Section 6.1 Symmetries and Algebraic Systems

Transcription

1 1 Section 6.1 and Algebraic Systems Purpose of Section: To introduce the idea of a symmetry of an object in the pla, which will act as an introduction to the study of algebraic structures, in particular algebraic groups. Abstraction and Abstract Algebra The ability to abstract is a unique feature of human thought, an ability not shared by lower forms of liing creatures. The ability to capture the essence of what we see and experience is so engraid in our mental processes, we er gie it a second thought. If the human mind did not hae the capability to abstract commonalities in daily liing, we would lie in a different world altogether. Suppose we lacked the capacity to grasp the essence of what makes up a chair. We would be forced to call eery chair by a different name in order to communicate to others what we are referring. The statement the chair in the liing room would hae no meaning unless we kw what exact chair was being referred. Parents point to a picture of a dog in a picture book and tell their infant, dog, and it is a proud moment for the parents when the child sees a dog in the yard and says, dog! The concept of number is a crowning achieement of human s ability to abstract the essence of size of sets. It is not cessary to talk about three people, three days, three dogs and so on. We hae abstracted among those things the commonality of threess so there is no ed to say add three goats plus fie goats, or three cats plus fie cats, we simply say three plus fie. The current chapter is a glimpse into some ideas of what is called abstract algebra. Before defining what we mean by an abstract algebra, you should realize you hae already studied some abstract algebras whether you know it or not, o being the integers. The integers are an abstract algebra, although you probably hae er called them abstract or an algebra. The integers are a set of objects, called integers, and binary operations of addition, subtraction, and multiplication, defid on the integers, along with a collection of axioms which the operations obey. Abstract algebra abstracts the essence of the integers and other mathematical structures, and says; let s not study just this or that, let s study all things which hae certain properties. (Not all that different from the infant when it says dog, realizing there are more dogs than in the picture book.) Abstract mathematics allows o to think about attributes and relationships, and not focus on specific objects that possess those attributes and relationships. The befits of abstraction are many; it uncoers deep relationships between different areas of mathematics by allowing o to rise up aboe the

2 2 nuances of a particular area and see things from a broader point of iew, like seeing the forest instead of the trees, so to speak. The main disadantage of abstraction is that abstract concepts are more difficult to learn and require more mathematical maturity before they can be understood and appreciated. In summary, abstract algebra studies geral mathematical structures with gien properties, the most important structures being groups, rings, and fields. Before we start our formal discussion of algebraic groups in the xt section, we motiate their study with the introduction of symmetries. We are all familiar with symmetrical objects, which we gerally think of as objects of beauty, and although you may not be prepared to gie a precise mathematical definition of symmetry, you know o when you see o. Most people would say a square is more symmetrical than a rectangle, and a hexagon more symmetrical than a square, and a circle is the most symmetrical object of all. egular patterns and symmetries are known to all cultures and societies. Although we gerally think of symmetry in terms of geometric objects, we can also include physical objects as well, like an atom, the crystalli structure of a miral, a plant, an animal, the solar system, or een the unierse. The concept of symmetry also embodies processes like chemical reactions, the scattering of elementary particles, a musical score, the eolution of the solar system, and een mathematical equations. In physics symmetry 1 has to do with the inariance (i.e. unchanging) of natural laws under space and time transformations. A physical law haing space/time symmetries establishes that the law is independent of translating, rotating, or reflecting the coordinates of the system. The symmetries of a physical system are fundamental to how the system acts and behaes. And once the word symmetry is mentiod, a mathematician thinks of a mathematical structure called a group. in Two Dimensions or a single (bounded) figure in two dimensions there are two types of symmetries 2. There is symmetry across a li in which o side of the object is the mirror image of its other half. This type of symmetry is called li symmetry (or reflectie or mirror symmetry). igure 1 shows an isosceles triangle with a li symmetry through its ertical median. 1 or the seminal work on symmetries, see Symmetry by Hermann Weyl (Princeton Uni Press) We do not include translation symmetries here since we are considering only single geometric objects.

