Idea Transcript
Section 6.1
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Symmetries
Section 6.1 Symmetries and Algebraic Systems Purpose of Section: To introduce the idea of a symmetry of an object in the Section plane, 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 living creatures. The ability to capture the essence of what we see and experience is so engrained in our mental processes, we never give it a second thought. If the human mind did not have the capability to abstract commonalities in daily living, we would live 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 every chair by a different name in order to communicate to others what we are referring. The statement “the chair in the living room” would have no meaning unless we knew 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 achievement of human’s ability to abstract the essence of size of sets. It is not necessary to talk about “three people”, “three days”, “three dogs” and so on. We have abstracted among those things the commonality of “threeness” so there is no need to say add “three goats plus five goats, or “three cats plus five cats”, we simply say three plus five. 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 have already studied some “abstract algebras” whether you know it or not, one being the integers. The integers are an abstract algebra, although you probably have never called them “abstract” or an “algebra.” The integers are a set of objects, called integers, and binary operations of addition, subtraction, and multiplication, defined 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 have 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 one to think about attributes and relationships, and not focus on specific objects that possess those attributes and relationships. The benefits of abstraction are many; it uncovers deep relationships between different areas of mathematics by allowing one to “rise up” above the
Section 6.1
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Symmetries
nuances of a particular area and see things from a broader point of view, like seeing the forest instead of the trees, so to speak. The main disadvantage 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 general mathematical structures with given properties, the most important structures being groups, rings, and fields.
Before we start our formal discussion of algebraic groups in the next section, we motivate their study with the introduction of symmetries. Symmetries Symmetries We are all familiar with symmetrical objects, which we generally think of as objects of beauty, and although you may not be prepared to give a precise mathematical definition of symmetry, you know one when you see one. 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. Regular patterns and symmetries are known to all cultures and societies. Although we generally think of symmetry in terms of geometric objects, we can also include physical objects as well, like an atom, the crystalline structure of a mineral, a plant, an animal, the solar system, or even the universe. The concept of symmetry also embodies processes like chemical reactions, the scattering of elementary particles, a musical score, the evolution of the solar system, and even mathematical equations. In physics symmetry1 has to do with the invariance (i.e. unchanging) of natural laws under space and time transformations. A physical law having 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 behaves. And once the word symmetry is mentioned, a mathematician thinks of a mathematical structure called a group. Symmetries in Two Dimensions For a single (bounded) figure in two dimensions there are two types of symmetries2. There is symmetry across a line in which one side of the object is the mirror image of its other half. This type of symmetry is called line symmetry (or reflective or mirror symmetry). Figure 1 shows an isosceles triangle with a line symmetry through its vertical median.
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For the seminal work on symmetries, see Symmetry by Hermann Weyl (Princeton Univ Press) 1980. 2 We do not include translation symmetries here since we are considering only single geometric objects.
Section 6.1
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Symmetries
Line Symmetry Figure 1 The drawings of insects in Figure 2 have a bilateral symmetry. Other figures are symmetric through more than one line, such as a square, which has four lines (or axes) of symmetry; its horizontal and vertical midlines, and its two diagonal lines. A regular n -gon has n lines of symmetry and a circle has an infinite number.
Figures with One Line Symmetry Figure 2 A second type of symmetry of a figure in the plane is rotational (or radial) radial symmetry. symmetry An object has rotational symmetry if the object is repeated when rotated certain degrees about a central point. The objects in Figure 3 have rotational symmetry about a point but no line symmetries.
Rotational Symmetries but No Line Symmetry Figure 3 The flower shaped figure at the left in Figure 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 have, respectively, rotational symmetries 3 and 4.
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Some objects have both reflective and rotational symmetries as illustrated by the regular polygons3 in Figure 4. The equilateral triangle has 3 rotational symmetries (rotations of 0 ,120 and 240 about a center point) and 3 reflective symmetries through median lines passing through the three vertices. A regular polygon with n vertices has n rotation symmetries (each rotation 360 / n degrees) and n lines of symmetry.
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Recall that a regular polygon is a polygon whose lengths of sides and angles are the same. Common ones are the equilateral triange, square, pentagon, and so on.
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Equal Number of Rotation and Line Symmetr Symmetries ymmetries Rotation Line Figure Symmetries Symmetries 3 rotations
3 line symmetries
0 ,120 , 240
4 rotations
4 line symmetries
0 ,90 ,180 , 270
5 rotations
0 , 72 ,144 , 216 , 288
5 line symmetries
Infinite number of rotation symmetries
Infinite number of line symmetries
Figures Having Both Rotational and Reflective Symmetries Figure 4 Margin Note: Many letters of the alphabet, such as A,B,C,D,… , have various rotation and line symmetries. Symmetry Symmetry Transformations Although you can think of symmetries as a property of an object, which it it, there is another interpretation which is more beneficial for our purposes. For us a symmetry is a function or what we often will call a mapping or transformation.
