Exploring the Relationship Between Hausdorff Dimension Definitions

In summary, the conversation discusses the two definitions of Hausdorff dimension, one by coverings and the other by the ratio of two logarithms. The first definition is found on Wikipedia and the second is illustrated with an example on a website. These are not two different versions, but rather different ways of understanding the same concept. The first definition involves covering a fractal set with sets of integer dimension, while the second definition uses triangles as an example.
  • #1
hedipaldi
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Hi,
I am trying to understand why do the two versions of Hausdorff (fractal) dimension are actually the same.I refer to the definition by coverings and the definition by ratio of two logarythms.

http://en.wikipedia.org/wiki/Hausdorff_measure
http://www.math.umass.edu/~mconnors/fractal/sierp/sierp.html

Thank's in advance
 
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  • #2
These are NOT two "versions" of the Hausdorff dimension. The first is the (Wkipedia) definition, the second is an example of calculating it in a particular case. The definition deals with covering the (fractal) set by sets with integer dimension and the triangles in the example are precisely that.
 
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What is fractal dimension?

Fractal dimension is a mathematical concept used to measure the complexity and self-similarity of a geometric pattern. It is a measure of how the detail in a fractal pattern changes with the scale at which it is viewed.

How is fractal dimension different from Euclidean dimension?

Unlike Euclidean dimension, which is a whole number and describes the size and shape of a regular geometric object, fractal dimension can be a non-integer value and describes the complexity and irregularity of a fractal pattern.

What is the Hausdorff dimension?

The Hausdorff dimension is a specific type of fractal dimension that measures the size or extent of a fractal pattern. It is named after German mathematician Felix Hausdorff and is often denoted as "dH".

How is fractal dimension calculated?

Fractal dimension can be calculated in a variety of ways, depending on the specific type of fractal being measured. One common method is the box-counting method, where the fractal pattern is divided into smaller and smaller squares and the number of squares needed to cover the pattern is counted.

What are some real-world applications of fractal dimension?

Fractal dimension has been used in a wide range of fields, including physics, biology, economics, and art. It has been used to model irregular shapes in nature, analyze financial market fluctuations, and create visually appealing artwork. Additionally, the concept of fractal dimension has been applied to computer graphics and image processing techniques.

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