Hausdorff dimension of the cantor set

Thread starter hedipaldi
Start date Sep 14, 2013
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Cantor Dimension Set

In summary, the Hausdorff dimension of the cantor set is equal to log2/log3, which indicates that the set has a non-integer dimension and is self-similar. It is calculated using a formula that considers the number of self-similar copies needed to cover the set at different scales. The significance of the Hausdorff dimension lies in its implications in mathematics and other fields, as well as its illustration of non-integer dimensions. It relates closely to the construction process of the cantor set, which involves self-similar iterations. The concept of Hausdorff dimension can also be applied to other fractal sets, but the exact value will vary.

Sep 14, 2013

hedipaldi

Hi,
Using the definition of Hausdorff measure:
http://en.wikipedia.org/wiki/Hausdorff_measure
I am looking for a simple proof that Hd(C) is greater than 0, where C is the Cantor set and
d=log(2)/log(3)
Thank's in advance

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Sep 15, 2013

A David

http://people.math.gatech.edu/~shenk/Research/FractalGeometry/p-series.pdf

FAQ: Hausdorff dimension of the cantor set

1. What is the Hausdorff dimension of the cantor set?

The Hausdorff dimension of the cantor set is equal to log2/log3, which is approximately 0.6309. This means that the cantor set has a fractal dimension between 0 and 1, indicating that it has a non-integer dimension and is a self-similar set.

2. How is the Hausdorff dimension of the cantor set calculated?

The Hausdorff dimension of the cantor set is calculated using the Hausdorff dimension formula, which takes into account the number of self-similar copies needed to cover the set at different scales. In the case of the cantor set, this involves dividing the length of each interval by 3 at each iteration, and then taking the limit as the number of iterations approaches infinity.

3. What is the significance of the Hausdorff dimension of the cantor set?

The Hausdorff dimension of the cantor set has significant implications in mathematics, particularly in the study of fractals and self-similarity. It also has applications in various fields such as physics, biology, and economics. Additionally, the Hausdorff dimension of the cantor set helps to illustrate the concept of a set having a non-integer dimension, which challenges traditional notions of geometry and space.

4. How does the Hausdorff dimension of the cantor set relate to its construction?

The Hausdorff dimension of the cantor set is closely related to its construction process, which involves removing the middle third of each interval at each iteration. This self-similar construction results in the set having a non-integer dimension, as the number of intervals and their lengths decrease at each iteration, but the overall structure remains the same.

5. Can the Hausdorff dimension of the cantor set be generalized to other fractal sets?

Yes, the concept of Hausdorff dimension can be applied to other fractal sets, as it is a measure of the scaling properties and self-similarity of a set. However, the exact value of the Hausdorff dimension will vary depending on the construction and properties of the specific fractal set being studied.