Back in Spring 1995, one of my SUNY Oswego students submitted the following one-sentence teacher evaluation: "The course was relatively easy until chapter 11 when I felt that the instructor was as lost as the students"! Chapter 11 -- typically associated with bankruptcy in the so-called 'real world' -- was in that case the symmetry chapter in Tannenbaum & Arnold's

Perhaps that anonymous student's not entirely unjustified comment was the best explanation for my decision to volunteer to teach

But where had

My initial intent was to write a student-oriented book, a text that our

Donald Crowe's 'repeated patterns', better known nowadays as frieze/border patterns and wallpaper patterns, may certainly be viewed as one of the very first mathematical (even if accidentally so) creations of humankind: long before they were recognized as the poor relatives of the three-dimensional structures so dear to modern scientists, these

Good question: this is the only chapter with some algebra (read analytic geometry) in a heavily geometrical book! The simple answer is that the General Education Committee of SUNY Oswego would not approve [Spring 1998] a mathematical course without some mathematical formulas in it... And it took me a while to come up with a constructive/creative way of incorporating some formulas into

So, if you are not algebraically inclined, don't hesitate to skip chapter 1 at first reading: the four planar isometries are indirectly reintroduced in the much more reader-friendly chapter 2, save for the general rotation, as well in chapters 3 and 4. (At the other end, some readers may be interested

I have no illusions: most of you are going to merely browse through my book, even if you happen to be a student whose GPA depends on it... Well, save for the potentially attractive figures, this book is not browser-friendly: its conversational style may be tiring to some, and the absence of 'summary boxes' depressing to others; and let's not forget a favorite student's remark to the effect that "it is odd that in a book titled

Both because

Despite the inclusion of patterns from

Yes and no: Escher's

More generally,

Granada's famed Moorish palace complex that inspired Escher is barely mentioned in

No. Consistent with the absence of bibliography, any discussion of the subject's historical development is absent from

This is an eminently legitimate concern: is it fair for a course that for most of its takers is their 'final' mathematical experience to be devoted to a single subject almost devoid of 'real world' applications? My response is that students may in the end understand more about what Mathematics is about by focusing on

Certainly not! In fact

Border patterns are planar designs invariant under translation in precisely one direction; wallpaper patterns are planar designs invariant under translation in two, therefore infinitely many, directions. This difference makes border patterns substantially easier to understand and classify. It is therefore natural to use border patterns as a stepping stone to wallpaper patterns. Further, border patterns may be seen as the building blocks of wallpaper patterns, and this is indeed an opportunity that

I would discourage this option, except perhaps early in high school, with the intention of covering wallpaper patterns the year after. I suspect nonetheless that several readers of

The Euclidean Geometry employed in

Yes, especially in case they happen to be visual learners. It is the teacher's responsibility to decide whether his/her students would benefit from a course based either partly or wholly on

The coverage of two-colored patterns in chapters 5 (border patterns) and 6 (wallpaper patterns) is a direct consequence of

The answer lies hidden in the previous paragraph! Assuming that it would be difficult for (my) students to understand

Finding the isometries of any given pattern is a great exercise for the student, and essential for the pattern's correct classification. But it is not possible to appreciate a pattern's structure and 'personality' without understanding the way its isometries interact with each other: any two pattern isometries combined -- that is, applied sequentially -- produce a third isometry that also leaves the pattern invariant; it is for this reason that mathematicians talk about border/frieze and wallpaper

As already indicated, chapter 7 offers a thorough coverage of isometry composition in a totally geometrical context -- perhaps the most thorough (as well as accessible) coverage of compositions of planar isometries to be found in any book. It is therefore possible to use chapter 7 for a largely self-contained (despite the references to pattern structure) introduction to planar isometry composition. At the other end, section 7.0

Finding the isometries of a border pattern is quite easy for most students. Wallpaper patterns are a different story, complicated by more than one possible direction for glide reflection, rotations other than half turn, etc. As indicated in passing in chapter 4, the determination of all the isometries mapping a 'symmetrical' set to a copy of it -- a 'recovery' process discussed in detail in chapter 3 -- can make the isometries of a complex wallpaper pattern much more visible and 'natural': quite often the isometries mapping a 'unit' of the pattern to a copy of it are extendable to the entire pattern! This is stressed in

A former student told me once that "this course put some order in his mind"; and several students report in their evaluations that

How did you know? You must have read the entire book! Yes, there is some discrimination ... in the sense that glide reflection is viewed as an isometry 'weaker' than reflection. This view is of course dictated by the fact that glide reflection, which may certainly be viewed as

