# isometric a

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ISOMETRICAA GEOMETRICAL INTRODUCTION TO PLANAR CRYSTALLOGRAPHIC GROUPSin memory of Professor Rinaldo Prisco

Fig. 28 Intersecting glide lines [Arthur L. Loeb, Color and Symmetry (Wiley 1971, Krieger 1978)]

GEORGE BALOGLOUSUNY OSWEGO, 2007ISBN 978-0-9792076-0-0

ti d semn rmona ka uen ti ka mga, |Aristotlhq Pltvnoq tayt lgei " d rmona stn orana tn fsin xoysa uean ka kaln ka daimonan? tetramerq d t dynmei pefykya do mesthtaq xei riumhtikn te ka rmonikn, faneta te t mrh atq ka t meguh ka a peroxa kat' riumn ka some t r an? n gr dys tetraxrdoiq Wyumzetai t mrh"Aristotle (?), Fragmenta Varia 47 (Pseudo-Plutarchus, De Musica)

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i

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 mysection of MAT 102 (SUNY Oswego's main General Education course for nonscience 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, graduallyabandoning 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

ii ( 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-sciencemajors 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 essentiallycompleted 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 (http://www.oswego.edu/~baloglou/103) 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

iii 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'sbook. 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.

iv 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

v 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,

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