#### Title

#### Document Type

Honors Capstone Project

#### Date of Submission

Spring 5-1-2010

#### Capstone Advisor

Claudia Miller

#### Honors Reader

Jeffrey Meyer

#### Capstone Major

Mathematics

#### Capstone College

Arts and Science

#### Audio/Visual Component

no

#### Capstone Prize Winner

no

#### Won Capstone Funding

no

#### Honors Categories

Sciences and Engineering

#### Subject Categories

Algebra | Algebraic Geometry | Applied Mathematics | Other Applied Mathematics

#### Abstract

** **

For my capstone project, I analyzed fractals. A fractal is a picture that is composed of smaller images of the larger picture. Each smaller picture is *self- similar*, meaning that each of these smaller pictures is actually the larger image just contracted in size through the use of the *Contraction Mapping Theorem* and shifted using *linear* and *affine* *transformations*.

Fractals live in something called a *metric space*. A *metric* *space*, denoted (*X, d*), is a space along with a distance formula used to measure the distance between elements in the space. When producing fractals we are only concerned with metric spaces in **R**^{2}, which is the collection of all ordered pairs. Also, we only use infinite sequences, in specific *Cauchy sequences*, which are sequences in which the distance between elements decreases as the sequence progresses.

Fractals live in a special metric space called (*H** *(*X*), *h*). Every element in (*H** *(*X*), *h*) is actually a *compact set*. A *compact* set is *closed*, *bounded*, and every infinite sequence has a subsequence that has a limit in the space. A space is *closed* if it contains all of its points and limit points. A space is *bounded* if you can draw a circle around the space enclosing all of its elements. When producing fractals, the *Hausdorff distance* is used to measure the distances between compact sets in (*H** *(*X*), *h*), which is the maximum distance between two compact sets in the space.

Fractals are produced using *contraction mappings*. This means that fractal pictures are contracted using a *contractivity factor*, between 0 and 1 that contracts the size of the image. The *Contraction Mapping Theorem* states that all contraction mappings or *iterated function systems*, which are sets of contraction mappings, have one fixed point to which the sequence of points in the fractal converge.

My final fractal is a picture of my initials, KMM. To make this picture, I used *linear* and *affine* *transformations*, *affine transformations* being *linear transformations* plus a shift. I drew my initials in a square. Using the Contraction Mapping Theorem, I then contracted the square to the origin, and then used shifts to move the picture to the twelve different sections to which each part belonged. I calculated the twelve different contraction mappings then used the program written by Professor Banerjee for Maple to apply The *Random Iteration Algorithm*. This algorithm starts with any point then randomly chooses one of the twelve contraction mappings, applies it to the point and finds a new point in the picture. It repeats this process infinite times, and the sequence of all points found converge to the fixed point, producing my final picture.

#### Recommended Citation

Mesznik, Kara, "Analyzing Fractals" (2010). *Syracuse University Honors Program Capstone Projects*. 353.

http://surface.syr.edu/honors_capstone/353

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.