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Page 64

		1.4.10— Definition of a Fractal

		A fractal is an object in space or a process in time that has a fractal dimension greater than its topological dimension.

		For example, the perimeter of the Koch curve has a fractal dimension of about 1.2619. The fractal dimension describes the space filling properties of the perimeter. The fractal dimension of the perimeter is between 1 and 2. Since the fractal dimension is larger than 1, the perimeter is so wiggly that it covers more than just a 1-dimensional line. But since the fractal dimension is less than 2, the perimeter is not so wiggly that it covers a 2-dimensional area.

		The topological dimension of the perimeter is 1. The topological dimension describes how the points on the perimeter are connected together. No matter how wiggly the perimeter is, it is still a line with a topological dimension equal to 1.

		Since the fractal dimension of the perimeter (1.2619) is greater than the topological dimension of the perimeter (1), the perimeter of the Koch curve is a fractal.

		The topological dimension tells us what kind of thing an object is, such as an edge, a surface, or a volume. When the fractal dimension is larger than the topological dimension, it means that the edge, surface, or volume has more finer pieces than we would have expected of an object with its topological dimension. It is more wiggly than we expected. That is why we keep seeing more smaller pieces when we examine it at finer resolution.

		The additional smaller pieces at finer resolution mean that the object covers more space than we would have expected of an object with its topological dimension. The topological dimension is an integer. The additional space covered means that the fractal dimension is an integer plus an additional fraction.

		Mandelbrot says that he coined the word "fractal" to reflect these central ideas that a fractal is: (1) fragmented into ever finer pieces and (2) has a fractional dimension.

 

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