Robert P. Munafo, 2008 Feb 1.

2. the Cantor middle-thirds set: A set of points on a line segment that is the result of an infinite number of steps, in which each step consists of removing the middle-third of the segment(s) remaining from the previous step. It has a Hausdorff dimension of 0.631:

step 0: _________________________________________________________________________________ step 1: ___________________________ ___________________________ step 2: _________ _________ _________ _________ step 3: ___ ___ ___ ___ ___ ___ ___ ___ step 4: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ (etc.)

A Cantor set has an infinite ("uncountable") number of points, the same number as a line segment or any other continuum. However (and paradoxically, by the intuition of Cantor's time) it has zero Lebesgue measure.

See also fatou dust, Fundamental Dichotomy, critical point.

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From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo.     Mu-ency index

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