Section 6.1
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Symmetries
Definition A symmetry of an object, say A , in the plane is a distance preserving mapping f (called an isometry) isometry that maps the object A onto itself4. That is, if f is a symmetry map, then f ( A ) = A .
Rotation Symmetry of 180 degrees
Note: Note If an object A in the plane 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. For unbounded figures such as wallpaper designs, floor tiles, and so on are often symmetries. We now look at some simple geometric shapes in the plane. Symmetries of a Rectangle Figure 5 shows a rectangle in which the length and width are not the same in which we have labeled the corners as A, B, C , D to help us visualize how the rectangle is rotated and reflected.
Figure 5 You can convince yourself that the rectangle has two rotation symmetries of 0 , 180 , and two line (or flip) symmetries where the lines of symmetry are the horizontal and vertical midlines. These four symmetries are illustrated in Figure 6.
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Symmetries in three dimensions are defined analogously.
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Section 6.1
Motion
Symbol
No Motion
e
Rotate 180
R180
Flip over Horizontal Median
H
Flip over Vertical Median
V
Symmetries First Position
Final Position
The Four Symmetries of a Rectangle Figure 6 So what do these symmetries have 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 define the product of two transformations as performing one symmetry followed by the other (i.e. a composition of functions). It is clear that since each symmetry leaves the object unchanged, so does the product of two symmetries. In other words, the product formed in this manner is also a symmetry. For example, if we perform a 180 rotation5, denoted by R180 , followed by H a flip through the horizontal midline, the result, as illustrated in Figure 7, is the same as performing the single symmetry V , a flip through the vertical midline. In other words, we have computed the product R180 H = V .
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It is our convention that all rotations are done counterclockwise.
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Symmetries
Product of Symmetries R180 H = V Figure 7 Note that the “do nothing” symmetry e , called the identity symmetry, symmetry and is analogous to 1 in normal multiplication of integers. Figure 8 shows the product of a few symmetries.
eH = He = H Ve = Ve = V R180 e = eR180 = R180 ee = e R180 R180 = e VV = e HH = e Products of Typical Symmetries Figure 8 Note also that two successive operations of any of the symmetries e, R180, V , H results in returning to the original position. For that reason we say each of these symmetries is its own inverse, inverse which are expressed in Figure 9.
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HH = e ⇒ H = H −1 VV = e ⇒ V = V −1 −1 R180 R180 = e ⇒ R180 = R180
ee = e ⇒ e = e−1 Inverses of Symmetries Figure 9 We can now make a multiplication table for the symmetries of a rectangle, which is shown in Figure 10. The product of any two symmetries lies at the intersection of the row and column of the symmetries. For example, the intersection of the row labeled R180 and column labeled H is V , which means
R180 H = V . The borders of the cells containing the identity symmetry e are darkened as an aid in reading the table.
e
R180
H
V
e
e
R180
H
V
R180
R180
e
V
H
H
H
V
e
R180
V
V
H
R180
e
Multiplication Table for Symmetries of a Rectangle Figure 10 Note: 1. Every row and column of the multiplication table contains one and exactly one 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 equivalently each symmetry is equal to its inverse. 3. The table is symmetric about the main diagonal which means the multiplication of symmetries is commutative. commutative i.e. AB = BA just like multiplication of numbers in arithmetic. We would call this a commutative algebraic system. 4.
The four symmetries e, R180, V , H of the rectangle along with the product
operation form what is called an algebraic group. group.
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Symmetries
We have seen how the symmetries of a rectangle form an algebraic structure of four elements, complete with algebraic identity, and where (in this example) every element has an inverse. Observe 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 inverse). Margin Note: There are objects with no line or rotational symmetries. Can you draw some? Symmetries of an Equilateral Triangle We now examine a figure that is “more symmetric” than the rectangle, the equilateral triangle. The equilateral triangle6 as drawn in Figure 11 has 6 symmetries, three rotational symmetries in which the triangle is rotated 0 ,120 , 240 about its center, and 3 line symmetries in which the triangle is reflected through lines passing through the vertices as drawn as dotted line segments.
Six Symmetries of an Equilateral Triangle Figure 11 Denoting these symmetry mappings as e, R120 , R240 , Fv , Fnw , Fne , where e is the identity (do nothing) symmetry, R120 , R240 are (counterclockwise) rotations of
120 , 240 respectively, Fv denotes the flip through the vertical median, Fnw denotes the flip around the northwest median, and Fne is the flip around the northeast median. Figure 12 illustrates the effect of these symmetries.
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Recall that an equilateral triangle is a triangle with the three sides (or three angles) the same.
