different types of symmetry concrete examples of symmetry groups tilings symmetry patterns monuments/buildings/ornamental art symmetry

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1 Apart from discussing different types of symmetry this lesson will also introduce the reader to concrete examples of symmetry groups. The basic notion of tilings will also be discussed. We shall also study symmetry and patterns in monuments/buildings/ornamental art by considering examples and case studies. The connection between symmetry and M C Escher's art will also be examined. The primary emphasis will be mainly on discussing symmetry of figures lying in a plane. There will only be a brief detour towards symmetry for three dimensional objects. Introduction: The word symmetry is most commonly used to convey a sense of regularity and of an evenly balanced object, which is proportional. Beauty is in the eye of the beholder, so the saying goes. We are not claiming that `more symmetric' implies a greater sense of beauty. Indeed, often, `asymmetry' brings with it a breaking of symmetry leading to many visual delights. However most people would be able to distinguish between two sets of objects and figure out which is more symmetric. Look at the sets of two objects pictured below. View these pictures and decide which is more regular, evenly balanced and proportional or in other words which of the two objects is more symmetric. ILLL, University of Delhi 195 of 314

2 figure 1 figure 2 figure 3 figure of 314 ILLL, University of Delhi

3 While we can visually and intuitively decide on which of the two objects is more symmetric, (in each of the above sets, the image on the left is more symmetric) it would be useful to have a precise idea of symmetry. In other words, we would like to establish the mathematical definition of symmetry for given objects. All the objects displayed above are what we call `finite figures. In this lesson while we will describe the mathematical notion of symmetry for finite figures, we shall also study symmetry for `infinite patterns. We give examples of infinite `strip patterns and `wallpaper patterns. Strip patterns are also known as frieze patterns. Frieze is a term used to a band of decoration running along the wall of a room and hence the usage of frieze pattern to describe strip patterns. Since the viewing space is finite, you have to imagine that in the case of strip pattern, the pattern extends along the strip infinitely on both sides. For the wallpaper pattern, you will have to imagine that the pattern extends in all directions indefinitely.!!!!!!!!! Strip pattern 1 Strip pattern 2 R R R R R R R R R R R R Strip pattern 3 Wallpaper pattern 1 ILLL, University of Delhi 197 of 314

4 Wallpaper pattern 2 We shall first learn about symmetry of `finite figures' before considering symmetry for `infinite figures'. With a little imagination in `extending to infinity, one can observe examples of strip patterns and wallpaper patterns in buildings, old monuments and in art and architecture, but more on that in a discussion towards the end of this lesson. Symmetry of finite figures: We will for the time being settle for a `working definition' of a finite figure. For us a finite figure will be a closed and bounded figure, which occupies a finite area (if it is two dimensional) or a finite volume (if it is three dimensional). With this `working definition' it is clear that strip patterns and wallpaper patterns are not finite figures. The most common type of symmetry that people encounter in any finite object is bilateral symmetry. This occurs when the left half of the object is identical to the right half. Typically, the exterior of the human body is an example where in general there is bilateral symmetry. figure 5 Examples of bilateral symmetry 198 of 314 ILLL, University of Delhi

5 figure 6 It is clear from the images in figure 6 that they do not possess bilateral symmetry as the left half is not identical to the right half. If you fold the figure along the line drawn in the middle you will find that the two halves of the figure do not coincide. Bilateral symmetry is an example of what is called `reflective or mirror symmetry'. Before we learn about various types of symmetry we should at least have a provisional definition of symmetry of a finite figure. Definition (intuitive) symmetry: A symmetry of a finite figure is the action you can perform on the finite figure such that if someone has closed their eyes during the performance they feel as though nothing has happened to the figure. In other words, it is in the exact same place and looks like it has not been touched. Let us learn more about the different types of symmetries possible for finite figures by considering an example. Click on the hyperlink below the picture to watch a short movie on a possible symmetry of an equilateral triangle. Hyperlink: equilateral ILLL, University of Delhi 199 of 314

