Psychology Undergraduate Students

MAT 1932, Spring 2005, Dr. Larry S. Liebovitch

The Mathematics and Science of Fractals

15094 M.W.F 11:00 - 11:50 AM (GCS 109 Boca Raton)

This course uses different teaching and learning styles for students who have been previously intimidated by mathematics.

It can be used as one of the mathematics courses (besides Statistics) needed to fulfill the mathematics requirement for undergraduate Psychology majors.

Using CD-ROM electronic curricula materials developed by grants from the National Science Foundation (DUE-9752226 and DUE-9980715).

Larry S. Liebovitch, Ph.D.
Florida Atlantic University
Center for Complex Systems and Brain Sciences
777 Glades Road, Boca Raton, FL 33431
telephone: 561.297.2239, fax: 561.297.2223

If you want to speak with me please telephone, DO NOT SEND E-MAIL (I am overwhelmed with e-mail which I do not have time to read.)


Course Policies

  1. Attendance: Students are expected to attend all scheduled classes. If you miss a class you are responsible for ALL the material covered during that class, including lecture material and rules and regulations about the course (such as these rules). Some of the material covered in the lectures is NOT in the CD-ROM.
  2. Withdraw: It is the responsibility of the student to withdraw from this class, should that status be desired - the instructor cannot withdraw students from the course. The instructor will not give the grade of "I" in lieu of a grade of "D" or "F". The grade of "I" will be considered only in exception cases (such as serious illness) for students who are presently performing at a C or higher level in the course.
  3. Homework Problems: All homework problems must be turned in on the dates assigned. Late homework problems will be accepted up to 1 week late, but they will be penalized. No homework problems will be accepted over 1 week late. You will have the option of redoing each homework assignment to improve your grade on it, BUT if it was not turned in on the date assigned, it will still be penalized. Just to repeat, ANY homework first turned in later than 1 week will count as zero. Homework problems must be submitted on 8.5 x 11 inch paper. Handwritten homework that is not readable is not acceptable. They should have a cover sheet with your full name. You are requested to make a copy of all material turned in for credit to guard against loss or damage.
  4. Grade:
    Homework 60%
    Journal 15%
    Exams 25%


The textbook will be the CD-ROM which will be distributed to the class.

Materials on the CD-ROM

  1. Modules: Each Module is a self-contained unit on one topic that includes an Instructor's Guide and links to its supporting Lecture Notes, Classroom Discovery Experiments, Discovery Applets, Discovery Spreadsheets, Homework Assignments, PowerPoint Slides, reference material in Fractals and Chaos Simplified for the Life Sciences, and Assessment Tools.
  2. Lecture Notes: Detailed notes of the classroom lectures.
  3. Classroom Discovery Experiments: Instructions, aims, and meanings of hands-on activities to be performed by groups of students in class. These include instructional video clips of the experiments.
  4. Discovery Applets: Instructions and results for self-discovery activities implemented in Java applets.
  5. Discovery Spreadsheets: Instructions and results for self-discovery activities implemented in Excel spreadsheets.
  6. Homework Assignments: Homework problems with worked solutions for the instructor.
  7. PowerPoint Slides: Illustrative slides for the instructor to show in class.
  8. Fractals and Chaos Simplified for the Life Sciences Hypertext version of the previous print textbook.

Lecture Contents

# Title
1 Introduction: Course Objectives
2 Pre-Assessment: Students' Knowledge of Mathematics
3 What is Mathematics?
4 A New Way to Count: The LOGARITHMIC Scale
5 More From Paper Folding: Logarithms
6 Using Symbols to Represent Paper Folds: L's and R's
7 Paper Folding Iterations: Real Folds in Real Paper and the Symbols L and R
8 Symmetry and Invariants: A More General Form of Self-Similarity
9 Invariants: Examples from Science
10 The Koch Curve: Introduction to the Snowflake
11 The Koch Curve: Area and Limits
12 The Koch Curve: Zeno's Animals and The Area of the Koch Curve
13 The Koch Curve: Length and Dimension
14 The Middle Third Cantor Set: Making a Fractal by Removing Pieces
15 The Sierpinski Triangle: Many Different Ways
16 Paradigms: How The Eye Works
17 Fractals in the Eye: Statistical Self-Similarity
18 What Graphs Tell Us: Discovering the Straight Lines
19 What Graphs Tell Us: What the Straight Lines Mean
20 What Professors Really Do
21 Dimension: Exponents and Graphs
22 Dimension: Slopes and Logarithms
23 Many Different Dimensions: Fractal, Topological, Embedding
24 Figuring Out the Fractal Dimension: Box Counting
25 Figuring Out the Fractal Dimension: The Scaling Relationship
26 More Scaling Relationships: How Nerves Work
27 Even More Scaling Relationships: The Mass and Density of Fractals
28 Newspaper Crumpling: Dimension, Density, and Scaling Relationships
29 Finding Fractals: Outside the Classroom
30 Making Fractals: Inside the Classroom
31 Statistics: When the Mean is Meaningless
32 Statistics: Bell Curve and Fractal PDFs
33 Fractal Statistics: How the Ear Works and More
34 Fractal Statistics: How the Heart Works and More
35 Chaos: Simple Things That Look Complicated
36 Chaos: An Exquisite Sensitivity
37 Concept Maps: The Big Picture
38 Post-Assessment: Students' Knowledge of Mathematics