One way or another, the teacher must stress the curious interplay between reflection and glide reflection outlined above, and also insist that the students use dotted (read dashed) lines for glide reflection axes and vectors and solid lines for reflection axes and translation vectors, as in the symmetry plans. (There are places in

Between the 'perfectly symmetrical' two-colored patterns of

A natural question arises: should such inconsistent colorings be avoided in teaching? Although I do cover this topic extensively in

The Conjugacy Principle states that the

This final chapter is devoted to my purely geometrical argument that there exist precisely 17 types of wallpaper patterns. It would clearly be beyond the scope of most General Education courses, and probably too sophisticated for the great majority of non-science majors as well. But it is largely self-contained -- totally self-contained in case section 4.0 and chapter 7 are assumed -- and requires mathematical maturity rather than knowledge. Interested instructors (or other readers) should probably teach/read it in parallel with

Tough question! The answer depends even on the way one defines a wallpaper pattern, and whether one believes that Group Theory has to be part of that definition in particular. Among thousands of visitors of

To be honest, a solid structural understanding of the seventeen types of wallpaper patterns was, and still is, more important to me than a rigorous/quick proof that there exist indeed precisely seventeen such types. Nonetheless, I suspect that what

[Note: the classification of border patterns in chapter 2 is even more 'informal' than that of wallpaper patterns, consistently with that chapter's introductory nature; the interested reader should be able to easily derive a more rigorous classification of border patterns based on symmetry plans.]

The main new idea is the reduction of complex (rotation + (glide) reflection) types to the three rotationless types with (glide) reflection (

Needless to say, the Conjugacy Principle shines throughout the classification!

Some readers may find a few interesting ideas lurking in my novel (non-group-theoretic) classification of two-colored patterns (which

Before trying to explore two-colored 'sparse crystals' (blocks not touching each other and therefore not obscuring colors) I would rather try to investigate compositions of three-dimensional isometries in a geometrical context (extending chapter 7) and classify the 230 crystallographic groups geometrically (extending chapter 8). I believe that both projects are feasible, and hope to pursue them now that

This question has been answered in Tom Wieting's

It would be nice if someone with more energy and knowledge sits down and writes a book on wallpaper patterns that could be used for a mathematics capstone course! Here is how this could be achieved: start with an elementary geometrical classification of wallpaper patterns like mine and then continue with the standard group-theoretic classification (available for example in Wieting's book)

[Conway's

No way! The figure on the cover is a tribute to the great crystallographer (and not only) Arthur Loeb and his

[Which

Responding to my May 2000 talk at a Madison conference honoring Donald Crowe, H. S. M. Coxeter -- in his 90's at the time, seated in a wheelchair barely ten feet from the speaker(s) -- remarked with a wry smile that "all the two-colored types had been derived in the 1930's by a textile manufacturer from Manchester [H. J. Woods] without using any Mathematics". The eminent geometer's remark captures much of the spirit in which

I like to say, in hindsight, that border and wallpaper patterns are "of

If you read between the lines above you already know that the teaching of

So a labor of love it was, and this is why I have largely preserved

My joy at having been able to preserve

My obvious desire to generate disciples for

For every book and completed project that sees the light of day there are several visions buried under perennial darkness: I happen to have the right personality for incompleteness, therefore I am almost ecstatic as these final lines are being written; repeatedly seduced as I was by those 'repeating patterns', the discipline often failed to match the excitement, the time and will appeared not to be there at times, the questions tended to dwarf the answers... While several friends and colleagues provided constant support, I believe that the project's completion and, I hope, success is primarily due to my

As made clear in the beginning of this Introduction, there would simply be no

Beyond Oswego, I am grateful to a number of mathematicians and others who provided links to

Back to Oswego, I am grateful to Alok Kumar, Ampalavanar Nanthakumar, and Bill Noun for their support and good advice; same applies to several other colleagues from Mathematics, Computer Science, Art and other departments (and also administration) at SUNY Oswego. Sue Fettes deserves special mention for her assistance with

In a somber tone now ... even though

-- Bob Deming, whose unpublished but highly effective notes on Linear Programming provided an early model for me on classroom-generated books

-- Jim Burling, who also taught

-- Gaunce Lewis (of Syracuse University), whose tragically untimely death was a haunting reminder of the fragility of intellectual pursuits

-- Don Michaels, who in his capacity as tireless news & web administrator contributed handsomely to the success of

Finally,

George Baloglou

Oswego, April 27, 2007