6 We can see that rotation by an angle of 90 degrees in the anti clockwise direction about the Z-axis, perpendicular to the surface of the equilateral triangle, passing through the centre of the triangle, is not a symmetry of the triangle. This is because the rotated triangle shown in blue (for convenience) is poking out from under the red (original) triangle. If we had closed our eyes during this procedure and only opened them after the rotation was completed we would be able to spot immediately that the triangle has changed! See the picture below. figure 7 A question we can ask is what are all possible symmetries of an equilateral triangle? As seen in the movie, a rotation of 120 degrees in the anti clockwise direction about an axis perpendicular to the surface of the equilateral triangle and passing through its centre will give us a (rotational) symmetry. Let us name this rotational symmetry as ". ROTATIONAL SYMMETRY: View the snapshots below to see what happens to the equilateral triangle on applying ". We then repeat the process by applying " again to the resulting triangle. The final image is of the original triangle having undergone a 240 degree anti clockwise rotation. In particular notice the new positions occupied by the vertices. figure of 314 ILLL, University of Delhi

7 Let us denote the `do nothing' 0 degree rotation by `I' and the 240 degree anti clockwise rotation described above by " 2 (to symbolize that it is the same as applying " twice in succession). In fact I, ", and " 2 are the only possible symmetries of an equilateral triangle that can be obtained by rotation of the equilateral triangle about (an axis through) a point. If we look at the effect of these (rotational) symmetries on the vertices of the equilateral triangle we note from figure 8 that: Rotation Vertices A#A I B#B C#C A#B " B#C C#A A#C " 2 B#A C#B Table 1 Another important point to note about the rotation ", is that if we apply " thrice then we have actually done nothing! That is, applying " thrice is equivalent to the `identity' rotation I or the `do nothing' 0 degree rotation. Since three is the least natural number such that applying " three times results in I, we say that the rotation " has order 3. Can you find the orders of the rotations I and " 2? (Q1.) Another property that we can notice about the non-identity rotations (" and " 2 ) is that only one point remains fixed in the equilateral triangle when these rotations are applied. This is the point in the centre of the equilateral triangle through which the axis of rotation (perpendicular to the surface of the triangle) passes. This fixed point is called rotocenter. As a convention we will consider rotations in the anti clockwise direction only. We could have decided on a convention of clockwise too. However it is important to stick to just one type, which for us shall be anti clockwise. It is clear from Table 1 that the rotational symmetries permute the vertices amongst themselves. Indeed, any symmetry of an equilateral triangle has to preserve the distance between any two points of the triangle. Hence every symmetry of an equilateral triangle has to permute the vertices of the triangle amongst themselves. ILLL, University of Delhi 201 of 314

8 REFLECTION OR MIRROR SYMMETRY: What are other possible symmetries of an equilateral triangle? Does it have bilateral symmetry? The answer to the second question is yes. See the pictures below. figure 9 The picture on the left shows the shaded portion obtained by reflecting the left half of equilateral triangle ABC along the line La. Since it matches with the right half we have a bilateral symmetry. We would get the same effect if we place a mirror along the line La. Alternatively if we folded the triangle along the line La then the two halves would coincide or match entirely. The picture on the right shows the actual effect of reflection along the line La. All points on the line La remain fixed. But points not on the line exchange places with their respective counterparts in the other half. Thus the vertex A remains fixed while the vertices B and C get exchanged. Note that to find the counterpart of a point P, we draw a perpendicular line from the point P to the line La and extend it to an equal distance on the other side. The point where the line ends now, say P, is the counterpart of the point P. Thus a `reflection along line La is `a function, which takes P to P and vice-versa. See the picture below illustrating this. figure of 314 ILLL, University of Delhi

