INTRODUCTION




Symmetry is a vast subject, significant in art and nature.

Mathematics lies at its root, and it would be hard to find a

better one on which to demonstrate the working of the

mathematical intellect. I hope I have not completely failed in

giving you an indication of its many ramifications, and in

leading you up the ladder from intuitive concepts to abstract

ideas.
-- Herman Weyl, Symmetry (Princeton, 1952)




Why and how Isometrica, and who would read it?


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 Excursions in Modern Mathematics: I had casually picked it as one of two 'optional' chapters in my section of MAT 102 (SUNY Oswego's main General Education course for non-science majors, consisting of various mathematical topics).


Perhaps that anonymous student's not entirely unjustified comment was the best explanation for my decision to volunteer to teach MAT 103, a General Education course devoted entirely to Symmetry, in Fall 1995: better yet, curiosity killed the cat -- once I started teaching MAT 103 I never took a break from it, gradually abandoning my passion for rigor and computation in favor of intuition and visuality.


But where had MAT 103 come from? Following a January 1991 MAA minicourse (Symmetry Analysis of Repeated Patterns) by Donald Crowe at the San Francisco Joint Mathematics Meetings, my colleague Margaret Groman developed (Fall 1992) a new course (Symmetry and Culture) in response to our General Education Board's call for courses fulfilling the newly introduced Human Diversity requirement: after all, was Professor Groman not an algebraist keenly interested in applications of Abstract Algebra (to symmetry for example), and had Professor Crowe not co-authored a book with anthropologist Dorothy Washburn titled Symmetries of Culture (Univ. of Washington Press, 1988)?


MAT 103 ceased to fulfill the Human Diversity requirement and was renamed Symmetries in Spring 1998, but it remained quite popular among non-science majors as a course fulfilling their Mathematics requirement; it also attracts a few Mathematics majors now and then. At about the same time I set out (initially in collaboration with Margaret Groman) to write a book -- not the least because Washburn & Crowe had temporarily gone out of print -- that was essentially completed in three stages: January 1999 (chapters 1-5), January 2000 (chapter 6), and August 2001 (chapters 7 & 8). Various projects and circumstances delayed 'official' completion until November 20, 2006 (the day a new computer forcefully arrived), with the first six chapters posted on my MAT 103 web site as of Fall 2003. In spite of my endless proofreading and numerous small changes, what you see here is very close in both spirit and content to the August 2001 version. [For the record, I have only added 'review' section 6.18 and subsections 1.5.3 & 4.17.4, and also added or substantially altered figures 4.73, 5.36, 6.121, 6.131, 7.44, and 8.3.]


My initial intent was to write a student-oriented book, a text that our MAT 103 students -- and, why not, students and also 'general' readers elsewhere -- would enjoy and use: this is why it has been written in such unconventional style, and in the second person in particular; in a different direction, this is why it relies on minimal Euclidean Geometry rather than Abstract Algebra. Looking now at the finished product, I can clearly see a partial failure: the absence of exercises and other frills (available to considerable extent through the MAT 103 web site), together with an abundance of detail (also spilling into the MAT 103 web site), may have conspired toward turning a perceived student's book into a teacher's book. Beyond students and teachers, and despite its humble origins, there may also be some specialists interested in Isometrica: I will attempt to address these three plausible audiences in considerable detail below; you may wish to skip these three sections at first reading and proceed to the end of the Introduction.





Comments for students and general readers



What is this book about, and how accessible is it?


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 planar crystallographic groups were being discovered again and again by repetition/symmetry-seeking native artists in every corner of the world. This book's goal is therefore the gradual unveiling of the structural and the mathematical that hides behind the visual and the artistic: so chapters 2 - 4, and even chapters 5 and 6, are more eye-pleasing than mind-boggling, while chapters 7 and 8 certainly require more of the reader's attention. It is fair to say that a determined reader can read the entire book relying only on some high school mathematics.



Why is Chapter 1 here to begin with?


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 MAT 103, simply by providing an analytical description -- and, quite unintentionally, classification -- of the four planar isometries (that is, the four possible types of distance-preserving transformations of the plane).


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 only in chapter 1, which is, I hope, a very accessible and engaging introduction to planar isometries, relying on neither matrices nor complex numbers.)




