Limbs #1 (detail)
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Limbs #4, (drawing)
Limbs, or the Discontinuous Structure of Matter (2017)
Self-similarity seems to be one of the fundamental geometrical construction principles in nature. For millions of years, evolution has shaped organisms based on the survival of the fittest. In many plants and also organs of animals, this has led to fractal branching structures. For example, in a tree the branching structure allows the capture of a maximum amount of sun light by the leaves; the blood vessel system in a lung is similarly branched so that a maximum amount of oxygen can be assimilated. Although the self-similarity in these objects is not strict, we can identify the building blocks of the structure, the branches at different levels. The distribution of craters on the moon obeys some scaling power laws, like a fractal. However, it is generally impossible to find hierarchical building blocks for these objects as in the case of organic living matter. There is no apparent self-similarity, but still the objects look the same in a statistical sense.
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Historically, the random variables were associated with or indexed by a set of numbers, usually viewed as points in time, giving the interpretation of a stochastic process representing numerical values of some system randomly changing over time, such as the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Examples of such stochastic processes include the Wiener process or Brownian motion process, which is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving molecules in the fluid. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such fluid there exists no preferential direction of flow as in transport phenomena.
In summary, many natural shapes possess the property that they are irregular but still obey some scaling power law, introducing some element of randomness into the otherwise rigorously organized classical fractals.