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Cabbage Leaves and Temporal Fractals


Fractal tumor on Wild Cabbage LeafI have always considered fractals in time to be related to self-similar music (such as a nested fugue), or just a plain-old self-similar time-series, such as stock market fluctuations, or the corn price fluctuations at the Chicago Mercantile Exchange, whose fractal nature was first noted by Mandelbrot.

Now there’s a different way to consider time-fractals - proposed by Carlos Escudero and colleagues of the Institute for Mathematics and Fundamental Physics in Madrid, in their Dynamic Scaling of Non-Euclidean Interfaces

Escudero "performs calculations of the dynamic scaling (how a surface changes in space and over time at several different scales) of growing structures, such as the kind of semiconductor films used in the microchip industry where, even under the most carefully controlled of conditions, rough (non-Euclidean) geometries can exist. He found that the moment-by-moment behavior of the surfaces are strongly effected by the fractal geometry."

Escudero is using his model to investigate the growth of tumor-like tissues in plants and the growth of semiconductor films. Interestingly, Escudero et. al "conclude that it is necessary to reexamine some experimental results in which standard scaling analysis was applied."

The Escudero approach is unique b/c it explicitly intertwines spatial and temporal fractals. Even so, I am not surprised that "moment-by-moment behavior of the surfaces are strongly effected by the fractal geometry," which seems obvious (albeit very hard to imagine measuring).

But I am intrigued by the apparent need to "reexamine some experimental results" I have some experimental diffusion data taken many years ago, of phosphorus in alloy steels. I applied a basic diffusion - law model to the data in order to extract diffusion coefficients, but the model fit was never quite right. I now wonder whether or not my results (taken using Auger Electron Spectroscopy) could benefit from a fractal-like analysis.

Stay tuned…