FRACTAL GRAPHICS

Ma 599

1. Course Description

Fractal geometry, basic definitions. Metric spaces, classification of subsets, the space of fractals. Transformations on metric spaces, contraction mappings, construction of fractals. Recursion and fractals. Sierpinski triangle, Hilbert curve, dragon curves, trees. Chaotic dynamics on fractals; fractal dimension. Programming principles using C or C++.

2. Goals of the Course

The purpose of this course is to introducing the C (or C++) programming language , which is rapidly becoming the preeminent applications language of the 1990s. The student is expected to become familiar with this language. As application of C programming, selected topics from the theory of factals graphics is also discussed.

3. Instructional Procedures

The following resources will be used:

Textbook

Computer facilities of the Rechzeh Math Resource Center

Procedures and methods will include:

a. Lectures covering the theory and areas of application

b. Guest Lecturers discussing their specialties

c. Assignment of programming projects

d. Independent projects

4. Course Content

A. Fundamentals of C programming ( or Object oriented programming principles using C++ up to the instructor )

a. Introduction to programming

The main( ) function

The printf( ) function

Programming Style

Top-Down program development

b. Data Types, Declarations, and Displays

Data Types

Arithmetic operators

Displaying numerical results

Variables and declarations

Integer qualifiers

c. Assignments, Addresses, and Interactive Input

Assignments operators

Storing and using addresses

Declaring Pointers

The scanf( ) function

d. Flow of Control

Selection

The if-else statement

Nested if statements

The switch statement

Repetition

The while statement

scanf( ) within a while loop

The for statement

The do statement

e. Function

Function and argument declarations

Standard library functions

Variable scope

Passing addresses

f. Arrays, Addresses, and Pointers

Single-dimensional arrays

Array initialization

two-dimensional arrays

g. Character Strings

String fundamentals

String definitions and pointer arrays

h. Structures

Single Structures

Arrays of structures

B. Fractal Graphics

a. Fractal Geometry

Basic Definitions

b. Metric Space

Classification of Subsets

The Space of Fractals

c. Transformations on metric spaces

Contraction mappings

Construction of fractals

d. Recursions and fractals,

Sierpinski Triangle

Hilbert Curve

dragon Curve

tree

e. Chaotic dynamics on fractals

fractal dimension

5. Evaluation Measures

Class participation 5%

Successful run of specified number of programs 25%

Presentation of a programming project to class 20%

Midterm examination 25%

Final examination 25%

6. Bibliography

A. Required Text

a. A First Book of ANSI C: Fundamentals of C Programming, second edition, by Gary Bronson and Stephen Menconi, published by WEST Publishing Company, 1993

b. The Art And Science of C: An Introduction to Computer Science, by Eric S. Roberts, published by Addison-Wesley Publishing Company,1993

c. Fractal Functions, Fractal Surfaces, and Wavelets, by Peter R. Massopust, published by Academic Press, 1994

B. Support Bibliography

a. The C Programming Language, second edition, by Kernighan and Ritchie, published by Prentice Hall, INC, 1988

b. Learning C, by Neill Graham, published by McGRAW-HILL, INC, 1992

c. C++ Inside & Out, by Bruce Eckel, published by McGraw-Hill, 1993

d. Fractal Vision: Put Fractals to Work for You, by Dick Oliver, published by SAMS Publishing Company,1992

e. Chaos and Fratals: The Mathematics Behind the Computer Graphics, by Robert L. Devaney and Linda Keen, published by AMS,1994

f. The Beauty of Fractals: images of complex dynamical systems, by Peitgen and Richter, published by Springer-Verlag Berlin Heidelberg , 1986