Any other reading tips, dear professor?


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 Isometrica there is no definition of isometry"! But those figures are there, slightly over one per page on the average, and most of them are interesting at worst and seductive at best (me thinks): so start by looking at appealing figures, then read comments related to them, then read stuff related to those comments, and ... before you know it you will have read everything! After all, this book talks to you -- are you willing to listen? (My thanks to another former student for this 'talking-to-me book' comment!)




Why is there no bibliography?


Both because Isometrica is totally self-contained and because suggestions for further reading are always made in the text (including this introduction) and in context. Moreover, Washburn & Crowe provides a rather comprehensive bibliography to which I would have little to add... But if you ask me for one book that you could or should read before mine, I would not hesitate to recommend Peter Stevens' Handbook of Regular Patterns (MIT Press, 1981): that is any math-phobic's dream book and, although I follow it in neither its 'kaleidoscopic' approach nor its 'multicultural' focus, several figures from Stevens have been included in Isometrica (with publisher's permission) as a tribute.




What is there for the non-mathematically inclined?


Despite the inclusion of patterns from Stevens, my book -- as well as MAT 103 in both its present and past forms -- fails to address in depth the cultural aspects of those patterns and the 'inner motives' of the native artists who created them: nothing like Paulus Gerdes' Geometry From Africa (Mathematical Association of America, 1999) or Washburn and Crowe's second book (with updated bibliography), Symmetry Comes of Age (Univ. of Washington Press, 2004). Still, I must mention a telling incident: a former student made once a deal with a quilt maker friend of hers involving the exchange of her copy of Isometrica for a quilt right after the MAT 103 final exam! In other words, mathematically oriented as it happens to be, Isometrica and its 'abstract' designs can still be a source of inspiration for many non-mathematically inclined readers.




Is Isometrica related to the work of Escher?


Yes and no: Escher's symmetrical drawings, for which he is well known, are certainly special cases of wallpaper patterns, which are Isometrica's main focus; but Escher's main achievement, the tiling of the plane by repeated 'real world' figures, is not discussed at all. Still, it is safe to say that those intrigued by Escher's creations are likely to be interested in Isometrica; conversely, Isometrica might be a solid introduction toward a serious reading of Doris Schattschneider's classic M. C. Escher: Visions of Symmetry (Abrams, 2004).


More generally, Isometrica is not a good source for tilings of any kind; a few obvious planar tilings are used as standard examples, but there is no mention of hyperbolic or spherical tilings, and likewise no discussion of Penrose and other aperiodic (non-repeating) tilings. Still, the curious reader may find Isometrica to be a good starting point for such topics. (The same applies to other 'popular', loosely related topics like fractals.)




How about Alhambra?


Granada's famed Moorish palace complex that inspired Escher is barely mentioned in Isometrica. For a detailed discussion of Alhambra's wallpaper aspects I would strongly recommend John Jaworski's A Mathematician's Guide to the Alhambra, currently available through the Jaworski Travel Diaries at http://www.grout.demon.co.uk/Travel/travel.htm.




Is Isometrica history-oriented at all?


No. Consistent with the absence of bibliography, any discussion of the subject's historical development is absent from Isometrica. For such information, and a broader view as well, the interested reader is referred to both the internet and such classics as Grunbaum & Shephard's Tilings and Patterns (Freeman, 1987) and Coxeter's Introduction to Geometry (Wiley, 1980).





Comments for teachers



Symmetry as a General Education course?


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 one subject and its development than by being briefly exposed to a variety of subjects. (Besides, even if I wrote Isometrica for a General Education course, it may certainly be used for other classes and audiences!)




Is Symmetry just about border and wallpaper patterns?


Certainly not! In fact MAT 103 does cover the isometries of the cube and the soccerball (and their compositions) toward the end, and students tend to enjoy these subjects at least as much as the rest of the course (especially when it comes to isometry composition, which is now greatly facilitated by finiteness). It is therefore fair to say that Isometrica may also be used for only part of a course devoted to symmetry or geometry; for example, one may spend just three to four weeks covering only chapters 2, 3, and 4, or merely two weeks on chapters 2 (border patterns) and 4 (wallpaper patterns).




What is the interplay between border patterns and wallpaper patterns?


