Combining Isometries- The Symmetry Group of a Square L.A. Romero August 22, 2017 1 The Symmetry Group of a Square We begin with a definition. Definition 1.1. The symmetry group of a figure is the collection of all isometries that leave the figure invariant. For example, Fig. 1 shows a square with its lines of reflection drawn in. It should be clear that the square looks the same if we rotate it by 0, 90, 180, or 270 degrees. It may appear silly to include a rotation by 0 degrees, since any figure looks the same if you rotate it by 0 degrees. However, much as our number system would be very clumsy without the number zero (how would your write 101?), talking about symmetry is clumsy if we do not include the identity element ( a transformation sending every element into itself) The square also looks the same if we rotate it by 360 or 360+90 degrees. However, we consider a rotation by 360 + 90 degrees to be the same as a rotation by 90 degrees. That is, we are only concerned with where points in the plane end up, not exactly how we got them to where they are. It should be clear that a square is also invariant under a reflection about a horizontal or vertical axis. It is also invariant under reflections about the diagonals. It can be shown that these are the only isometries that leave the square looking the same. It is useful to give each of these a symbol, which we will now do. I- The identity. This sends every element in the plane into itself. R 90 - This is a rotation by 90 degrees about the center of the square. R 180 - This is a rotation by 180 degrees about the center of the square. R 90 - This is a rotation by 90 degrees about the center of the square. This is the same as rotating by 270 degrees. S x - This a a reflection about the horizontal axis. S y This is a reflection about the vertical axis. S + This is a reflection about the diagonal that goes from the center to the upper right hand corner of the square. S - This is a reflection about the line that goes from the center to the upper left hand corner of the square. As when choosing what to call an unknown in algebra, this labeling of the symmetry operations is not universally used. That is, at a different time we mauy choose to label these using a different notation. For example, we might label the operations as g k, k = 1, 8. For example, letting g 1 = I, g 2 = R 90, etc. In this course we will not delve into the abstract mathematics of groups. However, we will point out that the symmetry group of the square (or any object ) forms what mathematicians call a group. In particular 1
Figure 1: A square with its lines of reflections drawn. There is an operation usually called multiplication. In our case the elements in our group are isometries, and multiplying two elements together means first applying one is isometry, then the other. For example S x S y means apply S y then S x. The multiplication is not necessarily commutative. For example, it turns out that S x S y is a different isometry than S y S x. The elements are closed under multiplication. That means that if you multiply any two elements in the group together, you get another element in the group. For a symmetry group, this is clearly true, since if an object looks the same when you apply an isometry A, and it looks the same when you apply an isometry B, then it must looks the same when you apply an isometry AB. There is an identity element I such that when you multiply any element by I, you get the element back. In our case, the identity is jsut the transformation that sends every element into itself. Every element has an inverse. The inverse of A is usually denoted by A 1. The inverse is an element such that AA 1 = I. It turns out that we also have A 1 A = I. As an example, R90 1 = R 90, and Sx 1 = S x. 2 Breaking the Symmetry of a Square The symmetry group of a square consists of 4 rotations (including a rotation by 0 degrees), and 4 reflections. We now give some examples of figures that are symmetric under a subset of these symmetry operations. Fig. 2 shows a figure that is symmetric under the same rotations as a square, but not under any of the reflections. Its symmetry group consists of the four rotations I, R 90, R 90, and R 180. The following definition will be useful when discussing symmetry. Definition 2.1. The rotation group of a figure is the set of all proper isometries that leave the figure invariant. For example, the square has the same rotation group as Fig. 2. Fig. 3) shows a figure that is symmetric under the rotations I and R 180 as well as the reflections S x and S y. 2
