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].