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Hofstadter Butterfly
The Hofstadter butterfly is a fractal structure that resembles the wings of a butterfly. The diagram describes the energy levels of an electron in a two-dimensional lattice under the influence of a magnetic field. This structure was discovered in 1976 by the American physicist Douglas R. Hofstadter and is a key concept in solid-state physics and quantum mechanics.
Image by Yasuyuki Hatsuda, Hosho Katsura and Yuji Tachikawa - Hofstadter's butterfly in quantum geometry,
New J. Phys. 18, 103023 (2016), © CC BY 3.0
Mathematical Description
The energy level scheme is derived from the so-called Harper equation,
a difference equation-based description of the system.
Physical Relevance
The Hofstadter butterfly is not only a mathematically interesting phenomenon
but also has practical significance in solid-state physics,
particularly for topological insulators, quantum Hall effects, and graphene structures.
After its theoretical prediction, the Hofstadter butterfly was experimentally observed
in real materials in 2013 using graphene superlattice structures.
This is a fascinating example of the connection between
quantum mechanics, solid-state physics, and fractal mathematics.
Butterfly Maschine
The usage is self-explanatory. Start with Run. After Stop, Next can also be used. The mathematical and physical background is very complex. However, Lars Huttar has provided a JavaScript implementation that enables fast calculations using polynomial iteration. I have added a color scheme for the figure with the option to select a palette and a background color. The background color dialog uses pickr.js, provided by Simon Reinisch under an MIT license. Additionally, I have made the calculation factor pt a parameter. This results in interesting butterfly fractals, but only in the case pt = 2 do they have the physical interpretation described above. You may even change the factor during the calculation.
Current q: 4