**Fractional Dimension,
Probability, and Form
**an excerpt from a paper entitled

by Jonathan J. Dickau

©'96,'99 - all rights reserved

Complex figures, and shapes, such as those assumed by some natural objects, and most lifeforms, are not well described by simple geometry. The postulates of Euclid, the laws of Trigonometry, and the regular polygonal solids will only take you so far in the study of real-life form. There are no perfect straight lines here, and even simple measurements involve variation dependent on the size of the measuring rod. The description of a perimeter for a lake or pond, for example, is complicated by the fact that the edge is irregular. If one examines a small section of the shoreline, in great detail, one will find areas which are nebulously defined because land and water interpenetrate each other. Even if the furthest outward extent (for example) is chosen, the exact measurement depends upon whether a small, or large measure is used.

A measuring rod the same size as the length, or width of the lake (at its widest points) would tell us very little about the actual perimeter, or the area encompassed. We could only make a crude estimate on this basis. Using progressively smaller measures, and trying to follow the actual shoreline ever more closely, ones measurement for the perimeter would increase. The value of this measurement would approach that for the length of an infinitely fine string, or thread, placed at the very edge, all the way around. This ideal can only be approximated in actual practice, but the study of Fractal Mathematics now gives us a means of refining our estimate.

Notice that I must stipulate that the thread be infinitely fine, and it must be infinitely flexible as well. Values obtained with a thick rope will be shorter than that obtained with a piece of twine, but both will exceed the measurement obtained with a yardstick. This is due to the way in which detail is followed; a string can follow the edge much closer than a yardstick can. Even if one assumes that the water and wind are absolutely calm, the exact determination of the shore also requires a degree of judgement, but if consistent rules are applied, the measured length of the shoreline grows larger as the scale of examination grows smaller. The area enclosed, however, does not appear to grow but approaches a constant. This relates to the concept of a fractional dimension.

At a small enough scale, both water, and sand grains become discontinuous as our measuring rod is shorter than their molecules. Prior to this point, the perimeter of the lake appeared to increase steadily with smaller measures, but the enclosed area appeared to approach a constant value. In point of fact, it is easy, to the point of being trivial, to define a perimeter which is outside the natural boundary at all points. It is also comparitively easy to define a boundary which is inside (by at least 1 unit) of the actual shore at all points. The task of exactly defining the boundary, however, approaches being impossible. It can only be specified in relative terms, with respect to a system of measurements.

This relative sense of measuremental definition is common in the world of mundane reality, even if it sometimes seems to fly in the face of the mundane world-view's common sense. The concept of scale, as it applies to ones frame of reference, is fundamental to any kind of measurement of solid form (Matter). The degree of convolution, or crinkliness around the edges, with respect to scale, defines the fractional, or Fractal dimension of an object. In a sense, forms with rough edges, variation, and detail at various levels of scale, exist between the theoretical boundaries of dimension. A smooth sphere or perfect circle is a form with an integral, or a non-fractal (that is a non-fractional, or a perfect integer), dimension. The surface of our planet has a mild fractal aspect to it, both Mars and Venus have deeper canyons and higher mountains, and thus their landscapes can be said to have a higher average fractal dimension.

Rocks, trees, and bushes, of a particular type, each have a characteristic fractal dimension associated with them. So do mountains, and mountain ranges, clouds, lightning bolts, river courses, and all manner of other things too. These objects are not just examples of things with irregular outlines said to possess a degree of complexity which gives them a fractal dimension. Their forms share many of the qualities that are observed in mathematically generated, or abstract, fractals. Some ferns, for example, are highly recursive in form, where the individual fronds resemble the overall form of the plant. The formulas for creating these shapes mathematically (using the technique of Iterated Function Systems developed by Michael Barnsley) are simple, and elegant. It makes one wonder if this relates to reasons why ferns have the particular form they do.

There seem to be reasons for objects of form to assume shapes which are geometrically stable, but this does not mean that these stable shapes are all boring, or stagnant. The forms produced thus far by researchers in the field of Fractal geometry include some of the most exotically, and articulately, beautiful forms ever seen, but this is only the beginning. The study of this field, as an independent subject, is not very old, and there is promise for many more beautiful forms, and several exciting discoveries to come. Even now, however, fractal models of a great variety of objects, and forces already exist.

It appears that there is an underlying fractal aspect to many forms in the physical world. In some cases (the fern for example), it appears that the fractal shape is the archetype for the form of the object, and the object but a model of the mathematically perfect shape. The probability field of the object appears to be a direct extension of this ideal shape, known as an attractor. It is as if this attractor actually pulls the leaves of a fern, or the branches of tree or a crystal, to expand in geometrically favorable directions. It is more probable that one will find the object at points on the attractor than elsewhere.

In the earlier sections, I indicated the existence of archetypal forms which helped shape the early universe, and define its dimensions. Fractal mathematics has broadened our concept of dimension, by allowing us to see what happens at the boundaries of distinction. Whether the distinction is between the lake and the shore, the mountain and the sky, or an amoeba and a drop of water, the boundary is the most interesting part. By allowing us to characterize the qualities of bounding surfaces, fractal math has given us a new vocabulary for describing our universe. It also suggests a creative dynamic which underlies certain structures and processes by enhancing the probability for a particular form of manifestation, or manifestation of form.

©'96,'99 Jonathan J. Dickau - all rights reserved

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Posted on

March 10, 1999