Fractals and Scales
Scale is the concept that makes measurements meaningful. For example, if you were told your Internet connection was 10, this number would not tell you anything. If it was 10 bytes per second, that would be exceptionally slow, while if it was 10 megabytes per second that would be fairly fast. The difference between a byte and a megabyte is that of scale.
Size and scale are closely related issues. In the normal course of measuring, you choose a scale that is appropriate to the size of the object being measured. For example, if you are measuring the length of a cat you would use inches but if you were measuring the length of a road you would use miles.
The scale you use can reflect the actual result of your measurement when you are measuring a length that is irregular, particularly perimeters. This was first observed by Lewis Fry Richardson. The reason is because a larger scale will only approximate the length, the smaller curves and bends will not be included in the measurement because they are smaller than 1 unit of the scale. As you use a smaller scale more of the perimeter is available to be measured, as Richardson saw regarding the British coastline.
Mandelbrot first explored the idea of a fractal in his work “How Long is the Coast Line of Britain.” The paper starts off with Richardson's observations and then goes on to discuss the properties of fractals. Fractals are objects that are self-similar as seen in these classic examples. As you can see, each smaller section of a fractal has the same outline as the overall factor. For fractals this continues infinitely as you zoom further in, which means that the further you zoomed in the larger the value you would get for the perimeter of the fractal.
This result becomes a practical concern for biologists because many biological systems are best represented by a fractal. Plants and animals contain a series of systems and sub-systems that have the same pattern on a decreasing scale. This is not a perfect representation of a fractal because there is a lower limit to size, the atom, but is much better described by fractal geometry than Euclidean geometry.
One place this comes into play is when biologists try to measure habits. The same tree will be much larger for an ant than it will for a blue bird because the ant is small enough to perceive the ridges of the bark as additional surface area. The consequence of not addressing the fractal nature of the ant's environment can be a significant underestimation of the number of individuals able to live in a given area.
Another evolving application for fractals in biology is DNA analysis. Biologists believe that DNA possess fractal attributes. This makes it resilient, because of the redundancy, and possibly traceable. The overall fractal patterns can be more easily observed than individual genes and biologists hope to be able to use them to discover the animal's ancestry. As mathematicians make further progress in describing and providing length estimations for fractals, biologists will continue to apply these tools to a variety of conditions in the natural world.