3 3 Li Symmetry igure 1 The drawings of insects in igure 2 hae a bilateral symmetry. Other figures are symmetric through more than o li, such as a square, which has four lis (or axes) of symmetry; its horizontal and ertical midlis, and its two diagonal lis. A regular n -gon has n lis of symmetry and a circle has an infinite number. igures with O Li Symmetry igure 2 A second type of symmetry of a figure in the pla is rotational (or radial) symmetry. An object has rotational symmetry if the object is repeated when rotated certain degrees about a central point. The objects in igure 3 hae rotational symmetry about a point but no li symmetries. otational but No Li Symmetry igure 3 The flower shaped figure at the left in igure 3 repeats itself when rotated 0, 120, 240 degrees about its center point so we say the figure has rotational symmetry of degree 3. The letter Z has rotational symmetry 2, and the two figures at the right hae, respectiely, rotational symmetries 3 and 4.

4 4 Some objects hae both reflectie and rotational symmetries as illustrated by the regular polygons 3 in igure 4. The equilateral triangle has 3 rotational symmetries (rotations of 0,120 and 240 about a center point) and 3 reflectie symmetries through median lis passing through the three ertices. A regular polygon with n ertices has n rotation symmetries (each rotation 360 / n degrees) and n lis of symmetry. 3 ecall that a regular polygon is a polygon whose lengths of sides and angles are the same. Common os are the equilateral triange, square, pentagon, and so on.

5 5 Equal Number of otation and Li Symmetr S ymmetries otation Li igure 3 rotations 0,120,240 3 li symmetries 4 rotations 0,90,180,270 4 li symmetries 5 rotations 0,72,144,216,288 5 li symmetries Infinite number of rotation symmetries Infinite number of li symmetries igures Haing Both otational and eflectie igure 4 Margin Note: Many letters of the alphabet, such as A,B,C,D,, hae arious rotation and li symmetries. Symmetry y Transformations Although you can think of symmetries as a property of an object, which it it, there is another interpretation which is more beficial for our purposes. or us a symmetry is a function or what we often will call a mapping or transformation.

6 6 Definition A symmetry of an object, say A, in the pla is a distance presering mapping f (called an isometry) that maps the object A onto itself 4. That is, if f is a symmetry map, then f ( A) = A. otation Symmetry of 180 degrees Note: If an object A in the pla is bounded (i.e. it is inside some circle with finite radius) then a translation of the object merely sliding it in some direction is not a symmetry since it alters the location of the figure. or unbounded figures such as wallpaper designs, floor tiles, and so on are often symmetries. We now look at some simple geometric shapes in the pla. of a ectangle igure 5 shows a rectangle in which the length and width are not the same in which we hae labeled the corrs as A, B, C, D to help us isualize how the rectangle is rotated and reflected. igure 5 You can conince yourself that the rectangle has two rotation symmetries of 0, 180, and two li (or flip) symmetries where the lis of symmetry are the horizontal and ertical midlis. These four symmetries are illustrated in igure 6. 4 in three dimensions are defid analogously.

7 7 Motion Symbol irst Position inal Position No Motion e otate lip oer Horizontal Median H lip oer Vertical Median V The our of a ectangle igure 6 So what do these symmetries hae to do with algebraic structures? Since a symmetry is a transformation (i.e. a function) which maps the points of an object back into itself, we can defi the product of two transformations as performing o symmetry followed by the other (i.e. a composition of functions). It is clear that since each symmetry leaes the object unchanged, so does the product of two symmetries. In other words, the product formed in this manr is also a symmetry. or example, if we perform a 180 rotation 5, denoted by 180, followed by H a flip through the horizontal midli, the result, as illustrated in igure 7, is the same as performing the single symmetry V, a flip through the ertical midli. In other words, we hae computed the product 180 H = V. 5 It is our conention that all rotations are do counterclockwise.

