Fractals Cassi Blum

What is a fractal? –noun Mathematics, Physics . a geometrical or physical structure having an irregular or fragmented shape at all scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractal dimensions) are greater than the spatial dimensions.

Some Common Fractal Terms Sets – a collection of objects or elements classed together Trees- a diagram in which lines branch out from a central point or stem without forming closed loops Curve – a continuous bending line, without angles Triangle – a closed plane figure having three sides and three angles Trigonometry – the branch of mathematics that deals with the relations between the sides and angles of plane or spherical triangles, and the calculations based on them Iteration – the act of repeating; a repetition Reflection – the replacement of each point on one side of a line by the point symmetrically placed on the other side of the line. Refraction – Infinite – unbounded or unlimited; boundless; endless

Some Common Fractal Terms Chaos Theory – the study of unpredictable and complex dynamic systems that are highly sensitive to small changes in external changes Pattern – a combination of qualities, acts, tendencies, etc. forming a consistent or characteristic arrangement Repetition – the act of repeating; repeated action, performance, production, or presentation Function – a relation between two sets in which an element of the second set is assigned to each element of the first set Self-Similar – having a likeness or resemblance in a general way Recursive Definition – a definition consisting of a set of rules that by repeated application of the rules, the meaning of the definiendum is uniquely determined in terms of ideas that are already familiar. Complex Plane – a plane of parts of which are complex numbers

  Fractals are created using a difficult string of mathematics called Chaos Theory. With most fractals, you'll find that they are infinite, no matter how much you zoom in on them. That's because they're self-similar, meaning it appears exactly the same at different scales.

Base-Motif Fractals Take any shape made of line segments (called the base) and substitute it with another shape (called the motif). After substituting for an infinite amount of time, you get a base-motif fractal. The process is called generator iteration .This is generally how base-motif fractals are classified.

Dusts and Clusters Created when you take base line segments and cut segments out of them. When cutting the same segments out of the new lines, you get points that form a fractal called a dust. A cluster is formed when you take a 2-dimensional figure instead of line segment and do the same thing.

Fractal Canopies Created by taking one line segment and splitting it into two parts. The process has to be continued indefinitely.

IFS Fractals IDS Fractals are created by using several geometric transformations to and transforming them into smaller figures. When continued indefinitely, you get a fractal.

Julia Fractal Sets The most famous type of fractal set. They are made with the formula z2+c, by using any complex number as the variables, and repeated. The process of using a formula to make a fractal is called formula iteration.

Mandelbrot Sets Another type of fractal made by using formula iteration. It uses the same formula (z2+c), but the variable z is set to 0+0i.

Nonstandard Fractals Some nonstandard fractals include Chain Fractals, “Star of David” Fractals, Cellular Automata Fractals, and Pascal’s Triangle. This isn’t an official classification, but they didn’t fit anywhere else.

Chain Fractals Chain fractals are created by starting with a simple ring, then substituting that ring with 20 rings. By repeating this process, you get a chain fractal.

Star of David Fractal This fractal is created in a base-motif fashion. Base-motif is taking a line segment (base)and replacing it with another type of segment (motif). When your bases are arranged like the star of David, your result looks like the picture shown.

Cellular Automata These fractals were made to resemble or simulate the growth of actual cells. The most popular of this types are called diffusion fractals. These fractals start from the center and spread outwards.

Pascal’s Triangle These are made my forming a triangle of numbers, with the sum of two of the numbers above it. These are very popular, even when not associated with fractals.

Pascal’s Triangle I found this too. This pictures shows all of the odd numbers highlighted in yellow. The result is a Sierpinski’s Triangle, which is another type of fractal. (related with dusts and cluster fractals.)

Peano Curves This name is given to any fractal that is two dimensional, meaning it takes up space on more than one plane.

Plasma Fractals Plasma fractals are unique because they have a random element in them, which gives them Brownian self-similarity. Plasma fractals are created by taking random numbers as the corner values of a rectangle…

Plasma Fractals To gather the data for the points inside the rectangle, you average the numbers of the two corners closest to the point. Chaos Theory is really no fun.

Fractals in Nature There are many things in nature that are self-similar, just like fractals. Romanesco broccoli are self-similar, and so are most fern leaves.

My Thoughts At first, this project looked boring to me. But as I continued to learn about fractals, I found myself to be really interested. There are so many different types, and I admire the creative minds that found them.

Sources http://library.thinkquest.org/26242/full/types/types.html http://pages.infinit.net/garrick/fractals/fractals2.html http://en.wikipedia.org/wiki/Fractal http://www.angelfire.com/art2/fractals/examples.html http://www.futuremind.ch/millenniumfractal/documentation/Fractals.html