9 From this example we can see that reflection takes place along a line. All points on that line of reflection remain fixed. In non-identity rotations, as seen earlier, there is only one fixed point namely the point through which the axis of rotation passes. This fixed point is called the rotocenter. This is the main difference between a reflection and a rotation for finite figures lying in a plane. Are there any other reflection or mirror symmetries of an equilateral triangle? The answer is yes. The pictures below show two more reflection symmetries that are possible. The effect of the reflection on the vertices is shown in each of the pictures on the right. figure 11 Let us give names to the reflections we have seen above. Let us call reflection along line La as! a, the reflection along line Lb as! b, and the reflection along line Lc as! c. The table below gives the effect that these reflection symmetries have on the vertices of the equilateral triangle. What happens if you apply a reflection twice? (Q2.) As we did in the case of the rotations, can we define the order of a reflection? (Q3.) Reflection! a! b! c Vertices A#A B#C C#B A#C B#B C#A A#B B#A C#C Table 2 From Table 2, it is clear that each reflection fixes exactly one vertex, namely the vertex through which the line of reflection passes and just interchanges the other two vertices. The lines La, Lb and Lc are also called lines of symmetry or axes of symmetry for the equilateral triangle ABC. The three reflections described above are the only ones possible for an equilateral triangle. So we have described up to now six different symmetries of an equilateral triangle. These are the three rotations I, ", and " 2 and the three reflections! a,! b, and! c. From the lesson on permutations you will learn that there are only 6 ILLL, University of Delhi 203 of 314

10 permutations possible of the letters A, B, C. Since any symmetry permutes the vertices it has to be one of the six possible permutations and so there can only be at most six symmetries of an equilateral triangle. But we have produced six symmetries already. Thus the collection I,!,! 2, " a, " b, and " c is the complete set of symmetries for an equilateral triangle. (Ref 1.) Once again we can see intuitively that since a symmetry leaves the finite figure looking exactly as it was, every symmetry of a finite figure has to preserve the distance between any two points of the finite figure. Using this we can infer that the only possible symmetries that can arise for a finite figure in a plane are reflections and rotations. Let us do a quick recap of the facts that we have learnt about I Symmetries of finite figures: 1. A symmetry of a finite figure is an action that leaves the figure looking unchanged and in exactly the same position. 2. There are only two types of symmetries possible for finite figures in a plane: reflections and rotations. 3. A reflection of a planar finite figure leaves all the points on the line of reflection or line of symmetry fixed.! Definition symmetry type D n : We say that a planar finite figure X has symmetry type D n if its (group of) symmetries consists of 2n elements of which exactly n are rotations and n are reflections. There are planar finite figures, which have a different symmetry type from D n. For example consider the picture of the Hindu Swastika given below. figure 17!

11 ! It is not difficult to see that this figure has a rotocenter at the point of intersection of its four arms but there are no lines of symmetry. Thus the symmetries of the above figure are only rotations. Indeed, we can have 0, 90, 180 and 270 degree rotations in the anti clockwise direction about the rotocenter. If we denote the 0 degree do nothing rotation by I and the 90 degree rotation by #, then we can describe the 180 degree rotation by # 2 and the 270 degree rotation by # 3. Note that # 4 = I and that # has order four. If we denote the above figure by X, then Sym(X) = {I, #, # 2, # 3 } and this group Sym(X) consisting of exactly four rotations and no reflections is called the cyclic group of order four and is denoted as C 4. Definition symmetry type C n : We say that a planar finite figure X has symmetry type C n if its (group of) symmetries consists of n rotations and no reflections. It can be proved mathematically that for any finite planar figure its symmetry type is always either C n or D n for some natural number n.! SYMMETRIES OF A STRIP OR FRIEZE PATTERN: Recall the examples that we gave of strip patterns. How are strip patterns created? There is a basic tile or motif, which is then repeated along a line at fixed intervals. The result is a strip or frieze pattern Consider strip pattern 2 shown below. Recall that we have to assume that the pattern extends infinitely along both sides. Is it possible to have a reflection symmetry for this strip pattern? What about rotational symmetries? Strip pattern 2 Since the strip pattern extends infinitely on both sides if we reflect the pattern along a line!!!!!!!!

