Talk:Smith–Volterra–Cantor set

Nowhere dense

The same content is present at nowhere dense but I felt this deserves a separate page. I'll be creating a page on Volterra's function in the near future if someone doesn't beat me to it  :-) - Gauge 06:07, 23 Aug 2004 (UTC)

There is a slight inconsistency here. Either the intervals removed at each step are the "middle quarter of the remaining intervals" or they are centred on a/2^n. But they cannot be both. The first leads to measure of 0.5, the second to measure of 0.53557368... --Henrygb 17:17, 25 Apr 2005 (UTC)

Thanks for noticing. I noticed that the intervals you gave seemed to be off, so I corrected them (hopefully :-)). - Gauge 01:42, 30 Apr 2005 (UTC)

You are right - somehow I multipled 3 by 2 and got 12. Thanks --Henrygb 22:53, 30 Apr 2005 (UTC)

Can somebody please help me understand?

Let the set be called S. By construction, S contains no intervals (i.e. S contains points that are seperate from each other.). And the measure of a single point is 0. So how can the total measure be 1/2? On the other hand, the total length of removal is 1/2, hence remaining length must be 1/2. Hence, S must contains intervals of length greater than 0. Can somebody please help me resolve this? 108.162.157.141 (talk) 01:53, 28 November 2013 (UTC)

Intuition must adapt to facts! Yes, it is hard. For now, your intuition tells you that the measure of a set is the sum of lengths of intervals. And your logic already tells you the opposite. Your intuition must adapt. It is a hard internal work. For even harder case, see Weierstrass function. Such is the life. Boris Tsirelson (talk) 06:18, 28 November 2013 (UTC)

Please, explain the hausdorf dimension

This article needs to explain the hausdorff dimension of the Generalized Cantor set, as listed on the wikipage List of fractals by Hausdorff dimension, which is shown as

$Dimension={\frac {-\log(2)}{\log \left(\displaystyle {\frac {1-\gamma }{2}}\right)}}$