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 Isometrica does not pass by; the subject is treated in depth in Shredded Wallpaper -- Bonita Bryson's 2005 honors thesis currently available at http://www.oswego.edu/~baloglou/103/bryson-thesis.pdf, which may also be used as a quick introduction to border and wallpaper patterns.




How about covering border patterns only?


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 Isometrica may limit their serious reading to chapter 2, which is probably the book's most successful and accessible chapter anyway!




How do border and wallpaper patterns relate to Euclidean Geometry?


The Euclidean Geometry employed in Isometrica is so minimal and elementary that a daring question emerges: would it actually be possible to develop the students' geometrical intuition through some informal exposure to border and wallpaper patterns before introducing them to Euclidean Geometry? Could the intense exposure to shapes and transformations enforced by the study of patterns facilitate the absorption of geometrical ideas and even arguments encountered in high school geometry? This might be a good research topic for Mathematics educators.




Could this be too easy for some students?


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 Isometrica, and how much time should be spent on it (if any). I have seen students who struggled for a D in MAT 103, as well as students who stated that it was the easiest course (in any subject) they have ever taken! Anyway, I do suspect that Isometrica could keep even the very best Mathematics/Science majors intrigued for a weekend (or at least a long Saturday afternoon), so please do not automatically give up on it simply because you happen to teach the best and brightest... [And do not forget that student's comment at the beginning of this Introduction -- it can be a treacherous subject!]




What is the role of color?


The coverage of two-colored patterns in chapters 5 (border patterns) and 6 (wallpaper patterns) is a direct consequence of Isometrica's debt to Washburn & Crowe already alluded to. But, while for Washburn and Crowe the study of the artistically/anthropologically important two-colored patterns was an end, for me it ended up being largely a mean: indeed a careful look at chapters 5 and 6 shows how the classification of two-colored patterns is largely used as an excuse to delve into the structure of (one-colored) border and wallpaper patterns, and the compositions of their isometries in particular.




Is Isometrica written top-down or bottom-up?


The answer lies hidden in the previous paragraph! Assuming that it would be difficult for (my) students to understand first 'abstract' (even if geometrically presented) composition of isometries (as treated in chapter 7) and then pattern structure based on that (top-down approach), I opted for an indirect, if not surreptitious, introduction to isometry composition departing from various classification issues in chapters 5 and 6 (bottom-up approach). My assumption is a questionable one, so a student-friendly top-down approach may indeed be presented in a future book! (In fact such an approach is currently being tested in Patterns and Transformations (MAT 203), an experimental SUNY Oswego course for honors students.)




What is the significance of isometry composition?


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 groups, the total absence of Group Theory from Isometrica notwithstanding.


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 alone shows how isometry composition can be studied 'empirically' in the context of multi-colored symmetrical tilings: that is in fact the way isometry composition is studied [since Spring 1997] in MAT 103, definitely making for the hardest part of the course -- likened once to "pulling teeth" by one of my best students! (To make 'isometry hunting' more fun, the instructor may even choose to initially hide from the students the helpful fact that, when it comes to isometry composition, rotations/translations and (glide) reflections act like positive and negative numbers in multiplication, respectively.)




What is the significance of isometry recovery?


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 MAT 103: students are initially encouraged to reconstruct the isometries, with the hope (or rather certainty) that they will gradually become more capable of seeing them; they are in fact told that "what you cannot see you may build", a guiding principle throughout the course! (A student's mother was thrilled enough by this principle to tell her daughter "now I do know that you are learning something in college" -- a very sweet comment indeed.) So, even though chapter 4 is almost entirely independent of chapter 3, I am strongly in favor of covering both.




How do students benefit from classifying patterns?


A former student told me once that "this course put some order in his mind"; and several students report in their evaluations that MAT 103 made them better thinkers. For such a visual, almost playful, course these comments may appear startling at first. But the classification process, especially of two-colored patterns, is very much a thinking process; for example, and very consistently with the guiding principle cited above, the classifier will often either detect or rule out an isometry based on logical rather than visual evidence.




What is the role of symmetry plans?