12 situated vertically in the middle of the space between any two faces we would have a symmetry. Similarly if we draw a vertical line that divides any face into left and right half, then reflection along this line would also leave the strip pattern looking exactly the same. Both are instances of reflective symmetry or mirror symmetry. See the picture below. The thick vertical line and the dotted line show us the two lines of (vertical) reflective symmetry mentioned above. Also note that there are infinitely many such lines of symmetry for Strip pattern 2. figure 18 The above strip pattern does not have any rotational symmetry or horizontal mirror symmetry. The Strip patterns 1 and 3 do not have any reflection symmetry or rotational symmetry. There are Strip patterns that have rotational symmetries or horizontal mirror symmetries. Some examples are given below. In the strip pattern shown below, the rotocenters are marked with small red circles. They are not part of the strip pattern. Further rotation by 180 degrees about an axis through any rotocenter and perpendicular to the plane of the pattern will leave the Strip pattern 4 looking unchanged. Strip pattern 4 Note that the basic motif that is repeated is.the rotocenters between the two `R s of the basic motif is an indication that the motif itself was created by a 180 degree rotation of one of the R s. Another `type of rotocenters has been marked between any two basic motifs. The next example is of a strip pattern with horizontal mirror symmetry but no rotational or vertical mirror symmetry. See the figure below. The line of reflection is shown by the dotted line. If we fold the strip along the dashed line then the two halves, upper and lower will coincide. Strip pattern 5 Note that there are examples of strip patterns that have none, some or all of the following ILLL, University of Delhi 211 of 314

13 types of symmetries: vertical mirror symmetry, horizontal mirror symmetry and rotational mirror symmetry. For example the strip pattern shown below has all the types of symmetries referred to above. The lines of symmetry as well as the red rotocenters are also shown in the figure below. Strip pattern 6 Is there any other type of symmetry that we can associate with a strip pattern? The answer is yes. There is another symmetry called translation symmetry. In fact, we will use this type of symmetry to define finite figures, strip patterns and wallpaper patterns. TRANSLATION SYMMETRY: Consider Strip Pattern 1. We saw that this strip pattern did not have any mirror symmetries or rotational symmetries. However if we shift this strip to the right by a distance t as shown in the figure below we will leave the strip looking exactly the same.!! t!!!!!!! figure 19 Definition translation: A translation of a planar or a three dimensional figure X, is an action (function) that moves all the points in X by the same amount of distance in the same direction. In the above strip pattern, the direction and basic distance moved, is shown by an arrow and the unit t. If we denote the strip pattern in figure 19 by X, then we can see that moving every point of X to the right by a distance t as shown will be a bijective function from X to X that preserves distances. Hence it is a symmetry of X. As a convention we shall consider moving the strip a distance t to the right as the positive direction. With this convention t 2 will mean moving the strip to the right as shown by a distance t once and then repeating the action again. Similarly t -1 will stand for moving the strip in the opposite left direction by a distance t. See the figure below of Strip pattern 4 to see an illustration of these conventions. 212 of 314 ILLL, University of Delhi

14 t 2 t t -1 figure 20 An important point to note is that a finite figure cannot have a non-trivial translation symmetry. (Note that the `do nothing symmetry can also thought of as a translation where t = 0.) When we perform a translation we move the given figure by a fixed distance in a prescribed direction. But then the finite figure no longer occupies the original position that it did. Thus a translation of a finite figure does not leave the finite figure looking exactly as it did in the exact same position. So a translation of a finite figure cannot be a symmetry. Indeed this important fact can also be used as a definition for a `finite figure. Definition finite figure: A finite figure is a shape that has no non-trivial translation symmetry. Definition strip pattern: A strip pattern is a pattern that has translation symmetry in only one direction. Strip patterns will always have translation symmetry by definition. They can have mirror symmetry (vertical and horizontal) and rotational symmetry. The `do-nothing symmetry will always be a symmetry of any strip pattern. It is considered both as a rotational (zero degree rotation) and as a translation (t = 0) symmetry. Can a strip pattern that has mirror symmetries have lines of symmetry other than vertical or horizontal lines? (Q 11.) Can there be a non-identity rotation for a strip pattern different from a! turn (that is, a 180 degree rotation)? (Q 12.) Once again for any given strip pattern X, let Sym(X) denote the set of symmetries of X. If we apply symmetries one after the other on Strip pattern X, the net effect is the same as another symmetry of the Strip pattern X. Further `combining symmetries in this manner follows the rules of closure, associativity, existence of identity and existence of inverses. Thus Sym(X) will form a group. Since by definition a strip pattern has translation symmetry in one direction, it is not difficult to see that Sym(X) will have infinitely many members. (Why?) For Strip pattern 5, can you list a sequence consisting of infinitely many symmetries? (Q 13.) ILLL, University of Delhi 213 of 314

