Why Are the Definitions of Hausdorff Dimension Equivalent?

  • Context: Graduate 
  • Thread starter Thread starter hedipaldi
  • Start date Start date
  • Tags Tags
    Dimension Fractal
Click For Summary
SUMMARY

The discussion clarifies that the two interpretations of Hausdorff dimension—one from Wikipedia and the other from a specific example—are not distinct versions but rather complementary aspects of the same concept. The first definition focuses on covering fractal sets with sets of integer dimensions, while the second illustrates this definition through a practical example involving triangles. Understanding this equivalence is crucial for grasping the foundational principles of fractal geometry.

PREREQUISITES
  • Familiarity with fractal geometry concepts
  • Understanding of Hausdorff measure
  • Basic knowledge of logarithmic functions
  • Experience with mathematical proofs and definitions
NEXT STEPS
  • Study the formal definition of Hausdorff measure in detail
  • Explore examples of calculating Hausdorff dimension using various fractals
  • Learn about the implications of Hausdorff dimension in real-world applications
  • Investigate the relationship between Hausdorff dimension and other dimensions, such as topological dimension
USEFUL FOR

Mathematicians, students of fractal geometry, and researchers interested in the properties and applications of Hausdorff dimension will benefit from this discussion.

hedipaldi
Messages
209
Reaction score
0
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
 
Last edited:
Physics news on Phys.org
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.
 
  • Like
Likes   Reactions: 1 person

Similar threads

Graduate Hausdorff dimension of the cantor set
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Is fractal nature dependent on the number of steps?
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Is the Long Line Locally Euclidean, Hausdorff, and Path Connected?
  • · Replies 3 ·
Replies
3
Views
2K
High School Clarification about Fractal Dimensions
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Hausdorff dimension of Hofstadter's Butterfly?
  • · Replies 5 ·
Replies
5
Views
3K
Graduate How to Convert a Complex Logarithm to a Complex Exponential
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Foliation of 4-dimensional connected Hausdorff orientable paracompact manifold
  • · Replies 13 ·
Replies
13
Views
5K
Graduate Finding base number with Hausdorff Dimension Approximation methods?
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Fractal dimension of the universe = 2?
  • · Replies 1 ·
Replies
1
Views
5K
Graduate How to find fractal dimension of Gosper Island
  • · Replies 2 ·
Replies
2
Views
4K