Polygons, Polyhedra, Patterns & Beyond

POLYGONS, POLYHEDRA, PATTERNS & BEYOND Ethan D. Bloch Spring 2015 i ii Contents To the Student v Part I POLYGONS & POLYHEDRA 1 1 Geometry Basics 3 1.1 Euclid and Non-Euclid . 3 1.2 Lines and Angles ................................ 12 1.3 Distance .................................... 21 2 Polygons 25 2.1 Introduction................................... 25 2.2 Triangles .................................... 28 2.3 General Polygons ................................ 39 2.4 Area....................................... 49 2.5 The Pythagorean Theorem. 63 3 Polyhedra 73 3.1 Polyhedra – The Basics . 73 3.2 Regular Polyhedra. 78 3.3 Semi-Regular Polyhedra. 81 3.4 Other Categories of Polyhedra. 84 3.5 Enumeration in Polyhedra. 87 3.6 Curvature of Polyhedra . 99 iv Contents Part II SYMMETRY & PATTERNS 105 4 Isometries 107 4.1 Introduction................................... 107 4.2 Isometries – The Basics . 110 4.3 Recognizing Isometries . 119 4.4 Combining Isometries. 125 4.5 Glide Reflections . 130 4.6 Isometries—The Whole Story . 136 5 Symmetry of Planar Objects and Ornamental Patterns 147 5.1 Basic Ideas ................................... 147 5.2 Symmetry of Regular Polygons I . 154 5.3 Symmetry of Regular Polygons II . 160 5.4 Rosette Patterns . 170 5.5 Frieze Patterns.................................. 178 5.6 Wallpaper Patterns . 187 5.7 Three Dimensional Symmetry . 204 6 Groups 215 6.1 The basic idea.................................. 215 6.2 Clock arithmetic. 218 6.3 The Integers Mod n ...............................223 6.4 Groups .....................................228 6.5 Subgroups.................................... 237 6.6 Symmetry and Groups. 241 Suggestions for Further Reading 245 Bibliography 251 To the Student I have written these lecture notes because I have not found any existing text for Math 107 that was adequate in terms of the choice of material, level of difficulty, organization and exposition. Much of the approach in Part II of these notes is inspired by the text I previously used for Math 107 (Farmer, David, “Groups and Symmetry,” AMS, 1996). I hope that these notes will fit the needs of this course, and will help you learn this material. If there are any errors in the text, or anything that is not clearly written, please accept my apolo- gies. I would very much appreciate your feedback on this text, both in terms of errors that you find and suggestions for changes or additions that you might have. Comments can be forwarded to me in person (after class, or in my office, Albee 317), or by email at [email protected]. Prerequisites It is assumed that anyone using this text has passed Part I of the Mathematics Diagnostic Exam. No particular background in mathematics beyond that is required. On some occasions we will make use of high school algebra (for example, the quadratic formula) and high school geometry (for example, the Pythagorean Theorem). For the most part, however, we will treat material that, while touching upon some very substantial ideas, does not require much in the way of algebra or geometry background. Precalculus (including trigonometry, logarithms, and the like) is not required. On a few occasions we will mention trigonometry, but those brief references can easily be skipped. What is needed to read this text is a willingness to learn new ideas, to think through subtleties, to work hard, and to use your imagination. Much of the material in the text is very visual, and making drawings, and mentally imagining geometric objects, is crucial. Some of the arguments vi To the Student in the text, though not requiring much in the way of technical background, are nonetheless quite tricky, and require careful attention to the details. Exercises Like music and art, mathematics is learned by doing, not just by reading texts and listening to lectures. In mathematics courses we assign exercises not because we want to put the students through some sort of mathematical boot camp, but because doing exercises is the best way to work with the material, and to see what is understood and what needs further study. Doing the exercises is therefore a crucial part of learning the material in this text. Exercises range from the routine to the difficult. When answering an exercise, you may use any facts in the text up till then (including previous exercises). One feature of the exercises in this text is worth mentioning. In many high school mathematics courses, the text has a variety of worked-out examples, and then the homework exercises are often virtually identical to the worked-out examples, but with different numbers. The students then do the exercises by simply mimicking the examples in the text. Students are often satisfied with these types of exercises, but from the point of view of intellectual growth, such exercises are sorely lacking. The point of learning mathematics is not to