8 8 Product of 180 H = V igure 7 Note that the do nothing symmetry e, called the identity symmetry, and is analogous to 1 in normal multiplication of integers. igure 8 shows the product of a few symmetries. eh = He = H Ve = Ve = V e = e = ee = e = e VV = e HH = e Products of Typical igure 8 Note also that two successie operations of any of the symmetries e, 180, V, H results in returning to the original position. or that reason we say each of these symmetries is its own inerse, which are expressed in igure 9.

9 9 HH = e H = H VV = e V = V 1 1 = e = ee = e e = e Inerses of igure 9 We can now make a multiplication table for the symmetries of a rectangle, which is shown in igure 10. The product of any two symmetries lies at the intersection of the row and column of the symmetries. or example, the intersection of the row labeled 180 and column labeled H is V, which means 180 H = V. The borders of the cells containing the identity symmetry e are darked as an aid in reading the table. e 180 H V e e 180 H V e V H H H V e 180 V V H 180 e Note: Multiplication Table for of a ectangle igure Eery row and column of the multiplication table contains o and exactly o of the four symmetries. It is like a Latin square. 2. In this example the main diagonal contains the identity symmetry e, which means the product of each symmetry times itself is the identity, or equialently each symmetry is equal to its inerse. 3. The table is symmetric about the main diagonal which means the multiplication of symmetries is commutatie. i.e. AB = BA just like multiplication of numbers in arithmetic. We would call this a commutatie algebraic system. 4. The four symmetries e, 180, V, H of the rectangle along with the product operation form what is called an algebraic group.

10 10 We hae seen how the symmetries of a rectangle form an algebraic structure of four elements, complete with algebraic identity, and where (in this example) eery element has an inerse. Obsere how this system is analogous to the integers with the operation of addition, except there are an infinite number of integers (and in the case of integers, only 0 is its own inerse). Margin Note: There are objects with no li or rotational symmetries. Can you draw some? of an Equilateral Triangle We now exami a figure that is more symmetric than the rectangle, the equilateral triangle. The equilateral triangle 6 as drawn in igure 11 has 6 symmetries, three rotational symmetries in which the triangle is rotated 0,120,240 about its center, and 3 li symmetries in which the triangle is reflected through lis passing through the ertices as drawn as dotted li segments. Six of an Equilateral Triangle igure 11 Denoting these symmetry mappings as e, 120, 240,,,, where e is the identity (do nothing) symmetry, 120, 240 are (counterclockwise) rotations of 120,240 respectiely, denotes the flip through the ertical median, is the flip around the denotes the flip around the northwest median, and northeast median. igure 12 illustrates the effect of these symmetries. 6 ecall that an equilateral triangle is a triangle with the three sides (or three angles) the same.

11 11 Motion Symbol irst Position inal Position No Motion e otate 120 Counterclockwise 120 otate 240 Counterclockwise 240 lip oer the Vertical Axis lip oer the Northeast Axis lip oer the Northwest Axis of an Equilateral Triangle igure 12

12 12 As we did for the rectangle, we can compute the multiplication, or Cayley table, for these symmetries. See igure 13. e e e e e 120 e e e Multiplication Table for of an Equilateral Triangle igure 13 We hae drawn the borders around the identity symmetry e so the table can be more easily read and interpreted. or easier reading, we hae shaded the northeast and northwest blocks of products inoling the three flip symmetries. Example 1 (Commutatie Operations) Are the symmetry operations for the equilateral triangle commutatie? In other words, does the order in which the operations are performed make a difference in the outcome 7? Solution To determi if the symmetries are commutatie, we exami the products in the multiplication table in igure 11 to see if they are symmetric around the diagonal elements. In this case the table is not symmetric for all symmetries so the symmetry operations are not commutatie. Note that although o product is commutatie; = = e. Example 2 (inding Inerse ) What are the inerses of each symmetry operation? Solution Note that e = = = = e which means the symmetries e,,, are their own inerses, that is Howeer, note that = and e = e =, , =, 1 n 1 n n =. =. The fact there is o and only 7 Not all mathematical operations are commutatie. Examples are matrix multiplication and the cross product of ectors. In daily life not all operations are commutatie either. Going outside and emptying the garbage is o example.