Washburn & Crowe facilitates the classification of individual two-colored patterns by way of step-by-step, question-and-answer flow charts; Isometrica reaches this goal through a complete graphic description of each two-colored type's isometries and their effect on color (preserving or reversing). This approach has the advantage of constantly and constructively exposing the students to the full isometry structure of the 7 border patterns (through 24 two-colored types and symmetry plans at the end of chapter 5) and the 17 wallpaper patterns (through 63 two-colored types and symmetry plans at the end of chapter 6). Quite clearly, similar symmetry plans could be used for the simpler tasks of classifying one-colored border patterns (chapter 2) and one-colored wallpaper patterns (chapter 4); but I prefer a purely non-graphical description of one-colored patterns in order to test/develop the students' reading skills a bit!




Does Isometrica discriminate against glide reflection?


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 deferred reflection, is harder to detect in a wallpaper (or border) pattern. Further, every wallpaper pattern reflection generates translation(s) parallel to it and, therefore, "hidden glide reflection(s)": reflection 'contains' glide reflection, but not vice versa (and despite the fact that every reflection may be viewed as a glide reflection the gliding vector of which has length zero). But a careful reading of section 8.1 shows that reflection and glide reflection are simply two equivalent 'possibilities'; and the 'shifting' processes introduced in sections 4.2 - 4.4 clearly indicate that reflection is the exception that verifies the rule (glide reflection).


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 Isometrica where some readers may disagree with my choice of solid or dotted lines; when a pattern reflection is combined with a parallel translation in order to create a 'hidden' glide reflection, for example, I use solid rather than dotted lines.)




What is the role of inconsistency with color?


Between the 'perfectly symmetrical' two-colored patterns of Washburn & Crowe and the randomly colored designs of the 'real world' lies a third, somewhat esoteric, class of two-colored patterns where, informally speaking, there is some order within their coloring disorders; more formally, some of their isometries happen to be inconsistent with color -- reversing colors in some instances and preserving colors in other instances -- but, otherwise, the coloring appears to be perfectly symmetrical, and with the two colors in perfect balance with each other in particular. Such inconsistently yet symmetrically colored patterns are largely absent from Washburn & Crowe, and for a good reason: it seems that native artists, driven perhaps by instinct or intuition, largely shunned them, producing either 'perfect' or 'random' colorings!


A natural question arises: should such inconsistent colorings be avoided in teaching? Although I do cover this topic extensively in MAT 103 and Isometrica, my answer is a reluctant "perhaps" -- especially to those teachers who may think that two-colored patterns would already strain their students considerably. On the other hand, anyone delving into this seemingly esoteric topic will be rewarded with many fascinating (both visually and conceptually) creations; the color inconsistencies involved will often transform a 'symmetrically rich' structure into a 'lower' type, illustrating the fateful principle that "coloring may only reduce symmetry". Anyway, those wishing to avoid the topic should be able to do so relatively easily, despite the presence of several color-inconsistent examples; and those venturing into it may be seduced enough to substantially enlarge Isometrica's collection of inconsistent colorings!




What is the role of the Conjugacy Principle?


The Conjugacy Principle states that the image of an isometry by any other isometry is an isometry of the same kind (with rotation angles or glide reflection vectors preserved modulo orientation); conversely, any two 'identical-looking' isometries are actually images of each other under a third isometry. In the context of wallpaper patterns, the Conjugacy Principle becomes an indispensable tool for their structural understanding and classification. Although formally introduced in section 6.4 (with the excuse of understanding the color effect of coexisting reflections and glide reflections) and applied throughout chapter 6, the Conjugacy Principle is thoroughly discussed and rigorously explained only in section 8.0 (paving the way for the classification of wallpaper patterns); it also appears in section 4.0 -- to the extent needed for the establishment of the Crystallographic Restriction (on rotation angles allowed for wallpaper patterns), which could admittedly wait until section 8.0.




What do we make of chapter 8?


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 Crystallography Now, a web page (http://www.oswego.edu/~baloglou/103/seventeen.html) devoted to a more informal presentation of my classification of wallpaper patterns.





Comments for experts



Does chapter 8 really offer a classification of wallpaper patterns?


