# Fractal geometry

### From Biomatics.org

Recurring geometrical architecture of structure and information.

By definition a Fractal is a set for which the Hausdorff Besicovitch dimension stricly exceeds the toplogical dimension. (Mandelbrot)

Fractals are recursively generated from a starting set together with a generator or rule to transform the set. In other words- in the first cycle the rule is applied to the initial set yielding a transformed set. The process is then repeated with the rule applied to the resulting set. For example in ontogeny a single cell has some rule for generating the next generation-the two cell stage. Eventually this process leads to an adult bio-mechanical organism.

Three common techniques for generating fractals are:

**Escape-time fractals**— These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal and the Lyapunov fractal.**Iterated function systems**— These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals.**Random fractals**— Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.

A genome sequence can be an informational fractal (see Baseparity). A fern can be a structural fractal.

Fractal Groups- a plausible definition of what a "fractal group" should be, examples of such groups and potential biological occurences.

The above "multiplication tables" demonstrate the fractal nature of this algebraic structure i.e. a fractal goup.

Above the same structure is seen in an amino acid.

The following sequence is obtained by replicating the previous object then connecting the corresponding points. The initial starting point is a single point.

Note the occurence of Pascal's triangle above, which then gives rise to Sierpinski's triangle, obtained from Pascal's triangle by coloring the even and odd entries two diffenent colors.

Fractal dimension

In fractal geometry, the **fractal dimension**, *D*, is a statistical quantity that gives an indication of how completely a fractal appears to fill space, as one zooms down to finer and finer scales. There are many specific definitions of fractal dimension and none of them should be treated as the universal one. From the theoretical point of view the most important are the Hausdorff dimension, the packing dimension and, more generally, the Rényi dimensions. On the other hand the box-counting dimension and correlation dimension are widely used in practice, partly due to their ease of implementation.

Although for some classical fractals all these dimensions do coincide, in general they are not equivalent. For example, what is the dimension of the Koch snowflake? It has topological dimension one, but it is by no means a curve-- the length of the curve between any two points on it is infinite. No small piece of it is line-like, but neither is it like a piece of the plane or any other. In some sense, we could say that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the question of whether its dimension might best be described in some sense by number between one and two. This is just one simple way of motivating the idea of fractal dimension.

## Hexagonal Fractals

Chemical structure of the fractal molecule. Art by: Courtesy Saw-Wai Hla

## Fractal Chemicals and Drugs

- Adrenergic beta-Antagonists [4] ¤
- Antigen-Antibody Complex [2] ¤
- Antineoplastic Agents [2] ¤
- Chromatin [2] ¤
- Collagen [2] ¤
- DNA [6] ¤
- Enzymes [2] ¤
- Glucose [2] ¤
- Ion Channel [3] ¤
- Liposomes [2] ¤
- Oligonucleotides [2] ¤
- Polymers [3] ¤
- Proteins [5] ¤
- Radioactive Fallout [2] ¤
- Receptors, Cell Surface [4] ¤
- Technetium Tc 99m Exametazime [3] ¤
- Water [2] ¤

Recursive Algorithms are fractal in nature.

Note also the similarity to graphical representations of boolean algebras and The geometry of logic

An Introduction to the fascinating patterns of Visual Math

Visit : Fractalogy: Fractal is not just a simple phenomenon. It is one of the most fundamental mathematical rules found in life and nature.