learn to imitate what the teacher or the book does, but rather to understand new concepts and be able to apply them. Hence, in this text, many of the exercises do not ask the students to mimic what is done in the text, but rather to think for themselves. Many (though not all) of the exercises in this text are, purposely, not identical to worked out examples in the text, but rather are to be solved by thinking about the material. Some of the exercises in this text require creativity and imagination, and others are open ended. Part I POLYGONS & POLYHEDRA 1 1 Geometry Basics 1.1 Euclid and Non-Euclid Ancient Greek mathematics was put into its ultimate deductive form by Euclid, who lived roughly around 300 BCE. In his work “The Elements,” Euclid took an already developed large body of geometry, and gave it logical order by isolating a few basic definitions and axioms, and then deducing everything else from these definitions and axioms. The statements that Euclid deduces from his axioms and definitions are called propositions (in modern textbooks they are often referred to as theorems, which means the same thing). We will look very briefly at some aspects of Euclid’s Elements. This massive work is divided up into 13 “Books;” our concern is primarily with Book I. Following [Har00] (though others emphasize this point as well), we stress that Euclid, and the ancient Greeks generally, viewed geometry rather differently than we currently do. The only quantities that were of interest were geometric ones, and of those, geometric quantities that could be constructed with straightedge and compass were of particular interest. Numbers for their own sake seemed less of interest (perhaps because they did not have a developed understanding of numbers, or perhaps that is why they did not develop such an understanding); a solid understanding of numbers came later in history. For example, consider the famous Pythagorean Theorem, which is demonstrated in Section 2.2. In contemporary language, this theorem states that if a right triangle has sides of length a and b, and hypotenuse of length c, then a2 +b2 = c2. So stated, this theorem tells us something about three numbers a, b and c: if these three numbers satisfy a certain condition (namely being the lengths of the sides and hypotenuse of a right triangle), then they must also satisfy the algebraic equation a2 + b2 = c2. To Euclid, such a statement about numbers would not have made sense. His version of the Pythagorean The- orem is: “In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.” (All quotes from Euclid are taken from 4 1. Geometry Basics [Euc56], which is the standard English translation.) Euclid’s version of the theorem is a statement about three squares, namely that one square (the one on the hypotenuse) “is equal” to two other squares (those on the sides) put together. Euclid is interested in the relation between these three squares, which are geometric objects, and not numbers. It might seem from a modern perspective that Euclid’s version of the Pythagorean is about area, and thus really does involve numbers, because the area of a planar figure (that is, a figure in the plane) is a number. In fact, Euclid does not mention the concept of area at all in his version of the Pythagorean Theorem. When Euclid says that two planar figures (such as squares) are equal, he is not making a statement about the numerical values of their areas (for example in square inches), but rather is saying that one figure can be cut up into triangles, and reassembled into the other figure. Thus, Euclid’s version of the Pythagorean Theorem is strictly about geometric objects. Today, we have the concept of assigning to each planar figure its area (which is a number), and we restate various geometric properties in terms of numerical properties, but that is not the way Euclid (and the other ancient Greeks) viewed things. See [Har00] for a thorough discussion of this issue, and more generally on what Euclid said, and how he understood geometry. (We highly recommend [Har00] as a companion to reading Euclid; though much of the book is aimed at a mathematically sophisticated audience, some parts are very accessible, and extremely insightful.) Without question, “The Elements” is one of the most important, and influential, works of mathematics ever written—it is arguably one of the most influential intellectual achievements of human civilization as a whole, not just of mathematics. Euclid’s treatment of geometry be- came the universally accepted method of doing geometry for almost two millenia, up till the 19th century. Moreover, not only was Euclidean geometry accepted as unquestionably true, but Euclid’s method of deductive reasoning was considered a model of logical argumentation, and an example of reasoning that produced theorems that were unquestionably true.

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