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Section 6.1
Motion
Symbol
No Motion
e
Rotate 120 Counterclockwise
R120
Rotate 240 Counterclockwise
R240
Flip over the Vertical Axis
Fv
Flip over the Northeast Axis
Fne
Flip over the Northwest Axis
Fnw
First Position
Symmetries of an Equilateral Triangle Figure 12
Symmetries
Final Position
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Section 6.1
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As we did for the rectangle, we can compute the multiplication, or Cayley table, for these symmetries. See Figure 13.
e
R120
R240
Fv
Fne
Fnw
e
e
R120
R240
Fv
Fne
Fnw
R120
R120
R240
e
Fne
Fnw
Fv
R240
R240
e
R120
Fnw
Fv
Fne
Fv
Fv
Fnw
Fne
e
R240
R120
Fne
Fne
Fv
Fnw
R120
e
R240
Fnw
Fnw
Fne
Fv
R240
R120
e
Multiplication Table for Symmetries of an Equilateral Triangle Figure 13 We have drawn the borders around the identity symmetry e so the table can be more easily read and interpreted. For easier reading, we have shaded the “northeast” and “northwest” blocks of products involving the three flip symmetries. Example 1 (Commutative (Commutative Operations) Are the symmetry operations for the equilateral triangle commutative? In other words, does the order in which the operations are performed make a difference in the outcome7? Solution To determine if the symmetries are commutative, we examine the products in the multiplication table in Figure 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 commutative. Note that Fnw Fne ≠ Fne Fnw although one product is commutative; R120 R240 = R240 R120 = e . Example 2 (Finding Inverse Symmetries) What are the inverses of each symmetry operation? Solution Note that e2 = Fv2 = Fne2 = Fnw2 = e which means the symmetries e, Fv , Fne , Fnv are
their
own
inverses,
that
is
e = e −1 , Fv = Fv−1 ,
Fne = Fne−1 , Fnv = Fnv−1 .
−1 −1 However, note that R120 = R240 and R240 = R120 . The fact there is one and only
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Not all mathematical operations are commutative. Examples are matrix multiplication and the cross product of vectors. In daily life not all operations are commutative either. Going outside and emptying the garbage is one example.
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Symmetries
one identity symmetry e in every row and column means that each symmetry has exactly one inverse symmetry. The collection e, R120 , R240 , Fv , Fnw , Fne of symmetries of the equilateral triangle along with the product operation defines the dihedral group of order 6 and denoted by D6 . In general the dihedral group of order D2 n represents the symmetries of a regular n -gon. Pure Mathematics: The story goes how Abraham Lincoln, failing to convince his Cabinet how their reasoning was faulty, ask them, “How many legs does a cow have?” When they said four, he then continued, “Well then, if a cow’s tail was a leg, how many legs does it have?” When they said five, obviously, 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 need to know what the things we are working with are, only the rules with govern them. We only need to know the axioms.
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Symmetries
Problems 1. Determine the number of line 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) F q Q g) G r) R h) H s) S i) I t) T j) J u) U k) K v) V 2. Draw a figure that has the following symmetries. a) b) c) d) e) f) g)
no rotational and no line symmetries 1 rotational and no line symmetries no rotational and 1 line symmetry 1 rotational and 1 line symmetry 2 rotational and no line symmetries no rotational and 2 line symmetries 4 rotational and no line symmetries
3. (Multiplication Tables) For the following objects determine the symmetries and compute the multiplication table for the symmetries. Find the inverse of each symmetry. a)
Section 6.1 b)
c)
d)
e)
f)
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Symmetries
Section 6.1
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Symmetries
g)
4. (Common Symmetries) Symmetries)8 The following logos have an equal number of line and rotation symmetries. These are the symmetries of a regular n gon. Symmetries of this type are called dihedral symmetries. symmetries Find 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)
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We thank Annalisa Crannel of Franklin and Marshall College for providing these logos.
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Section 6.1
Symmetries
5. (Symmetries of Solutions of Differential Equations) The solutions of the differential equation dy dx = y is the one-parameter family y = ce x 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. (Find the Symmetries) Draw a geometric object whose symmetries a, b, c, d have the following multiplication tables. a)
e e a
e a
a a e
b)
e e a b
e a b
a a b e
b b e a
c)
e e a b c
e a b c
a a b c e
c c e a b
b b c e a
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 (Symmetries of a Parallelogram) Describe the symmetries of a parallelogram that is neither a rhombus nor a rectangle.
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Section 6.1
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7. (Symmetries of an Ellipse) Describe the symmetries of an ellipse that is not a circle. 8. (Multiplication Tables) Find the multiplication table for the symmetries of the star in Figure 15. Once you determine the pattern for the table, it goes fairly fast.
Figure 14
9. (Representation of D2 with Matrices) Show that the matrices
1 0 −1 0 0 1 0 −1 e= , A= , B= , C= 0 1 0 −1 1 0 −1 0 satisfy the following multiplication table of the dihedral group D2 where the group operation is defined 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