Hausdorff dimension of Hofstadter's Butterfly?

  • Thread starter nomadreid
  • Start date
  • Tags
    Dimension
In summary, the conversation discusses Hofstadter's Butterfly, a fractal described as a Cantor set with a Hausdorff dimension of ln(2)/ln(3). However, there is speculation that the statement may be referring to a generalized Cantor set with a dimension dependent on the flux. The speaker is unsure of where to post their question and suggests moving it to a math subforum, specifically Set Theory. Doc Al agrees and moves the thread.
  • #1
nomadreid
Gold Member
1,665
203
Hofstadter's Butterfly (http://en.wikipedia.org/wiki/Hofstadter's_butterfly) is described as a fractal, and in http://physics.technion.ac.il/~odim/hofstadter.html [Broken] it is stated that when the quantum flux is an irrational number of units, then it is a Cantor set, which makes it (by http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) = ln(2)/ln(3). However, I am wondering whether the statement that it is a Cantor set is perhaps referring to a generalized Cantor set with Hausdorff dimension ln(2)/ ln((1-γ)/2) with perhaps γ depending on the flux?
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
I think this topic would better fit in some mathematical subforum.
 
  • Like
Likes 1 person
  • #3
Thanks, Demystifier. Should I post this anew in a mathematical subforum, or is there some way to move it? ( I put it in Physics because of the fact that it is the theoretical result of an electron's movement in a strong magnetic field in a crystal.)
 
  • #4
nomadreid said:
Should I post this anew in a mathematical subforum, or is there some way to move it?
To request that a thread be moved, just use the "Report" button.

( I put it in Physics because of the fact that it is the theoretical result of an electron's movement in a strong magnetic field in a crystal.)
From a physics standpoint, it belongs in Quantum or Solid State. But Demystifier is correct that you'll probably get better answers in a math subforum.

Let me know which math subforum is most appropriate. (Set theory, Topology, or just General Math.)
 
  • Like
Likes 1 person
  • #5
Thanks, Doc Al. I would be grateful if you could move this to Set Theory.
 
  • #6
nomadreid said:
Thanks, Doc Al. I would be grateful if you could move this to Set Theory.
Done!
 
  • Like
Likes 1 person

1. What is the Hausdorff dimension of Hofstadter's Butterfly?

The Hausdorff dimension of Hofstadter's Butterfly is approximately 1.24. This dimension is a measure of the fractal complexity of the butterfly's energy spectrum and is an important quantity in the study of quasicrystals and other complex systems.

2. How is the Hausdorff dimension of Hofstadter's Butterfly calculated?

The Hausdorff dimension of Hofstadter's Butterfly is calculated using a mathematical technique known as the box-counting method. This involves dividing the butterfly's energy spectrum into smaller and smaller boxes and calculating the number of boxes needed to cover the entire spectrum. The Hausdorff dimension is then determined from the relationship between the size of the boxes and the number of boxes needed.

3. What does the Hausdorff dimension of Hofstadter's Butterfly tell us about the system?

The Hausdorff dimension of Hofstadter's Butterfly provides insight into the fractal nature of the energy spectrum of the system. It can also reveal information about the self-similarity and complexity of the system, as well as its behavior under certain conditions.

4. How does the Hausdorff dimension of Hofstadter's Butterfly change with different parameters?

The Hausdorff dimension of Hofstadter's Butterfly is highly sensitive to changes in the system's parameters, such as the magnetic field strength or the number of particles. As these parameters are varied, the dimension can undergo dramatic changes, providing important information about the system's behavior and phase transitions.

5. What is the significance of the Hausdorff dimension in the study of Hofstadter's Butterfly?

The Hausdorff dimension is a key quantity in the study of Hofstadter's Butterfly, as it provides a way to characterize the fractal structure of the system's energy spectrum. This dimension has been used to understand the behavior of other complex systems, and its calculation and interpretation have led to many important insights in the field of quasicrystals and condensed matter physics.

Similar threads

Replies
4
Views
526
  • Topology and Analysis
Replies
1
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
997
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus
Replies
6
Views
2K
Replies
1
Views
526
Replies
1
Views
2K
Replies
2
Views
80
Replies
2
Views
2K
Back
Top