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where pn = (3/2)n on the 2n intervals of length ln = (1/3)n and is identically zero everywhere else. An example of such a function Mn(x) is shown in Figure 2.2. The

Figure 2.2 Intermediate stage (n = 6) in the construction of the Cantor singular function or 'Devil's staircase' (top) and schematic illustration of the self-affinity of the Cantor singular function or 'Devil's staircase' (bottom); the enlargement is identical to the original, but the enlargement (scaling) factors are different in the x and y directions.

Figure 2.2 Intermediate stage (n = 6) in the construction of the Cantor singular function or 'Devil's staircase' (top) and schematic illustration of the self-affinity of the Cantor singular function or 'Devil's staircase' (bottom); the enlargement is identical to the original, but the enlargement (scaling) factors are different in the x and y directions.

limit of Equation (2.1) as n ^^ is a singular function, discovered by Cantor [11], and usually referred to as the 'Devil's staircase'. It is a continuous, nonconstant function that is (nearly) horizontal all over except on an uncountable set (the Cantor set), of Lebesgue measure equal to zero. At each point in this uncountable set, the derivative of the Cantor singular function is given by a Dirac S distribution.

The Cantor singular function has many of the features of the Cantor set (listed above), except (i), i.e. it does not contain exact copies of itself at different scales. Inspection of Figure 2.2 (top) shows that the shape of the function Mn(x) in the interval [0, 1/3] is similar to that of the whole function (in the interval [0, 1]), scaled down in the x direction (abscissa) by a factor of 1/3. However, it is also apparent that the scaling factor in the y direction (ordinate) in Figure 2.2 (top) is not 1/3 but 1/2 (see Figure 2.2, bottom). A set or function is said to be self-affine when its scaling factors are different in different directions. Self-similarity requires these scaling factors to be identical in all directions. Therefore, the Cantor singular function or 'Devil's staircase' is not self-similar, but self-affine.

Following in Cantor's steps, another assault on the concept of dimension was made simultaneously by Peano [12] and Hilbert [13], in two short but influential articles. Both describe polygons that appear at first glance to be perfectly innocent, but nevertheless happen to fill a square more and more completely, so that, in the limit, they pass through every single point in the square.

The construction of Peano's original curve begins with a single line segment, the initiator (stage 0 in Figure 2.3). It is substituted by the generator (stage 1), which touches (but does not cross) itself at two points labeled A and B in Figure 2.3. If each straight line segment in stage 1 is replaced by a properly scaled-down generator,

Stage 0

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