Fractal time is certainly a notion with some intuitive appeal. It comes up indirectly in the study of
Elliott waves, which is the idea that the long-term trends of market prices are similar to the medium-term trends, which are again similar to the short-term and immediate-term trends:
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Elliott waves are fractals in the popular sense of the term, i.e.: they exhibit self-similarity at every level, but no study has yet been able to identify or specify a
mathematical fractal function yielding useful results for interpreting historical trends or predicting them into the future.
A similar idea comes up more scholastically in the
metabolic theory of ecology (MTE). Wikipedia sums it up better than I could:
Quote:MTE is based on an interpretation of the relationships between body size, body temperature, and metabolic rate across all organisms. Small-bodied organisms tend to have higher mass-specific metabolic rates than larger-bodied organisms. Furthermore, organisms that operate at warm temperatures through endothermy or by living in warm environments tend towards higher metabolic rates than organisms that operate at colder temperatures. This pattern is consistent from the unicellular level up to the level of the largest animals on the planet.
The theory posits that major characteristics of life and ecosystems from life-spans to populations and endless other traits are a fractal function of organism size, temperature and metabolic rate (which is itself a function of the organism's surface area).
In their seminal paper on the topic, ecologists
West, Brown & Enquist describe MTE in a richly evocative conclusion:
Quote:Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional. Quarter-power scaling laws are perhaps as universal and as uniquely biological as the biochemical pathways of metabolism, the structure and function of the genetic code, and the process of natural selection. The vast majority of organisms exhibit scaling exponents very close to 3/4 for metabolic rate and to 1/4 for internal times and distances. Theseare the maximal and minimal values, respectively, for the effective surface area and linear dimensions for a volume-filling fractal-like network. On the one hand, this is testimony to the power of natural selection, which has exploited variations on this fractal theme to produce the incredible variety of biological form and function. On the other hand, it is testimony to the severe geometric and physical constraints on metabolic processes, which have dictated that all of these organisms obey a common set of quarter-power scaling laws. Fractal geometry has literally given life an added dimension.
Hope this helps!