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 Crystallography Now, only one was kind enough to tell me that my classification is "more intuitive than others, but not at all rigorous", his main point being that "two wallpaper patterns are of the same type if and only if their isometry groups are isomorphic". Fair enough, but is it reasonable to be able to characterize such simple structures, known to humankind for thousands of years, only in terms of advanced mathematical concepts? How would Euclid describe -- and perhaps classify -- the seventeen types in the Elements, had he included them there? (Just a thought!)


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 Isometrica offers could easily be turned into a formal proof by replacing isomorphism of isometry groups by a properly defined 'isomorphism' of symmetry plans. Such an isomorphism would certainly distinguish between solid lines (reflection) and dotted lines (glide reflection) or between hexagonal dots (sixfold centers) and triangular dots (threefold centers), etc. Under such an approach, any two symmetry plans consisting only of round dots (half turn centers) should represent the same type of wallpaper pattern (p2); even more frighteningly, any two wallpaper patterns having nothing but translations would be of the same type (p1) on account of their 'blank' symmetry plans, and so on. More interestingly, the reader is invited to compare the way this symmetry plan approach distinguishes between p4g and p4m (section 8.3) or between p31m and p3m1 (section 8.4) to the way the traditional group-theoretic approach reaches the same goals: rather than looking at their generator equations, Isometrica focuses on the two possible ways in which their (glide) reflections may 'pass through' their lattices of rotation centers.


[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.]




Any new ideas in the proposed classification?


The main new idea is the reduction of complex (rotation + (glide) reflection) types to the three rotationless types with (glide) reflection (pg, pm, cm) via the characterization of the latter in terms of their translations. So section 8.1, where the said characterization is achieved, may seem endless, but the derivation of the remaining types in the subsequent sections is swift and rather elegant (I hope).


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




Any other surprises prior to chapter 8?


Some readers may find a few interesting ideas lurking in my novel (non-group-theoretic) classification of two-colored patterns (which assumes the classification of one-colored patterns), and in the exploitation of symmetry plans in sections 6.9 and 6.11 - 6.12 in particular. Others may be delighted at the various ways of passing from one border or wallpaper type to another: although such 'transformations' are included in Isometrica mostly for educational purposes, they are bold commentaries on the ever-elusive structure of patterns, too!




Can Isometrica's ideas be extended to the three dimensions?


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 Isometrica has been completed; anyone interested in competing with me may like to start with Isometries Come In Circles (my 'mostly two-dimensional' novel derivation of three-dimensional isometries, currently available at http://www.oswego.edu/~baloglou/103/circle-isometries.pdf).




What happens when more than two colors are involved?


This question has been answered in Tom Wieting's The Mathematical Theory of Chromatic Plane Ornaments (Marcel Dekker, 1982). I was ambitious enough to investigate multicolored types in the context of maplike colorings of planar tilings, and also without the group-theoretic tools employed by Wieting; more specifically, I was interested in the interplay between tiling structure and coloring possibilities. That was not necessarily a hopeless project, and I did/do have some interesting ideas, but I had to finally admit that my attempts -- during the summers of 2000 and 2005 -- were not that realistic: several hundred multicolored tilings later a projected ninth chapter (initially numbered as seventh) had to be abandoned, and this fascinating, literally colorful, project was postponed indefinitely... [Section 9.0 (i.e., introduction only) is available at http://www.oswego.edu/~baloglou/103/isometrica-9.pdf, but has not been included in Isometrica; it concludes with a 'four color' conjecture on 'symmetrically correct' coloring of tilings.]




Any other future projects related to Isometrica?


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) and Conway's topological classification, developing/reviewing all needed mathematical tools along the way. The success of such a project (and course) would probably depend on the author's ability to delve into the hidden interplay among the three approaches.


[Conway's orbifold approach may be found, together with broadly related topics, in Geometry and the Imagination -- informal notes by John Conway, Peter Doyle, Jane Gilman, and Bill Thurston currently available at http://www.math.dartmouth.edu/~doyle/docs/gi/gi.pdf; look also for The Symmetry of Things, by John Conway, Heidi Burgiel, and Chaim Goodman-Strauss (AK Peters, forthcoming).]




Can we judge this book by its cover?


