The French mathematician, Benoit Mandelbrot was the first to recognize that the geometry of many of Nature’s objects revealed a similar pattern regardless of the scale it was examined on. The more you magnify the image, the more the structure appears the same. Mandelbrot introduced the term "self-similar" to describe such objects. "In 1975, Mandelbrot coined the word fractal as a convenient label for irregular and fragmented self-similar shapes.

The mathematics of fractals is amazingly simple in that it consists of repeating "operations" of additions and multiplication’s. In the process, the result of one operation is used as the input for the subsequent operation; the result of that operation is then used as the input for the next operation, and so on. Mathematically, all the "operations" use the exact same formula, however, they must be repeated millions of times to get the solution. The manual labor and time required to complete a fractal equation prevented mathematicians from recognizing the "power" of Fractal Geometry until the advent of powerful computers enabled Benoit Mandelbrot to define this new math.

In classical geometry the points, lines, surface areas and cubic structures all represent dimensions expressed in whole integers, 0-, 1-, 2-, and 3-dimensions, respectively. Fractal geometry is employed to model images that are more "interdimensional." For example a curved line is a 1-dimensional object. In fractals the curve can zig-zag so much that it actually comes close to filling the plane. If the curve of the line is relatively simple it is close to a dimension of 1. If the line’s curves are so tightly packed that they fill the space, the line approaches 2-dimensions. Fractal Geometry fills in the spaces between whole number dimensions.

A structural characteristic of fractals is relatively simple to understand: fractals exhibit a reiterated pattern of "structures" nested within one another. Each smaller structure is a miniature, but not necessarily an exact version of the larger form. Fractal mathematics emphasizes the relation between the patterns seen in the whole and the patterns seen in parts of that whole. For example, the pattern of twigs on a branch resembles the pattern of limbs branching off of the trunk. Fractal objects can be represented by a "box" within a "box," within a "box," within a "box," etc. If one knows the parameters of the first "box," then one is automatically provided with the basic pattern that characterizes all of the other (larger or smaller) "boxes."

As described in the Mathematics of Human Life article by W. Allman (cited in reference section), "Mathematical studies of fractals reveal that the branching-within-branching structure of a fractal represents the best way to get the most surface area within a three-dimensional space...." While the cell membrane is in reality a 3-dimensional object, its molecular bilayer possesses a constant and uniform thickness. As such the thickness of the membrane can be ignored and the membrane can be modeled as a 2-dimensional "surface-area" structure. Since evolution is the modeling of the membrane’s awareness (related to its surface area), the efficiency of modeling provided by fractal geometry would most likely reflect that chosen by Nature.