15 We look at a case study on the Taj Mahal. In the exercises we will have questions on analysing symmetries of some temples from the southern part of India. The idea is to look for instances of symmetry that abound in these structures. One point to be kept in mind is that in real life the examples of symmetry may not fit our definitions exactly. We may also need to ignore some parts and only look at certain portions to be able to discuss notions of symmetry. In the examples we shall try to trace or identify examples of reflection, rotation, translation, and glide reflection. We shall also try to identify strip patterns, wallpaper patterns and tilings. The pictures used in the case study of the Taj Mahal are either from the website or are pictures which are in the public domain. This website gives a wonderful virtual tour of the Taj Mahal. The pictures from the above mentioned website have been downloaded by using the link on the website titled `downloadable assets for schools and non-commercial use. The Taj Mahal: figure 34 In the above picture of the Taj Mahal the dashed line is a line of reflective symmetry. (You have to ignore the plants and the stone seat in the picture.) Similar instances of bilateral symmetry can be seen in the following monuments in the Taj Mahal complex: ILLL, University of Delhi 227 of 314

16 figure 35 (Mosque) figure 36 (Taj gateway with garden) The next figure that follows is a floor plan of the Taj Mahal. Let us consider these for instances of symmetry. figure 37 (Floor plan of Taj Mahal) 228 of 314 ILLL, University of Delhi

17 We will examine the above figure through its different layers. The two rectangles in the interior represent tombs and we will have to ignore them to make sense of the various symmetries present. For the rest of this discussion assume that the two rectangles are invisible. The tombs are surrounded by two octagonal structures. Below is an actual picture of the cenotaph and its tombs taken from above. The inner octagonal jali wall is clearly visible. figure 38 (Octagonal cenotaph, Taj Mahal) Recall symmetries of finite figures. An octagon is a regular polygon with eight sides will have symmetry type D 8. Its group of symmetries will contain eight reflections and eight rotations. Four lines of symmetry will be the lines joining mid points of opposite sides and the other four will be lines joining diagonally opposite vertices. See the diagram below. figure 39 ILLL, University of Delhi 229 of 314

18 Further, let! represents a rotation of 45 degrees in the anticlockwise direction about an axis perpendicular to the plane of the octagon and passing through the blue rotocenter. Then the eight rotations are given by: I,!,! 2,! 3,! 4,! 5,! 6 and! 7. Let us go back to the floor plan in figure 37. What else can we observe? The series of pictures below show what happens to the floor plan on anticlockwise rotation by 0 degrees, 90 degrees, 180 degrees and 270 degrees about the centre. Note the position of the rectangular tombs is the only indication that such a rotation has been performed. If they are ignored then we have four rotational symmetries. figure 40 Apart from rotations the entire floor plan (with tombs ignored) also has four reflections as shown in the picture below. figure 41 Finally, let us consider just one corner of the floor plan. (Consider only the shaded blue areas.) Here we get only one mirror symmetry. The line of reflection is shown in the figure below. 230 of 314 ILLL, University of Delhi

