Fractals

Joe CzuprynMTH 491

I. Introduction

II. Dimension

III. Brief history

IV. Specific fractals and their properties

V. Appearances and applications

Presentation Outline

Introduction

Self-similarity – when broken into smaller and smaller pieces, the new pieces look exactly the same as the original

Dimension - how much an object fills a space

Introduction (cont.)

S represents the scaling factor and is always a natural number.

N represents the number of smaller, self-similar figures (for a scaling factor S) needed to create the larger figure.

Dimension

Dimension Of A Line

a b c

Dimension Of A Square

Dimension Of A Cube

1600s - Gottfried Leibniz

1883 - Georg Cantor

1904 – Helge von Koch

1915 – Vaclav Sierpinski

Early 1900s – Gaston Julia and Pierre

Fatou

Early History

Polish-born, French mathematician

Fractals: Form, Chance and Dimension (1975)

The Fractal Geometry of Nature (1982)

Benoit Mandelbrot

In the nth step, 3(n-1) triangles will be removed.

Sierpinski Triangle

Sierpinski Triangle - Dimension

δ will be used to refer to the side length of the equilateral triangle.

In the nth iteration, 3 * 4(n-1) triangles are added.

Koch Snowflake

Koch Snowflake - Dimension

Area of an equilateral triangle =

Formula for a geometric series with common ratio r:

Koch snowflake – Area

Koch snowflake – Area (cont.)

Koch snowflake – Area (cont.)

Using geometric series

Each iteration increased the length of a side to (4/3) its original length.

Thus, for the nth iteration, the overall perimeter is increasing by (4/3)n.

Koch Snowflake - Perimeter

Divergent Sequence

The perimeter is then considered to be infinite!

How does this apply to Mandelbrot’s “How long is the Coast Line of Britain?” problem?

Koch Snowflake – Perimeter (cont.)

Trees and plants

In the human body: Blood vessels Alveoli in the lungs

Naturally Occurring Fractals

Used by Boeing to generate some of the first 3-D computer generated images

Currently being used to make antennas smaller in cell phones

Fractals And Technology

Fractal patterns exist in a healthy human heartbeat

May give doctors a way to detect small tumors/early stages of cancer

Fractals And Medicine