. Dewdney in 1985 when he published instructions in Scientific American on how to produce this remarkable fractal on home computers. Most programmers would have only had monochrome monitors at that time, but with the dramatic increase in computational power, graphic designers have been able to create ever more amazing animated journey’s through this fractal, and many others besides. Even though there is now an unimaginable amount of computational power compared to the 80s, one of my favourite high-definition recent animations took the designer 2 days to set and 6 months to render.

The following sequence of diagrams shows a journey through the Mandlebrot set. Although the mathematics use the concept of a complex number, the main equation used to generate such complexity is surprisingly simple:

zn+1 = zn2 + c

A complex number is one which has two parts, a real part, which are the set of positive and negative numbers we normally use, and an imaginary part which is a number which has a negative square, an example being i2 = −1. The solutions to Mandlebrot’s equation are plotted on a screen, the surface of which is called a complex plane.

Figure A: Mandlebrot set

Figure A starts from a perspective where we are looking at the plane from above. Some areas are coloured, and there is a large black shape in the middle. In order to explore this shape, we can imagine that we are in a space ship, and are going to travel down to this shape, so the direction of travel is into the plane.

Figure B: Mandlebrot set

When we approach the edge of the black shape, we see in figure B that in fact, like the coast of Britain, it is made up of other shapes the same form as itself.

What we can then do is explore certain ares of this shape further. What we discover is a strangely beautiful universe, one where we never quite seem to land.

Figure C: Mandlebrot set

Figure C shows a much deeper part of the fractal, as the original figure is expanded in size to levels comparable to the entire size of our own galaxy and universe.

The graphic designers Team Fresh have created an animation of a Mandlebrot set lasting almost 14 minutes. In this zoom, they magnify the initial complex plane to a magnitude of 6.066e+228 (2760).

It is not possible to imagine the final size of magnification, the following examples from HD-Fractals below give you an idea of how large the initial image would be if magnified to a range of different magnitudes:

1E6 Vancouver Island
1E9 Jupiter’s radius
1E12 Earth’s orbit
1E18 distance to Alpha Centauri
1E21 Milky Way galaxy
1E42 size of proton to the universe

1 to 6E228 is like expanding a proton to 700000000000000

00000000000000000000000000000000000000000000

00000000000000000000000000000000000000000000

00000000000000000000000000000000000000000000

0000000000000000000000000000000000000000 times the size of the visible universe.

Another way of thinking about the level of magnification over 14 minutes is that it is the equivalent of travelling faster than the speed of light.

Having understood the mathematics of fractals, the next logical question is to ask how they relate to life. If you look at figure A again, you will see that inside the shape there is a back area of simple order. The extreme outside areas represent a mathematical chaotic domain. It is on the boarders of these two areas where the really interesting explorations take place, and as we saw in the previous figure with period doubling, Mandlebrot sets also demonstrate the same period-doubling behaviour on the route from order to chaos. The buds of the set form at bifurcation points at the edge of chaos. This behaviour mirrors real organic systems such as insect populations, again as we have already discussed.

If we look at the principle behind the generation of a fractal, it is based on a very simple equation. This simplicity gives rise to a shape of unimaginable complexity, so could the same principle be at work in nature, which as we have seen with the case of genes, is able to generate complex and different forms from almost similar genetic material. This is an interesting line of questioning to take, because at first it appears to be reductionist. However, as Briggs and Peat point out, the equation which generates the Mandlebrot set is not plotting a shape as we are used to in Euclidean geometry. The shape is actually providing the starting point for an evolving shape which emerges from an iterative feedback process.

In real life, organic forms are nowhere near as orderly as these mathematical fractals. But if an element of randomness is added, then it is possible to create very life-like forms. we will examine a much more simple mathematical fractal, one which we will use to create a fern.

