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In real life nothing is perfect. We see complexity in nature; the coastlines of a map, branching of a tree, behaviour of a colony, snowflakes, mammalian circulatory systems. The survival of the species literally depends on their ability to detect these patterns. Humans are also extremely talented in pattern recognition. We can easily distinguish a naturally drawn tree from a bad, “unnaturally perfect” tree drawing. The same question arises: how can we capture these motifs if it is that complex? Could there be some simple governing laws? Is there a consistent mathematical definition leading to these complex structures? Can we somehow learn it from our observations? How can we learn them? These questions will constitute the core of our discussions in the following chapters.
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#### Déjà vu in nature: Self-similar structures
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<span dir="">Last century also witnessed the distillation of studies on “self-similarity” into “fractals”, which is a very good example of the possibility to express complex observations with simple rules. Fractals are infinitely large self-similar objects, created by iterative constructs. From a geometric point of view, we consider two objects as similar if they have the same shape in a given plane, even if they have different sizes. This is also true for dynamical problems and we can find various applications of fractals in science, technology, art, architecture, even law.</span>
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<span dir="">Mathematical models are expressed in terms of fractal-like, repeating patterns to describe complex processes. Let us consider a real life example: the circulatory system in our bodies. The resources, such as air and food, are taken through one inlet point, which are to be distributed to every cell constituting us. The investment to create a direct link between the inlet point and the target for each element (cell) is practically impossible. It works by a complex branching network optimized for efficient transport. Here the evolutionary challenge is the minimum amount of pipeline needed for resource management. If the distribution system is modelled as a network of self-similar branches, it was found to be scaling with the famous ¾ power law for the metabolic rates (West et al.), which is a characteristic scaling for all organisms. In short, the metabolic rate was shown to be limited by the efficiency with which the organisms distribute resources.</span>
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<span dir="">From this lecture’s perspective, emergence of complex patterns from simple laws is of critical importance, as we are aiming to utilize available observations (data) to find such coherent patterns to analyse and interpret make predictions the system of interest.</span>
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_Reading materials:_
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* <https://en.wikipedia.org/wiki/Self-similarity>
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* <https://en.wikipedia.org/wiki/Symmetry>
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* <https://www.wired.com/2010/09/fractal-patterns-in-nature/>
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* [Power laws and self-similarity](https://www.sciencedirect.com/science/article/pii/S0307904X1400047X)
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* [The fractal nature of nature: power laws, ecological complexity and biodiversity](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1692973/)
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* [Common power laws for cities](https://www.pnas.org/content/117/12/6469)
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* [A General Model for the Origin of Allometric Scaling Laws in Biology](https://www.science.org/doi/abs/10.1126/science.276.5309.122) |
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