No way! The figure on the cover is a tribute to the great crystallographer (and not only) Arthur Loeb and his Color and Symmetry (Wiley, 1971), which offers an alternative geometrical study of wallpaper patterns. More specifically, it is a humorous reminder of Loeb's nifty derivation of the composition of two intersecting glide reflections (and that mysterious parallelogram associated with them): this important problem forms the pinnacle of my discussion of isometry composition in chapter 7, and it seems to be absent from all other books that could have discussed it; my approach is not as direct as Loeb's, but it has its own methodological advantages (such as requiring a thorough discussion of the composition of a glide reflection and a rotation, a topic not directly addressed by Loeb).


[Which Isometrica figure would be on the cover if I didn't choose to attract the reader's attention to Loeb's work and genius? Tough question, but the winner is figure 8.19 (on the 'ruling' and unexpected mirroring of half turn centers by glide reflection): in addition to capturing Isometrica's spirit, it could lead to an alternative and probably quicker discussion of half turn patterns in section 8.2. And a close second would no doubt be figure 8.39, which dispenses of the patterns with threefold/sixfold rotation and reflection by showing that their only 'factor' can be a cm.]





Further comments, acknowledgments, dedications



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 Isometrica has been written, as well as the subject's precarious position between Art and Mathematics. At another level, Coxeter's remark serves as a reminder of the interplay and struggle between rigor and intuition, between structure and freedom, which has certainly left its mark on Isometrica.


I like to say, in hindsight, that border and wallpaper patterns are "of limited interest to many people" -- not artistic enough for artists and not mathematical enough for mathematicians... Further, and contrary to the pleasant illusions created by Stevens or Washburn & Crowe or Isometrica, symmetry itself is an exception rather than a rule in the real world: I was rather flattered to hear from two former students that they think of me when they run across symmetrical figures during their New York City strolls, but how frequent, and how important after all, are such symmetrical encounters? How meaningful is abstract beauty in an increasingly tormented world? I have been caught telling friends that it is not enough for me to hear my students say that they enjoyed my course (and, by extension, book), I actually need to hear -- even if occasionally -- that it changed their life, or, less arrogantly on my part, that "it caused them see the world a little differently" (this is quoted verbatim from a former student's recent e-mail).


If you read between the lines above you already know that the teaching of MAT 103 and the writing of Isometrica have certainly changed my life: I knew that since the first week of classes in Fall 1995, when I came up with an assignment calling for the creation of the seven border pattern types using vertical and horizontal congruent rectangles -- an assignment that looks trivial now but kept me up late that night (because the idea of 'multidecked' border patterns is not 'natural' to our minds, perhaps). Moreover, there I was, someone with absolutely no prior interest in drawing or Design, spending many hours and nights creating 'new' patterns, first by hand, then on a computer ... gradually discovering how such patterns and concepts could form a gateway to mathematical thought for students as interested in Mathematics as I once was in Design! [The term "design" is used quite narrowly here, and intentionally so: Graphic Design majors who take MAT 103 tend to find its patterns rather inspiring!]


So a labor of love it was, and this is why I have largely preserved Isometrica in its original form: perhaps my preferred strategy or tactics for presenting this incredibly flexible material have changed since 2001, but I chose to preserve my initial insight and the writing adventure that ensued. For the same reason, combined with various personal circumstances, Isometrica is going straight to the internet rather than some constricting publishing house: the software packages employed (MathWriter and SuperPaint) were already ancient when I started, the English may seem awkward here and there, the figures are somewhat primitive and often imperfect, the overall format is kind of kinky, but you are getting the real thing, and for free at that! [You may in particular get a good sense of the struggle and discovery process that went on as the exposition revs up through the chapters: even if there is a "royal road to geometry" ... I often fail to follow it ... keeping in mind that "the shortest approach is not always the most interesting"!]


My joy at having been able to preserve Isometrica's desired form is offset by the sadness of having left so much out: my plans of including everything bypassed by 'first insight' in the form of exercises had to be abandoned, but I am still hoping of creating additional web pages -- probably linked to the online version of this Introduction -- in the future, covering extra topics in detail (and color); and if this hope never materializes, with the future of MAT 103 as inevitably unclear as is, I trust that enough material has been included here to inspire others toward new mathematical ideas and/or artistic creations. [Please forgive this desperate optimism about Isometrica being read and even expanded, but it is my firm belief that its informal and adventurous style is going to win it some lasting friends!]


