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.
* 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;
NS(90)sep.
@ Mathematical: Mandelbrot 82 [I], PRS(89); Halsey et al PRA(86); Falconer
90, 95.
@ Fractal geometry and calculus: Le Méhaute 90; Parvate et al
a0906 [integrals and derivatives along fractal curves]; > s.a. integration.
@ Fractal surfaces: Russ 94.
Measures of Fractality > s.a. dimension; fractals
in physics.
$ Fractal dimension:
Defined as
dfr:= d ln M(R) / d ln R .
* Lacunarity: Related to departure from translational invariance and size distribution of holes.
Examples, Types > s.a. cell
complex.
* Julia sets: Precursors.
* Mandelbrot set: The most famous example, given by
M:= {c in C | Pcn(0) 
0
as n → 
}, with Pc: C' → C', z 
Pc(z)
= z2 + c, C':= C 
{}
.
* Cantor dust: A fractal curve such that the length between two points on it is given by
L = 
1–ln2/ln3 B → 0 as → 0 ,
where is a unit of scale and B a constant.
* Cantor set: The only
perfect, totally disconnected, metric topological space;
Can be realized in many homeomorphic ways, e.g., by the "middle third" construction;
There is a continuous projection 
from
it to any
compact
metric topological space [@ Hocking & Young 61].
* Koch curve: A fractal curve; If is a length scale and A a
constant, the length between two points on it is
L = 1–ln4/ln3A → , for →
0
.
* Peano curve: A fractal curve which can be written as a Lindenmayer
system.
@ 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].
Applications > see fractals in physics.
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send feedback and suggestions to bombelli at olemiss.edu – modified 4
jun 2009