My introduction to fractal geometry was through the works of M.C. Escher, the Dutch artist who cut exquisite images of overlying patterns into wood to create works of beauty.
Escher used hyperbolic geometry for his woodcuts and he did not know what fractals were; his work is referred to as a “tessellation.” The honeycomb of bees is a fine example of a tessellated natural structure.

Imagine an equilateral triangle. Now, imagine smaller equilateral triangles perched in the center of each side of the original triangle–you have a Star of David. Now, place still smaller equilateral triangles in the center of each of the star’s 12 sides.

Repeat this process infinitely and you have a Koch snowflake, a mind-bending geometric figure with an infinitely large perimeter, yet with a finite area. This is an example of the kind of mathematical puzzles that this book addresses.

Fractual Geometry
The Fractal Geometry of Nature is a mathematics text. But buried in the deltas and lambdas and integrals, even a layperson can pick out and appreciate Mandelbrot’s point: that somewhere in mathematics, there is an explanation for nature. It is not a coincidence that fractal math is so good at generating images of cliffs and shorelines and capillary beds.

A “simple” explanation of fractal geometry from ThinkQuest’s online library:

While the classical Euclidean geometry works with objects which exist in integer dimensions, fractal geometry deals with objects in non-integer dimensions. Euclidean geometry is a description lines, ellipses, circles, etc. Fractal geometry, however, is described in algorithims — a set of instructions on how to create a fractal.

The world as we know it is made up of objects which exist in integer dimensions, single dimensional points, one dimensional lines and curves, two dimension plane figures like circles and squares, and three dimensional solid objects such as spheres and cubes. However, many things in nature are described better with dimension being part of the way between two whole numbers. While a straight line has a dimension of exactly one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it curves and twists. The more a fractal fills up a plane, the closer it approaches two dimensions. In the same manner of thinking, a wavy fractal scene will cover a dimension somewhere between two and three. Hence, a fractal landscape which consists of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with many average sized hills would be much closer to the third dimension.