The Cantor Set and Mathematica
CantorSet.ma package definitions
Cantor0.ma an historical preface
Cantor1.ma several approaches to the Cantor set and generalizations
Cantor2.ma the Cantor function and generalizations
Cantor3.ma Lebesgue measure, dimension
a:AUTHORS
Steven R. Dunbar
Department of Mathematics and Statistics
University of Nebraska-Lincoln
Lincoln, NE 68588-0323
(402)-472-7236
srd@mathcml.unl.edu (non-NeXT mail)
sdunbar@mathlab01.unl.edu (NeXT mail)
David Fowler
Department of Curriculum and Instruction
University of Nebraska-Lincoln
Lincoln, NE 68588-
(402)-472-3347
dfowler@cantor.unl.edu (NeXT mail)
b:CATEGORY for NeXT PUBLIC DOMAIN CD-ROM FOR EDUCATION:
Mathematics (NeXT Mathematica Notebooks)
c:WHAT THE APPLICATION DOES:
This series of Mathematica Notebooks provides a visual and analytic
introduction to the Cantor set, the Cantor function and some
generalizations. The Cantor set is a standard example in advanced
undergraduate and graduate level courses in mathematical analysis
and topology. The Cantor set is particularly well-suited for
visualizing and experimenting with such mathematical topics as
limits, alternate number bases and series, iterated-function-systems,
symmetry relations, derivatives, Riemann-Stieltjes integrals,
Lebesgue measure, fractal dimension, Hausdorff dimension, similarity
dimension, complex numbers, connectedness, and topological groups.
The method of the Notebooks is a modified version of the famous
Moore method of teaching advanced mathematics. That is, we present
the basic definitions and theorems and then present some examples.
The examples are usually visual and invite experimentation and
alteration. The intent is to provide new ways of thinking about
concepts which are usually presented abstractly. Then we present
some exercises, divided into three categories: Basic Exercises,
Try for Yourself, and Have Fun With Mathematica. The Basic Exercises
are just that, straight-forward applications of the definitions
and concepts. The Try for Yourself exercises are more difficult
and challenging. The Have Fun with Mathematica exercises are
intended to extend the Notebooks by providing further visualization
of the concepts.
d:HOW THE APPLICATION IS USED AT UN-L
At UN-L the series of Notebooks is used as a directed reading course
for advanced undergraduates and beginning graduate students in
mathematics. The students have either completed or are taking
concurrently a standard course in analysis or topology, based on
for example W. Rudin's Principles of Mathematical Analysis. The
Notebooks are intended as a supplement to the course, providing
visualization and alternate viewpoints of concepts introduced in
the standard course. A grade for the reading course is mastery
based, that is, a student contracts for a grade based on the number
and difficulty of Basic Exercises, Try for Yourself, and Have Fun
with Mathematica problems to be completed. The problems are to be
written up in the rigorous style of a typical course in analysis
and graded. Incorrect or incomplete proofs are returned and
rewritten until correct. The intent of the Notebooks is to stimulate
thinking about the concepts which will lead to being able to write
correct proofs.
The Notebooks can also be used for self-study.
e:SOFTWARE VERSIONS:
The Notebooks were developed under NeXT Release 2.1, Mathematica Kernel Version 2.0 and NeXT Front End Version 2.0
f:INSTALLATION INSTRUCTIONS:
The Notebooks all use the Mathematica package CantorSet.ma for
basic definitions and functions. This package must be loaded into
the notebooks with the Mathematica Get[ ] command. Depending on
the local defaults for package locations, it may be necessary to
first set the directory containing CantorSet.ma with the Mathematica
command SetDirectory.ma
g:OTHER COMMENTS
This series of Notebooks is still under construction and modification.
Some inconsistencies in notation, function definitions, and some
evidence of the construction may still be apparent. Indeed, some
of the Mathematica code may have bugs in it, or even may not work!
Please accept these Notebooks as a preliminary and incomplete at
this stage. Beginning about August 24, 1992 this series of Notebooks
will undergo initial use by a group of mathematics graduate students.
We expect to learn from their experience and considerably improve
the format and content of the Notebooks. In addition, at least
two more Notebooks are under construction at this time:
Cantor4.ma,Complex Cantor Sets; and Cantor5.ma: The Cantor Set as
Topological Group.
Report any bugs, inconsistencies, errors, comments, suggestions,
and ideas to the authors above.
A detailed explanation of our motivation and philosophy for this
series of Notebooks is explained in "Cinematic Thinking and
Mathematica Notebooks", Steven R. Dunbar, David Fowler, Mathematica
in Education, Vol. 1, Number 3, page 1.