Fern Fractal – reproduced with permission from Philip Franses

The mathematics used are extremely simple difference equations which were discovered by Newton. They describe how systems evolve how systems evolve from one step to another by including a difference based on the existing values of the variables. This does appear to be a quality of nature, whereby a change in one variable is governed by a change in another variable. To see this in action, we can use the following equations:

x.= (0.2 * x) – (0.26 * y)
y.= (0.23 * x) + (0.22 * y) + 1.6

If we imagine that we start with the values of both x and y being 0, the new value of x will be 0 and the new value of y will be 1.6. If we run the equations again, the new values will be x = -0.416 and y = 1.952. The next step is to imagine that these equations change at random via the following rules:

Choice 1 (1% of the time)
x. = 0
y.= (0.16 * y)
Choice 2 (7% of the time)
x. = (0.2 * x) – (0.26 * y)
y.= (0.23 * x) + (0.22 * y) + 1.6
Choice 3 (7% of the time)
x. = (-0.15 * x) + (0.28 * y)
y.= (0.26 * x) + (0.24 * y) + 0.44
Choice 4 (85% of the time)
x. = (0.85 * x) + (0.04 * y)
y.= (-0.04 * x) + (0.85 * y) + 1.6

When these equations are run through around 30,000 times and the results plotted in space on an x and y axis, the result is a fractal picture of a fern. It is fractal because at every scale, you see the same recurring pattern. We can describe the relationship between x and y as being entangled. A change in the value of x affects y, and a change in the value of y affects x. In order to understand one implication of these equations, we have to return to our newly developed concept of seeing.

With a mechanical worldview, when we create models we tend to do so based on fixed objects, such as x and y, particles etc, which we see as separate. But another worldview is one where what we see as primary is change and relationship. This is a huge leap to make in our seeing, and it took me an extremely long time to even begin to understand this notion. I think it is only when you begin to experience change and relationship, when you move from an inorganic world of dead matter to one which is alive at all levels of reality, that this change of perspective can begin to make sense.

This new or expanded fractal way of seeing may not in fact be new at all. In Holonomics we look at the history of science to examine if we can understand how science as we now know it has impacted on our way of seeing. Working with fractals, holograms, and the living processes of the metamorphosis of plants can certainly seems strange to us, especially if we approach complexity with mechanistic minds which limits our ways of perceiving nature.

Desborough Mirror, British Museum reproduced with permission under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0) licence

It is fascinating to look at some of the many different Celtic mirrors which have been discovered in the British Isles, and which date back as far as 300BC. These mirrors are usually named after the place where they were found, such as the Desborough Mirror, once of the best preserved. The mirror has an extremely complex clover-like pattern on its back, which is remarkably similar to many patterns generated using fractal mathematics. What is interesting about the Desborough Mirror is that although at first sight it seems symmetrical, in fact it is not. This is significant because as we have seen, ordered systems can and do move to a more dynamic state on the edge of chaos, before finally moving into highly chaotic states. It is interesting to explore whether or not the designers of the mirror were aware of this quality of natural systems were aware of this behaviour, sometimes known as symmetry-breaking, and attempting to reflect it in their art.

When the world becomes dematerialised, less solid, then there becomes space for entering into the mindset of great thinkers such as David Bohm, who was able to conceive of reality as a single whole, which while itself does not consist of separate objects, has at its very foundation dynamical flowing movement. This can certainly seem paradoxical, and it is perhaps the reason why  profound thinkers on complexity such as Philip Franses and Basil Hiley, a long-time colleague of Bohm talk about the study of complexity being more akin to alchemy and a pilgrimage.

Related Articles

Exploring Chaos, Fractals and Bifurcation

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Mandlebrot Sets reprinted with kind permission from Dr. Wolfgang Beyer, http://www.wolfgangbeyer.de

2 responses to “Reflections on Fractals

    • The Mandlebrot set is created based on iterations using complex numbers. The black space represents solutions which pretty much remain stable after many iterations. White represents solutions which approach infinity rapidly. And so the colours represent different levels of oscillation between stability and approaching infinity. In terms of which colours to chose and how to map, there are a number of different solutions.

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