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 ∈ \(\mathbb C\)
| *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; 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* = *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).

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

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send feedback and suggestions to bombelli at olemiss.edu – modified 31 oct 2018