13 13 o identity symmetry e in eery row and column means that each symmetry has exactly o inerse symmetry. The collection e, 120, 240,,, of symmetries of the equilateral triangle along with the product operation defis the dihedral group of order 6 and denoted byd 6. In geral the dihedral group of order D 2n represents the symmetries of a regular n -gon. Pure Mathematics: The story goes how Abraham Lincoln, failing to conince his Cabit how their reasoning was faulty, ask them, How many legs does a cow hae? When they said four, he then continued, Well then, if a cow s tail was a leg, how many legs does it hae? When they said fie, obiously, Lincoln said, That s where you are wrong. Just calling a tail a leg doesn t make it a let. This story may be true in the real world, but in the world of pure mathematics it is wrong. If you call a cow s tail a leg, then it is a leg. In pure mathematics, we do not ed to know what the things we are working with are, only the rules with goern them. We only ed to know the axioms.

14 14 Problems 1. Determi the number of li and rotational symmetries of the following letters. a) A l) L w) W b) B m) M x) X c) C n) N y) Y d) D o) O z) Z e) E p) P f) q Q g) G r) h) H s) S i) I t) T j) J u) U k) K ) V 2. Draw a figure that has the following symmetries. a) no rotational and no li symmetries b) 1 rotational and no li symmetries c) no rotational and 1 li symmetry d) 1 rotational and 1 li symmetry e) 2 rotational and no li symmetries f) no rotational and 2 li symmetries g) 4 rotational and no li symmetries 3. (Multiplication Tables) or the following objects determi the symmetries and compute the multiplication table for the symmetries. ind the inerse of each symmetry. a)

15 15 b) c) d) e) f)

16 16 g) 4. (Common ) 8 The following logos hae an equal number of li and rotation symmetries. These are the symmetries of a regular n gon. of this type are called dihedral symmetries. ind the rotational and reflection symmetries of the following figures. Hint: Don t forget the identity mapping which is a rotation of zero degrees. a) b) c) d) e) 8 We thank Annalisa Cranl of ranklin and Marshall College for proiding these logos.

17 17 5. ( of Solutions of Differential Equations) The solutions of the x differential equation dy dx = y is the o-parameter family y = ce where c is an arbitrary constant. Show that the transformation x = x + h, y = y where h is an arbitrary real number maps the family of solutions back into the family of solutions, and hence is a symmetry transformation of the differential equation. 6. (ind the ) Draw a geometric object whose symmetries a, b, c, d hae the following multiplication tables. a) e a e e a a a e b) c) e a b e e a b a a b e b b e a e a b c e e a b c a a b c e b b c e a c c e a b d) e a b c d e e a b c d a a b c d e b b c d e a c c d e a b d d e a b c 6 ( of a Parallelogram) Describe the symmetries of a parallelogram that is ither a rhombus nor a rectangle.

18 18 7. ( of an Ellipse) Describe the symmetries of an ellipse that is not a circle. 8. (Multiplication Tables) ind the multiplication table for the symmetries of the star in igure 15. Once you determi the pattern for the table, it goes fairly fast. igure (epresentation of D 2 with Matrices) Show that the matrices e, A =, B, C 0 1 = = = satisfy the following multiplication table of the dihedral group D 2 where the group operation is defid as matrix multiplication. This means the group can be represented by matrices where the group operation is matrix multiplication. e A B C e e A B C A A e C B B B C e A C C B A e Dihedral Multiplication Table