Fractals| Fractals |
In General
* Idea: A physical quantity
is called a fractal if it depends on the size of the scale used to measure
it; A fractal is often self-similar at different scales, containing structures
nested within one another.
* History: Cantor; 1885,
H Poincaré; 1918, F Hausdorff; 1960s, Search for analyticity and regularity
properties; 1975, B Mandelbrot, "kinematical" description of fractal
geometry (he coined the term "fractal").
* Status: 1996, Fractal
phenomena are observed in many fields (dielectric breakdown patterns, ...),
and it would be nice to have a theoretical framework for treating fractals,
comparing them, etc; The concept of fractal dimension has been defined, but
for the rest a theoretical basis is lacking.
@ General references: in Gleick 87;
Mandelbrot NS(90)sep.
@ Mathematical: Mandelbrot 82 [I],
PRS(89);
Halsey et al PRA(86);
Falconer 86, 03.
@ Fractal geometry and calculus: Le Méhaute 90;
Strichartz 06 [differential equations on fractals];
Parvate et al a0906
[integrals and derivatives along fractal curves];
Muslih & Agrawal JMP(09) [scaling method for volumes, areas, solid angles, and applications];
Calcagni ATMP(12)-a1106 [and fractional spaces];
> s.a. fourier transforms; integration;
laplace operator; vector calculus.
@ Fractal surfaces: Russ 94.
Measures of Fractality
> s.a. dimension; fractals in physics.
$ Fractal dimension: Defined as
\[ d_{\rm fr}:= {\rm d}\ln M(R)\, /\, {\rm d}\ln R\;.\]
* Lacunarity: Related to
departure from translational invariance and size distribution of holes.
Examples, Types
> s.a. Apollonian Gasket; cell complex.
* Julia sets: Precursors.
* Mandelbrot set: The most famous
example of a fractal, given by
M:= {c ∈ \(\mathbb C\) | Pcn(0) ≠ 0 as n → ∞}, with Pc: \(\mathbb C\)' → \(\mathbb C\)', z \(\mapsto\) Pc(z) = z2 + c, \(\mathbb C\)':= \(\mathbb C\) ∪ {∞} .
* Cantor dust: A fractal curve such that the length between two points on it is given by
L = ε1−ln2/ln3 B → 0 as ε → 0 ,
where ε is a unit of scale and B a constant.
* Cantor set: The only
perfect, totally disconnected, metric topological space; It can be realized
in many homeomorphic ways, e.g., by the "middle third" construction;
There is a continuous projection π from it to any compact metric
topological space [@ Hocking & Young 61].
* Koch curve: A fractal curve;
If ε is a length scale and A a constant, the length
between two points on it is
L = ε1−ln4/ln3A → ∞, for ε → 0 .
* Other fractal curves: The Peano
curve, a fractal curve which can be written as a Lindenmayer system; The graphs of the
Weierstrass Functions and
Takagi Function, and of white noise (with fractal dimension 2).
@ Mandelbrot set: Metzler AJP(94)sep [perplex];
Shishikura AM(98) [Hausdorff dimension of boundary = 2].
@ Other examples: Weiss PRS(01) [Cantor set];
Anazawa et al PhyA(04) [with typical scale];
> s.a. Sierpinski Carpet.
Applications > s.a. fractals in physics.
* Examples: Crystal growth, forest fires, fibrillations.
@ Geology / geophysics:
Turcotte 97 [1st ed r PT(93)may];
issue CSF(04).
@ Physiology:
Bassingthwaighte et al 94;
West & Deering PRP(94);
Brú et al PRL(98) [tumor growth].
main page
– abbreviations
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send feedback and suggestions to bombelli at olemiss.edu – modified 31 oct 2018