What is...an inverse fractal?
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- Опубликовано: 3 окт 2024
- Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What is...an inverse fractal? Or: Zooming out.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
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TeX files for the presentation.
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Thumbnail.
Picture created using latex tikz
Main discussion.
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Background material.
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Computer talk.
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demonstrations...
Pictures used.
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Picture from • What is...the drunken ...
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RUclips and co.
• Fun with Hausdorff Dim...
• What is Hausdorff Dime...
• Sierpinski's Triangle ...
#geometry
#fractals
#mathematics
I think what you’re trying to get at is describing the stability of a system through some iterative process where the system is some dynamical system and the iterative process can be described in combinatorics. Since this happens in general on different objects stable categories should be considered.
One way to approach this is to also consider coalgebras. See “The Seirpinski Carpet as a Final Coalgebra” (2021) by Noquez and Moss and “Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra” by Bhattacharya and Moss.
Oh, thanks. The references with the final coalgebra look interesting 🙂
@@VisualMath Yup. A similar process is described by partially order sets called posets. Posets satisfy an ascending chain condition which crucially stabilizes. This applies to Noetherian rings with for example fields and graded rings with for example polynomial rings.
Posets have associated algebras called incidence algebras with some nice properties like zeta functions and Möbius inversion.
Here's another two: the graphs of the n-ary versions of the AND and OR logic operators form 'inverse fractals.'
Ah, I didn't knew those examples. Thanks for sharing ☺