19 figure 42 Now let us turn our attention to the motifs, decorative borders and Jali work that is ubiquitous in the Taj Mahal complex. figure 43 (motif in mosque, Taj Mahal) Let us concentrate on the writings in the central portion as shown in the figure below. In order to look for rotational symmetry we will again have to ignore the writing in the very centre of the figure. figure 44 The series of eight snapshots below show the eight rotational symmetries of the above figure. Only the writing in the very centre is an indication that such rotations have taken place. ILLL, University of Delhi 231 of 314

20 figure 45 The next example is a decorative border. These borders (if continued indefinitely along both sides) give us examples of strip patterns. See the figure below consisting of the basic motif, the frieze pattern with some lines of symmetry, a basic translation and some rotocenters marked. figure 46 t The figure below is the same strip in figure 46 but after a half turn or a 180 degree rotation. There are some minor differences which we have ignored can you spot them? 232 of 314 ILLL, University of Delhi

21 (Q 22.) (We do not mean lines of symmetry or rotocenters here!) Does the strip pattern in figure 46 have a glide reflection? figure 47 It should be mentioned here that common examples of frieze patterns can be found in cloth borders or in saris, in balcony grills, stairways and in nature amongst many other instances. figure 47 Below we have some examples of periodic tilings from floor patterns in the Taj Mahal. figure 48 ILLL, University of Delhi 233 of 314

22 MAURITS CORNELIS ESCHER AND SYMMETRY: M C Escher figure 49: Self-portrait (c. MCE) In the discussion that follows we shall introduce the reader to a few of Escher s wonderful artwork and its connections to symmetry. All of Escher s artwork are copyrighted and we permission to use at most five of his works in our lesson. You can see most of his artwork in the picture gallery section of Some other sites that do give a good account of Escher s life and work are and The following page in the second website titled Escher on display has several pictures on how Escher s art has been used to decorate buildings. Without much ado let us begin with an analysis of some of Escher s artwork. We begin with an artwork titled Möbius Strip 1 made in The series of three snap shots shows the figure as it is, under a 120 degree rotation about its centre and finally a 240 degree rotation. Only the square shape of the frame shows that such a rotation has been done. figure 50 (c. MCE) 234 of 314 ILLL, University of Delhi

23 We had learnt that any finite figure has only two symmetry types C n or D n. Does Mobius Strip I have any reflections? (Q 23.) Which category does Möbius Strip I fall in? (Q 24.) The next work is titled Regular Division-65. Again the series of snapshots below show the four rotational symmetries. Are there any reflections in this case? (Q 25.) What is the symmetry type of this artwork? (Q 26.) figure 51 (c. MCE) Around 1936, Escher visited Alhambra, a Moorish palace and fortress in southern Spain. The use of geometry in the decorations in the palace inspired him. It is believed that with careful analysis (distinguishing colours) one can find examples of all 17 wallpaper patterns in Alhambra. Below is an example of a tessellation from Alhambra. figure 52 Mathematics continued to influence Escher s art and in the 1950s Escher wrote several papers on describing how he achieved tessellations using planar figures by a method called the `regular division of the plane. His metamorphosis series in particular showed the actual process of how shapes changed from one figure to the other. The next piece that we analyse is titled Regular Division-70 (Butterflies). The original picture is shown below and then we have put together four copies of the picture to show how it can be used to cover the plane. ILLL, University of Delhi 235 of 314

24 horizontal and vertical translations are marked. Are there any lines of reflection in the above pattern? (Q 28.) A few words in the end: In the end, what is true is that symmetry is all around us in both the physical world and the temporal realm. This is evident from the few examples given below out of an `unending list. Enjoy the `hexagonal grid of a honeycomb built by bees and the symmetries inherent in flowers, starfish, in a carpet made for Shah Jahan in 1646, in paintings and artwork, in temples and totem poles. Find a picture of a snowflake. Is there symmetry? (Q 29.) Honeycomb Flowers Starfish Shah Jahan s Carpet (c. MMA) Vitruvian Man by Leonardo Da Vinci ILLL, University of Delhi 237 of 314

25 Vitthala temple in Hampi Totem Pole 238 of 314 ILLL, University of Delhi