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*_{fr}:= d ln *M*(*R*) / 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, given by

*M*:= {c ∈ \(\mathbb C\) | *P*_{c}* ^{n}*(0) ≠ 0
as

* __Cantor dust__: A fractal curve such that the length between two points
on it is given by

*L* = *ε*^{1–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;
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* = *ε*^{1–ln4/ln3}*A*
→ ∞, 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)#2.

@ __Physiology__: Bassingthwaighte et al 94; West & Deering PRP(94);
Brú
et al PRL(98)
[tumor growth].

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