Introduction to Fractal Geometry

- The Wonder of Noninteger Dimension -

$Lastupdate: Tue Dec 6 22:44:26 2005 $

Lecturer : Dr. Tohru Ikeguchi (Associate Professor, Department of ICS)
TA : Mr. Ryosuke Hosaka
Date : April 15, 2004 16:00 - 17:30

Abstract:

We often face a number of complex objects in our daily life. For example, shapes of clouds in the sky, leaves on trees, coast lines, snow frakes, rivers, thunderstomes ridge lines, stones, blood vessels, lungs, and brains. Fractal is one of the interesting and essential concepts to describe the complexity of these shapes.

In this lecture, first we show that it is impossible to measure the complexities of these shapes with the convensional measures, such as length, area, volume. Next, we introduce a new measure, which is called a ``fractal dimension'' to solve the issue, and also explain how to evaluate the fractal dimensions. Finally, we show several examples of fractals in real engineering applications.

Contents

  1. Complex Objects
  2. What is Fractal?
  3. Examples of Fractals
  4. Cantor Set
  5. How to Characterize Fractals
  6. What is Dimension of a Set of Points?
  7. Covering and Dimension
  8. Dimension of the Cantor Set
  9. Noninteger Dimension
  10. Applications of Fractals

Course Materials

  1. Slides are here (pdf).

Quiz

  1. Calculate the dimensions of von Koch curve and Sierpi\'nski gasket.
  2. Give examples of fractals.

Links

  1. Tohru Ikeguchi Laboratory
$Lastupdate: Tue Dec 6 22:44:26 2005 $
Email: tohru[at]ics.saitama-u.ac.jp.
Copyright (C) 2004 Tohru Ikeguchi, Saitama University.