**4.1 Triadic Cantor set and the random Cantor set**

The archetypal fractal is what is known as Cantor triadic set. We start by describing the fundamental construction. Consider a unit interval. Let us delete the middle third but leave the end points. We repeat the procedure with the two segments left and so on as shown in Fig. 3 infinitely many times. At the end we obtain an uncountable set of points of measure zero. This means adding all these points together and we obtain a zero length. However, from the point of view of transfinite set theory something very profound is left, namely a transfinite points set with a finite dimension, the so-called Hausdorff-Besicovitch dimension [2]

$$d\_{\mathbb{C}} = \frac{\ln 2}{\ln 3} \cong 0.63$$

Mauldin and Williams replaced the orderly triadic construction by a random construction. In their original paper [10] they said they used a uniform probabilistic distribution. The Mauldin-Williams theorem which states that with the probability equal to one, a one dimensional randomly constructed Cantor set will have the Hausdorff-Besicovitch dimension

$$d\_{\mathbb{C}}^{(0)} = \frac{\sqrt{5} - 1}{2}$$

The Menger-Urysohn dimension of all Cantor sets is zero, while the empty set has the dimension minus one [11].