My obvious desire to generate disciples for Isometrica has a non-obvious implication: despite the copyright notices at the beginning and ending of each chapter, I do allow the reproduction of my book for educational purposes; if for example a teacher anywhere in the world wishes to have hard copies (of either Isometrica in its totality or some of its chapters) for his/her students, then it is fine with me to have that school's printing service produce such copies, even if at a reasonable cost and marginal profit. So please do not write to me for permissions (concerning either Isometrica or various web pages related to it): I would love to have feedback from you, but giving me credit for the materials you have used is all that I am asking 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 MAT 103 students and their enthusiasm. At the risk of being oblivious to the small but precious contributions of many, I would like to single out and thank five former students for their encouragement and inspiration: Terry Loretto (Fall 1995), Dreana Stafford (Spring 1999), Michael Nichols (Fall 1999), who also provided crucial assistance with SuperPaint in January 2000, Richard Slagle (Fall 2003), and Bonita Bryson (Spring 2004), who also wrote the aforementioned honors thesis (on the tiling of wallpaper patterns by border patterns).


As made clear in the beginning of this Introduction, there would simply be no Isometrica without Margaret Groman's original vision; I am equally grateful to her for her constant encouragement and suggestions for improvement. Likewise, I am indebted to Mark Elmer, who has also taught MAT 103 several times, for his careful reading of Isometrica and useful observations. Beyond MAT 103, I am grateful to my friend and collaborator Phil Tracy, who has also read Isometrica and discussed it with me in considerable detail; and likewise to my colleagues Chris Baltus, Fred Barber, Joseph Gaskin, Michel Helfgott, and Kathy Lewis for their mathematical camaraderie over the years.


Beyond Oswego, I am grateful to a number of mathematicians and others who provided links to Isometrica's early ambassador, Crystallography Now, or offered useful feedback: Helmer Aslaksen, Andrew Baker, Dror Bar-Natan, Bryan Clair, Marshall Cohen, Wis Comfort, David Eppstein (Geometry Junkyard), Sarah Glaz, Andreas Hatzipolakis, Dean Henderson, William Huff, Loukas Kanakis, Nikos Kastanis, Barbara Pickett, Doug Ravenel, Jim Reid, Saul Stahl, Tohsuke Urabe, Marion Walter, Eric Weisstein (Wolfram MathWorld), Mark Yates, and others -- notably family and friends in Thessaloniki, contributors to the sci.math newsgroup, and participants of my January 2003 Symmetry For All MAA minicourse -- who should forgive me for having overlooked their input. I am also grateful to George Anastassiou, Varoujan Bedros, and Fred Linton for their advice on technical and 'legal' matters; along these lines, special thanks are also due to my friend and non-mathematical collaborator Nick Nicholas.


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 MathWriter (in its final years). Finally, many thanks are due to Patrick Murphy, Jean Chambers & David Vampola, and Julia & Matthew Friday for many a pleasant evening -- followed at times by all-night Isometrica writing and, inevitably, drawing -- in tranquil Oswego.


In a somber tone now ... even though Isometrica was dedicated from the beginning to the memory of our colleague Ron Prisco (Margaret Groman's Abstract Algebra teacher forty years ago, among other things), who passed away before even I started writing it but "had a lot of faith in my work", I would like to honor here the memory of a few local friends whom we lost during the last couple of years:


-- 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 MAT 103 a couple of times, organized our seminar, and was a fatherly figure for a number of younger colleagues


-- 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 MAT 103


Finally, Isometrica owes a lot to my late father, Christos Baloglou (1919 - 2002): a high school geometer who also taught Descriptive & Projective Geometry to Aristotle University Engineering students in the 1960's and published Scattered Drops of Geometry in 2001, he certainly influenced me to study Mathematics. My whole symmetry project may be seen as a Sisyphean effort to annul his lovely -- and, less obviously, loving -- verdict on it: "Son, this is not Mathematics"!


George Baloglou

Oswego, April 27, 2007


It is like being able to enjoy music without learning the notation -- Dani Novak (Ithaca College), May 20, 2007



It's a beautiful book; thanks for your tremendous work! -- Janos Pach (New York University), August 30, 2007






Photo (8/29/07) by Jim Russell -- from a 9/19/07 Campus Update article ( Artful Geometry)