Fractal+Dimension

We generally understand the term "dimension" to mean a component in Eucliidean space having 1, 2, 3, or 4 dimensions. Fractal dimension is non-Euclidean. The Hausdorff Dimension is use to describe fractal object, having an irregular geometric shape.

Hausdorff Dimension
We can reduce an object to its linear analog and divide it in 1/r sections. The measure N of given division (r) is expanded by N=r^D. We can therefore calculate the original length by L = N*r. . However, the length increases as the step sizes (r) are reduce because N increases proportionally. We know that L is function of the the step size i.e L(r). In essence, the term "length" is meaningless unless a formal unit of "step size" (r) is standardized, hence the SI unit for length, the meter.



Richardson Effect
Consider measuring the length of a coastline with decreasing rulers. While it is supposed that measuring a coastline with finer step sizes will generate a final true, in reality, the fractal nature of the coast produces more and more features that increase it's length. This increasing length at finer step sizes is known as the //Richardson Effect//. Richardson made this discovery while researching coastlines of continents.



Mandelbrot
Richardson's plot of coastlines on a log(L(r)) = m*log(r) + b. Mandelbrot defined the fractal dimensional increment such that m = (1-D). A smooth coastline like South Africa would have a D ~ 1, while rougher coasts would yield D > 1 and possibly non-integer values, unlike Euclidean dimensions. Traditionally, mathematics was only applied to smooth surfaces. Mandelbrot is created with applying mathematics to rough surfaces, especially complicated natural phenomena such as tree branching, bifurcation of blood vessels, cloud patterns and other natural phenomena. Mandelbrots mathematical contribution is associated with his first analyses of the Juila sets while working at IBM. With a computer, he was able to calculate many Julia sets. He developed his own set which is plot coordinates for all possible Julia sets. The result is a Mandelbrot set where the image has self-similar features - repeated features at finer length scales.



Fractal geometry has been applied to explain natural phenomena. West, Brown and Enquist used fractal geometry to predict the mass-energy relationship between large and small biological animals, where E = m^(3/4). It turns out that large animals like elephants require less fuel per unit mass than small animals like mice. Fractal geometry has been used so that measuring the branching distribution of a single tree can be compared to the tree size distribution of a forest and thereby predict the CO2 consumption of an entire forest. Fractal geometry has also been used by engineers to create fractal antennae -- small antennae capable of receiving a wide range of frequencies, first introduced by Cohen.



Relation to Fracture Mechanics
Fractal dimension is related to the tortuousness of a fracture surface. This can be determine is virtually three dimensions: 1D: tortousity of lines and interfaces (D*) 2D: crack branching patterns measured from the crack branching coefficient (CBC) 3D: torosity of coastlines of out of plane fracture surface features measured of slit island method.

Fractal dimension fundamentally relates the fracture toughness, elastic modulus and atomic spacing by K_IC - E * a^(0.5) * D*

Fractals in Nature
Common fractal constructions using fractals:
 * Koch curve
 * Sierpinski Triangle
 * Menger Sponge
 * Barnsley Fer