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Finding Patterns in the Infinitely Complex
Topic: Fractals
by Katelyn, 2019 Cohort
Consider the human lung- the smooth outer surface conceals an exquisite and complex internal structure. The trachea branches into primary bronchi, which branch into secondary then tertiary bronchi, which each branch into bronchiole. These branch into lobules, which branch into alveolar sacks, which branch into
Many natural systems exhibit intricate structures which our geometric toolkit struggles to describe. For instance, a river system which branches into increasingly smaller streams; or a maple tree, whose trunk splits to branches, full of leaves, each with a branching structure of veins.
As we magnify to each scale, from trachea to bronchi to bronchiole to lobule, we see that each level is a smaller copy of the one before it. This is a characteristic feature of fractals, which lungs, trees and river systems all have in common. Fractals are patterned structures which repeat at different scales. Fractals extend geometry to precisely describe complexity by looking at the relationships between pieces, rather than specifying points.
Generating Roughness#
Fractals have evolving symmetry. Starting with an initial structure a generator function can be infinitely applied. Take a triangle, then place evenly on each side another triangle a third of the original size; this transforms it into a 6-point star. Doing this again creates a spiky, 48-point shape. Repeating this process infinitely yields a snowflake (or Koch curve). Although this snowflake appears far more complex than the triangle, it is merely a series of increasingly small, exact copies of the initial structure.
A series of small, but slightly different structures, can also form a fractal; for instance, increasingly small, slightly different-sized forks can create a lightning bolt, or a river system.
The infinitely intricate fractal structure is ‘rough’, rather than ‘smooth’ in the way a single fork or triangle is. The upshot is that fractal structures are too complex to be one-, two- or three-dimensional
Nature is ‘rough’. Structures like lungs, trees, snowflakes, or lightning, are not ideally smooth. Unless we ignore their complexity and assume smoothness, our geometry struggles to describe them. Fractal dimensions are a tool describe this ‘roughness’, as non-smooth shapes lie between dimensions. For instance, if a triangle is two- dimensional, our Koch snowflake is slightly less than two-dimensional, with a value of
Fractals evolve from simplicity to this ‘rough’, intricate structure through infinite iteration. Fractals are dynamic and sometimes chaotic structures, created by an unending feedback loop driven by recursion. As a result, fractal analysis applies to a diverse range of dynamic, chaotic systems, including cloud formation, DNA or market trends.
A complex fractal design, generated from a simpler structure repeated at scale, can be analysed in parts. This method applies to any complex system which repeats at scale. For instance, a complex multi-level organisation such as a government could be analysed at the level of departments, their constituent sections, or small working-groups.
Fractals demonstrate that we are surrounded by similarity. Looking at similarities, or subtle differences, between and within each level helps us better understand complex systems and problems.
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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
Consider the human lung- the smooth outer surface conceals an exquisite and complex internal structure. The trachea branches into primary bronchi, which branch into secondary then tertiary bronchi, which each branch into bronchiole. These branch into lobules, which branch into alveolar sacks, which branch into
Many natural systems exhibit intricate structures which our geometric toolkit struggles to describe. For instance, a river system which branches into increasingly smaller streams; or a maple tree, whose trunk splits to branches, full of leaves, each with a branching structure of veins.
As we magnify to each scale, from trachea to bronchi to bronchiole to lobule, we see that each level is a smaller copy of the one before it. This is a characteristic feature of fractals, which lungs, trees and river systems all have in common. Fractals are patterned structures which repeat at different scales. Fractals extend geometry to precisely describe complexity by looking at the relationships between pieces, rather than specifying points.
Generating Roughness#
Fractals have evolving symmetry. Starting with an initial structure a generator function can be infinitely applied. Take a triangle, then place evenly on each side another triangle a third of the original size; this transforms it into a 6-point star. Doing this again creates a spiky, 48-point shape. Repeating this process infinitely yields a snowflake (or Koch curve). Although this snowflake appears far more complex than the triangle, it is merely a series of increasingly small, exact copies of the initial structure.
A series of small, but slightly different structures, can also form a fractal; for instance, increasingly small, slightly different-sized forks can create a lightning bolt, or a river system.
The infinitely intricate fractal structure is ‘rough’, rather than ‘smooth’ in the way a single fork or triangle is. The upshot is that fractal structures are too complex to be one-, two- or three-dimensional
Nature is ‘rough’. Structures like lungs, trees, snowflakes, or lightning, are not ideally smooth. Unless we ignore their complexity and assume smoothness, our geometry struggles to describe them. Fractal dimensions are a tool describe this ‘roughness’, as non-smooth shapes lie between dimensions. For instance, if a triangle is two- dimensional, our Koch snowflake is slightly less than two-dimensional, with a value of
Fractals evolve from simplicity to this ‘rough’, intricate structure through infinite iteration. Fractals are dynamic and sometimes chaotic structures, created by an unending feedback loop driven by recursion. As a result, fractal analysis applies to a diverse range of dynamic, chaotic systems, including cloud formation, DNA or market trends.
A complex fractal design, generated from a simpler structure repeated at scale, can be analysed in parts. This method applies to any complex system which repeats at scale. For instance, a complex multi-level organisation such as a government could be analysed at the level of departments, their constituent sections, or small working-groups.
Fractals demonstrate that we are surrounded by similarity. Looking at similarities, or subtle differences, between and within each level helps us better understand complex systems and problems.