ISOMETRICA INTRODUCTION -- AUGUST 2001
Professor Groman introduced MAT 103 ("Symmetry and
Culture") in Fall 1992, as a course fulfilling both the Human
Diversity and the Mathematics requirements of the General Education
program at SUNY Oswego. In time MAT 103 became more
mathematical, with two-colored wallpaper patterns introduced in Fall
1995, colorings of tilings in more than two colors in Fall 1996,
compositions of isometries in the context of colored tilings in Spring
1997, and algebraic description of planar isometries
(distance-preserving transformations of the plane) in Fall 1998. In
view of the restructuring of SYNY Oswego's General Education program
in the late 90's, it became clear that we could achieve only one of
the course's original goals: MAT 103 was approved as a
Mathematics course ("Symmetries") in the Knowledge & Foundations area
of General Education in Spring 1998, and the writing of these notes
began immediately afterwards.
MAT 103's evolution and renaming does not mean that we
have completely abandoned the course's cultural component: chapter 2
(border patterns) and chapter 4 (wallpaper patterns) are spiced with
symmetrical designs conceived all over our small world and during
several historical periods; and I hope to soon create a web page
presenting such material in color and in context, part of
http://www.oswego.edu/~baloglou/103 (where various
developments, additional material, exercises and examples will also be
posted).
If you happen to be a professor, and a Mathematics professor in
particular, reading this, please keep in mind that our work is mainly
addressed to General Education students with a weak mathematical
background and is meant to be read by them. This does not mean that
there will be no challenges here for the Mathematics major or even
instructor, especially after the addition of exercises: indeed I hope
that the final version of this work will be appropriate as a textbook
or supplement for a college or high school geometry course.
If you are a student, and a MAT 103 student in
particular, please keep in mind that teaching symmetry at this level
is essentially a novel enterprise. Although MAT 103
itself has been a very successful course so far, the success of this
book-in-the-making depends a lot on your reaction and comments. I have
made every effort to produce something readable and even pleasant, but
it is inevitable that there are shortcomings to be discovered and
improvements to be made: if ever in doubt when going over these notes,
please let your instructor know! And if you are not that studious or
mathematically inclined, keep in mind that these notes may be read at
several levels: you can always start from the pictures, then move into
those comments that elaborate on the pictures, then read everything
else -- it works!
A few comments on the mathematical content of this work now. The main
goal is an alternative, completely geometrical and elementary,
classification of wallpaper patterns: this is done in chapter 9
[8], relying on techniques and ideas from chapter 8 [7],
where isometries and their pattern-creating interactions are studied
in some depth; some of these techniques and ideas arise naturally in
chapter 6, where investigation of two-colored possibilities for
wallpaper patterns is used as an incentive to informally delve into
their structure. Coloring is introduced in chapter 5, and in the
simpler context of two-colored border patterns, the investigation of
which in chapter 2 serves as a prelude for the introduction to
wallpaper patterns in chapter 4. Isometries themselves are studied in
chapters 1 and 3, both of which may in fact be covered after, or in
parallel with, chapter 2 or even chapter 4. Chapter 7
[9] is the only one that remains incomplete at present:
it aims at an investigation of wallpaper pattern structure in the
context of multi-colored tilings, and will probably be posted on the
MAT 103 web site during the 2001-2002 academic year.
The overall approach is very geometrical and rather informal. An
algebraic description of isometries is offered as a way of connecting
MAT 103 to the broader mathematical world, but is not
essential to the main goal. Groups and fundamental regions are
certainly mentioned, but are not at all central in the presentation.
Pattern classification flowcharts have been replaced by symmetry
plans, which, in our opinion, allow for a better view and
understanding of the patterns. Border patterns are viewed as building
blocks of wallpaper patterns, and the process of passing from one
pattern type to another is dynamic. Color is used to both appeal to
the artistically inclined reader and reveal the group structure hidden
behind the patterns to the mathematically minded.
I am indebted to Margaret Groman for her encouragement and many
conversations and insights over the last six years, as well as
collaboration during the first stages of the project. I have also
received valuable suggestions from Mark Elmer, who used an earlier
version of these notes during the Fall 2000 semester, and from my
former colleague and collaborator Phil Tracy. Special thanks are due
to Fred Barber, Jim Burling, Doug Deal, Phil Downum, Matthew Friday,
Jack Narayan, Sue Weber and Helen Zakin for their assistance at
crucial moments. And I would like to express my gratitude to a number
of former students who helped me to improve MAT 103 in a
variety of ways; in particular, Michael Nichols has provided valuable
assistance with the graphics.