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Fractals: Tackling Complex Problems by Acknowledging their Infinite Complexity

Topic: Fractals
by Finnlay, 2020 Cohort

“A fractal is a mathematical object with the property that repeating patterns will appear as you magnify a certain point. This property is called “self-similarity”, and importantly this can present itself in two ways. Scale invariant, where a single pattern repeats itself and the fractal is identical at any level of magnification. Or scale variant, where patterns reoccur but the shape of the fractal can change with magnification. This is what we call “statistical self-similarity”, a fractal may not be identical at any level of magnification in a literal sense, but in a statistical sense they are. Meaning various mathematical properties and patterns are visible regardless of magnification. Importantly, fractals are infinite, these patterns be they varying or not will repeat endlessly.

Both forms of fractal are extremely common in nature. A fern whose leaves are smaller versions of the whole is a common example of a scale invariant fractal. An example of a scale variant fractal pattern is the coastline of a nation, where similar patterns can be found as we look at the coastline in more and more detail. Benoit Mandelbrot, the founder of fractal geometry, famously used the coast of England to motivate the importance of fractal thinking, as it could be used to obtain a more precise measurement.

When faced with large or infinite complexity we have two options. Generalise and disregard information at some level of depth as unimportant. Or formulate a description of the relationship between the levels of depth instead of attempting to look at all the data. The latter is the fractal approach.

Medical Science#

Recognition of the merits of fractal analysis, along with the proliferation of high-power computing, has been the basis for innovation in the field of medical research. Cells and tissues exhibit irregularity and self-similarity under increasing levels magnification, a trait which is characteristic of fractals. Thus these areas lend themselves well to fractal analysis.

In morphology, the study of structure and form, fractal geometry has allowed researchers to characterise both normal and pathologic cells and tissues. For just one example take the study of cancer tissues. Using a property called fractal dimension as a characterisation factor, researchers have been able to identify the size of breast and prostate tumours more precisely.

Architecture#

Architects understood the advantage of fractal thinking before the term fractal was even coined. A prime example is the Eiffel tower, whose structure is self-similar. When designing the tower Gustave Eiffel chose to use trusses made up of smaller beams rather than large solid beams. The branch points where trusses and sub-members connect in the tower demonstrate fractal behaviour. What architects have known for a long time is that replacing a beam with a truss made up of smaller sub-beams increases the strength, to what level of granularity you can repeat this process is dependent on the available manufacturing techniques. This repeated branching of beams is a fractal-like structure and this approach allowed Gustave Eiffel to construct a tower that was incredibly light, but also extremely strong.

Explore this topic further#

• Fractals Everywhere Ch1{.link-ext target=”_blank”}
• Wikipedia page{.link-ext target=”_blank”}
• Benoit Mendelbrot “Fractals and the Art of Roughness”{.link-ext target=”_blank”} (Ted Talk)
• Keith McGreggor “Fractal Thinking”{.link-ext target=”_blank”} (TED Talk)

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Disclaimer#

This content has been contributed by a student as part of a learning activity.
If there are inaccuracies, or opportunities for significant improvement on this topic, feedback is welcome on how to improve the resource.
You can improve articles on this topic as a student in "Unravelling Complexity", or by including the amendments in an email to: Chris.Browne@anu.edu.au

“A fractal is a mathematical object with the property that repeating patterns will appear as you magnify a certain point. This property is called “self-similarity”, and importantly this can present itself in two ways. Scale invariant, where a single pattern repeats itself and the fractal is identical at any level of magnification. Or scale variant, where patterns reoccur but the shape of the fractal can change with magnification. This is what we call “statistical self-similarity”, a fractal may not be identical at any level of magnification in a literal sense, but in a statistical sense they are. Meaning various mathematical properties and patterns are visible regardless of magnification. Importantly, fractals are infinite, these patterns be they varying or not will repeat endlessly.

Both forms of fractal are extremely common in nature. A fern whose leaves are smaller versions of the whole is a common example of a scale invariant fractal. An example of a scale variant fractal pattern is the coastline of a nation, where similar patterns can be found as we look at the coastline in more and more detail. Benoit Mandelbrot, the founder of fractal geometry, famously used the coast of England to motivate the importance of fractal thinking, as it could be used to obtain a more precise measurement.

When faced with large or infinite complexity we have two options. Generalise and disregard information at some level of depth as unimportant. Or formulate a description of the relationship between the levels of depth instead of attempting to look at all the data. The latter is the fractal approach.

Medical Science#

Recognition of the merits of fractal analysis, along with the proliferation of high-power computing, has been the basis for innovation in the field of medical research. Cells and tissues exhibit irregularity and self-similarity under increasing levels magnification, a trait which is characteristic of fractals. Thus these areas lend themselves well to fractal analysis.

In morphology, the study of structure and form, fractal geometry has allowed researchers to characterise both normal and pathologic cells and tissues. For just one example take the study of cancer tissues. Using a property called fractal dimension as a characterisation factor, researchers have been able to identify the size of breast and prostate tumours more precisely.

Architecture#

Architects understood the advantage of fractal thinking before the term fractal was even coined. A prime example is the Eiffel tower, whose structure is self-similar. When designing the tower Gustave Eiffel chose to use trusses made up of smaller beams rather than large solid beams. The branch points where trusses and sub-members connect in the tower demonstrate fractal behaviour. What architects have known for a long time is that replacing a beam with a truss made up of smaller sub-beams increases the strength, to what level of granularity you can repeat this process is dependent on the available manufacturing techniques. This repeated branching of beams is a fractal-like structure and this approach allowed Gustave Eiffel to construct a tower that was incredibly light, but also extremely strong.