Fractals occupy fractional dimensions between whole numbers

Evidence

Mandelbrot shows that data of this kind describe curves similar to those of continuous functions for which no derivative exists. A simplified model of this is Koch’s curve; it is self-similar, and it can be shown that the dimension of self-similarity in which it is constructed is not a whole number but log 4/log 3. It would be justified to say of such a curve that it is located in a space whose “number of dimensions” is between one and two, and thus that it lies intuitively somewhere between a line and a flat surface. Because their relevant dimension of self-similarity is a fraction, Mandelbrot calls objects of this kind fractals. — Jean-François Lyotard, The Postmodern Condition, 414

Categorization

• DEFINE: Precise definition of fractal dimensionality (log 4/log 3 for Koch’s curve) — the mathematical basis for Lyotard’s charge that continuous function is losing preeminence.