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Results tagged with fractals
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Fractals deal with special sets that exhibit complicated patterns in every scale. Fractal sets usually have a Hausdorff dimension different from its topological dimension. Examples include Julia sets, the Sierpinski triangle, the Cantor set. Fractals naturally appear in dynamical system, such as iterations in the complex plane, or as strange attractors to continuous dynamical systems, (see Lorentz attractor).
4
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Accessible literature on fractional dimensions of subsets of $\mathbb R^n$
Erin Pearse's Introduction to dimension theory and fractal geometry may well be suited for this purpose. It introduces the various ways to define and measure a fractional dimension (box counting, Mink …
- 188k
10
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Accepted
How to plot this fractal
The source info (War in the Age of Intelligent Machines) identifies the fractal as a Julia set, iterates of $z\mapsto z^2+z_0$. It has evidently been distorted (warped) to give it a 3D appearance. I p …
- 188k
5
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The Koch snow flake, Holder exponents of conformal mappings
U.R Freiberg and M.R. Lancia, Energy Form on a Closed Fractal Curve (2004):
The Koch snow flake is the union of three Koch curves of Hausdorff dimension $D=\ln 4/\ln 3$ and Hölder exponent $\beta=\lo …
- 188k
1
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Laplacians and Renormalization
The renormalization factor of the Sierpinski gasket is derived in Spectral Decimation Functions and Forbidden Eigen Values in the Graph of Level Sierpinski Triangles. It depends on the contraction rat …
- 188k
13
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Unexpected occurrences of the Sierpinski triangle
The moves leading to the solution of the Towers of Hanoi puzzle form a Sierpiński triangle, as nicely described in this blog:
It is worth pausing a moment to think about this. The Tower of …
6
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Precise location of the Mandelbrot Bulb Attachment to the main Cardioid
in the parameterization where the main cardioid is a circle, the bulbs are attached at rational angles $\phi=2\pi m/n$: see R.L. Devaney, The Mandelbrot bulbs.
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10
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Accepted
Relationship between fractal dimension and Hurst exponent
In principle, fractal dimension and Hurst exponent are independent of
each other: fractal dimension is a local property, while the
long-memory dependence characterized by the Hurst exponent is …
- 188k
7
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What one really can do with fractals built from L-systems?
The original use of L-systems is to model the fractal-like forms that appear in plant growth. Johan Knutzen gives a nice overview with pictures in Generating Climbing Plants Using L-Systems.
In compu …
- 188k
2
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Is there a survey of recent work relating to the Hausdorff dimension of sets defined through...
I would guess you'll find many recent pointers in the publication list of Lars Olsen, and in that of Godofredo Iommi, for example:
Applications
of multifractal divergence points to sets of numbers d …
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