Is it possible to choose any subset of the operations and find a figure whose symmetry group con these and only these operations. For example, could we find a figure whose symmetry group is I, R 90? Clearly this is impossible, since if a figure looks the same when you rotate it by 90 degrees, it must look the same when you rotate it by 180 degrees, since you can rotate by 180 degrees by rotating it twice by 90. If it looks the same after you rotate it once, it will also look the same after you rotate it twice. It follows that if the symmetry group has I and R 90 it must also have R 180 (and also R 90 ). Would it be possible to have a figure that has the same rotation group as a square, but is only invariant under the reflections S x and S y (not S + and S )? No matter how hard you try, you will not be able to find such a figure. In the next section We explain why you cannot do this. Figure 2: A figure that has the rotational symmetry of a square, but none of the reflectional symmetry. 3 Combining a Rotation and a Reflection What is the result of combining the rotation S x and R 90? In particular, what is R 90 S x? We can use our general theorems on isometries to note that the resulting operation must be a reflection about a line passing through the center of the square. This follows from the fact that: R 90 S x is the combination of a proper and an improper isometry, and hence must be an improper isometry. R 90 S x clearly leaves the center of the square fixed, and hence it is not a glide reflection. This means that it must be a reflection, and one whose line of reflection passes through the center of the square. This implies that any line that gets mapped into itself must in fact be the line of reflection. We will now find the line of reflection using Fig. 4. Suppose that we first reflect about the horizontal line, and then rotate by 90 degrees. We can see that the green line will get mapped into itself by these operations. In particular, the reflection will send the green line into the orange line. When we rotate by 90 3
Figure 3: A figure that is invariant under the symmetry group consisiting of the operations I, R 180, S x, and S y. It has the same symmetry group as a rectangle. degrees we then send the orange line back into the green line. It follows that the combination of these two operations sends the green line into itself, and hence the green line is the line of reflection. It should be noted that if we reversed the order of the operations ( S x R 90 ) we would send the orange line into itself. These are special cases of the following general theorem. Theorem 3.1. Let S be a reflection about a line l passing through the origin, and R be a rotation by α degrees. Then RS is a reflection about a line l obtained by rotating l by α/2 degrees. Also, SR is a reflection about a line l obtained by rotating l by α/2 degrees. This theorem shows that if a figure has the rotation group of a square, and it has a line of reflection, then it must in fact have lines of reflection obtained by rotating this line of reflection by any multiple of 45 degrees. This explains why a figure that has the rotational symmetry of the square must have all of the lines of reflection that a square has. 4 Combining Reflections In the last section we saw that any pattern that has the rotational symmetry of a square, and some line of reflection l, must have lines of reflection inclined at multiples of 45 degrees to l. This shows that any such pattern must have all of the lines of reflection of the square. We now ask if it would be possible to have a pattern that has the rotational symmetry of the square, has all of the lines of reflection of the square and also has some additional lines of reflection. We will now see that this would be impossible. We begin by considering what happens when we combine two reflections whose lines of symmetry pass through the center of the square. As an example, suppose we reflect about the green line in Fig 4, followed by a refleciton about the horizontal line. Using our notation this is the isometry S x S +. We would like to know what this is. Using general principles we know that Since S x S + is the combination of two improper isometries, it must be a proper isometry. 4
Figure 4: If you reflect about the horizontal line, and then rotate by 90 degrees,the green line gets sent into itself. This shows that R 0 S x is a refleciton about the green line. Since both S x and S + leave the center of the square fixed, the isometry S x S + must leave the center of the square fixed. Any proper isometry that leaves a point fixed must be a rotation about that point. It follows that S x S + must be a rotation about the center of the square. Once we know that S x S + is a rotation about the center of the square, all we need to do is to track a single point and see how much it gets rotated by. We will do this by tracking the green line. Clearly the transformation S + sends the green line into itself. The transformation S x now sends the green line into the orange line. It follows that S x S + sends the green line into the orange line. That is, it is a rotation by 90 degrees. We could use a similar argument to show that S + S x is a rotation by 90 degrees. These are both specific examples of a general theorem. Theorem 4.1. Suppose you reflect about a line l and then about a line l. If α is the angle between the two lines, then this a rotation about the point of intersection of the two lines by 2α. the direction of rotation is from the first line to the second. This theorem now shows that we cannot make a figure that has rotational symmetry of a square and that has any lines of reflection other than those of the square. If it had any additional lines of reflection, then two of the lines of reflection would have to make an angle of less than 45 degrees with each other. By combining these two reflections together, we could get a rotation about the origin by less than 90 degrees. This would give us a rotation that was not in the rotation group of the square, and hence the figure would not have the same rotation group as the square. 5