Fractals...Math found in Nature

Fractals are exciting and integrate math and science and art. The overlap of areas in the humanities and sciences appears to make fractals very popular. A movie follows which shows a well-known fractal and gives a link to more learning about fractals. Then, there is a portion of a research article showing one application of fractals in forestry. There are many more applications. This post is to give you an idea about fractals and encourage you to pursue the fractals in your own area of interest whether art or science or applied science. Enjoy the movie.



"Biologists have traditionally modelled nature using Euclidean representations of natural objects or series. Examples include the representation of heart rates as sine waves, conifer trees as cones, animal habitats as simple areas, and cell membranes as curves or simple surfaces. However, scientists have come to recognize that many natural constructs are better characterized using fractal geometry. Biological systems and processes are typically characterized by many levels of substructure, with the same general pattern being repeated in an ever-decreasing cascade. Relationships that depend on scale have profound implications in human physiology (West and Goldberger 1987), ecology (Loehle 1983; Wiens 1989), and many other sub-disciplines of biology. The importance of fractal scaling has been recognized at virtually every level of biological organization (Fig. 1; Section 5).

"Fractal geometry may prove to be a unifying theme in biology (Kenkel and Walker 1993), since it permits generalization of the fundamental concepts of dimension and length measurement. Most biological processes and structures are decidedly non-Euclidean, displaying discontinuities, jaggedness, and fragmentation. Classical measurement and scaling methods such as Euclidean geometry, calculus and the Fourier transform assume continuity and smoothness. However, it is important to recognize that while Euclidean geometry is not realized in nature, neither is strict mathematical fractal geometry. Specifically, there is a lower limit to self-similarity in most biological systems, and nature adds an element of randomness to its fractal structures. Nonetheless, fractal geometry is far closer to nature than is Euclidean geometry (Deering and West 1992).

"The relevance of fractal theory to biological problems is dependent on objectives. To the forester interested in estimating stand board-feet, a Euclidean representation of a tree trunk (as a cylinder or elongated cone) may be quite adequate. However, for an ecologist interested in modelling habitat availability on tree trunks (say, for small epiphytes or invertebrates), fractal geometry is more appropriate. Using a fractal approach, the complex surface of tree bark is readily quantified. A forester's diameter tape ignores the surface roughness of the bark, giving but a crude estimate of the circumference of the trunk. For an insect 10 mm in length, the 'distance' that it must travel to circumnavigate the trunk is much greater than the measured diameter value. For an insect of length 1 mm, the distance travelled is greater still. This has consequences on the way that the tree trunk is perceived by organisms of different sizes. If the bark has a fractal dimension of D = 1.4, an insect an order of magnitude smaller than another perceives a length increase of 10D-1 = 100.4 = 2.51, or a habitat surface area increase of 2.512 = 6.31. By contrast, for a smooth Euclidean surface, D = 1 and both insects perceive the same 'amount' of habitat. The higher the fractal dimension D, the greater the perceived rate of increase in length (or surface) with decreasing scale."
Source: http://www.umanitoba.ca/faculties/science/botany/LABS/ECOLOGY/FRACTALS/fractal.html