Escher s Tessellations: The Symmetry of Wallpaper Patterns. 30 January 2012
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1 Escher s Tessellations: The Symmetry of Wallpaper Patterns 30 January 2012 Symmetry I 30 January /32
2 This week we will discuss certain types of drawings, called wallpaper patterns, and how mathematicians classify them through an analysis of their symmetry. About 100 years ago, it was shown that there are only 17 different symmetry types of a wallpaper pattern. Symmetry I 30 January /32
3 This week we will discuss certain types of drawings, called wallpaper patterns, and how mathematicians classify them through an analysis of their symmetry. About 100 years ago, it was shown that there are only 17 different symmetry types of a wallpaper pattern. This classification was first done in three dimensions, when crystallographers were studying how symmetry determined chemical properties of crystals. The mathematical ideas in the two-dimensional and three-dimensional classification are very similar, and are easier to visualize in two dimensions. Symmetry I 30 January /32
4 This week we will discuss certain types of drawings, called wallpaper patterns, and how mathematicians classify them through an analysis of their symmetry. About 100 years ago, it was shown that there are only 17 different symmetry types of a wallpaper pattern. This classification was first done in three dimensions, when crystallographers were studying how symmetry determined chemical properties of crystals. The mathematical ideas in the two-dimensional and three-dimensional classification are very similar, and are easier to visualize in two dimensions. What can this mean, as there are no limit to the number of designs of wallpaper? Symmetry I 30 January /32
5 This week we will discuss certain types of drawings, called wallpaper patterns, and how mathematicians classify them through an analysis of their symmetry. About 100 years ago, it was shown that there are only 17 different symmetry types of a wallpaper pattern. This classification was first done in three dimensions, when crystallographers were studying how symmetry determined chemical properties of crystals. The mathematical ideas in the two-dimensional and three-dimensional classification are very similar, and are easier to visualize in two dimensions. What can this mean, as there are no limit to the number of designs of wallpaper? The artist M. C. Escher created many very interesting drawings of wallpaper patterns. We will use some of them to illustrate the ideas we will discuss. Symmetry I 30 January /32
6 M. C. Escher Symmetry I 30 January /32
7 Maurits Cornelis Escher ( ) is best known for his mathematically oriented art, including his tessellations. He was not trained in mathematics, and has commented that he neither was a mathematician nor even that he knew much mathematics. However, he had to learn a considerable amount of mathematics in order to produce his tessellations. He even came up with his own classification of the possible wallpaper patterns, which is more detailed than the one we will discuss. We are going to ignore color in discussing symmetry, while Escher considered color important. Let s look at some of his art. Symmetry I 30 January /32
8 Horsemen Symmetry I 30 January /32
9 Lizards Symmetry I 30 January /32
10 How are these two pictures similar? How are they different? Escher viewed these as pieces of pictures which go on forever in two directions. We will use this viewpoint. Symmetry I 30 January /32
11 How are these two pictures similar? How are they different? Escher viewed these as pieces of pictures which go on forever in two directions. We will use this viewpoint. One thing different about these pictures is that the first has no rotational symmetry while the second does. The first has some sort of reflectional symmetry while the second one does not. Symmetry I 30 January /32
12 How are these two pictures similar? How are they different? Escher viewed these as pieces of pictures which go on forever in two directions. We will use this viewpoint. One thing different about these pictures is that the first has no rotational symmetry while the second does. The first has some sort of reflectional symmetry while the second one does not. Symmetry I 30 January /32
13 One thing common to these pictures is that the picture is built from drawing a piece of the picture, and then repeating that piece by shifting it horizontally and vertically. Symmetry I 30 January /32
14 One thing common to these pictures is that the picture is built from drawing a piece of the picture, and then repeating that piece by shifting it horizontally and vertically. The following picture shows a piece which, when shifting it appropriately, creates the entire picture. Symmetry I 30 January /32
15 Symmetry I 30 January /32
16 Clicker Question Can you find a piece of the picture when shifting it repeatedly will produce the full picture? Imagine the picture going on forever. A Yes B No Symmetry I 30 January /32
17 There is more than one way to do this. One is to draw the four sided figure connecting the top fins of four fish. It may be hard to see that this works because are seeing such a small part of the (infinite) picture. Symmetry I 30 January /32
18 Here is another example; the four-sided figure can be repeated over and over to fill out the picture. Symmetry I 30 January /32
19 Symmetry of a Picture To develop further some sense of the idea of symmetry, let s look at a series of somewhat less professional pictures before we return to Escher pictures. Symmetry I 30 January /32
20 First Example Symmetry I 30 January /32
21 First Example While these are clearly two different pictures, they have the same symmetry. In both cases we can translate the picture horizontally and vertically by appropriate amounts and have the picture superimposed upon itself. Again, think of these pictures as a piece of an infinite picture. Symmetry I 30 January /32
22 Second Example Symmetry I 30 January /32
23 Second Example Besides translational symmetry, each of these pictures has rotational symmetry. We can rotate each by 180 degrees and have the picture superimposed upon itself. Again, these two pictures have the same symmetry. Symmetry I 30 January /32
24 Third Example Symmetry I 30 January /32
25 Third Example These two pictures do not have the same symmetry. Both have translational symmetry in two directions. However, the first has no rotational symmetry while the second does. Symmetry I 30 January /32
26 Fourth Example Symmetry I 30 January /32
27 Fourth Example These two also do not have the same symmetry, since the second has reflectional symmetry while the first does not. We can reflect the second across a vertical mirror placed appropriately to have the picture superimposed upon itself. Symmetry I 30 January /32
28 We have focused only on pictures which have translational symmetry in two directions, and will continue to do so. These pictures are the so-called wallpaper patterns. Symmetry I 30 January /32
29 We have focused only on pictures which have translational symmetry in two directions, and will continue to do so. These pictures are the so-called wallpaper patterns. In order to quantify the notion of symmetry, mathematicians associate to such a picture a collection of objects to which we refer as isometries. Symmetry I 30 January /32
30 Isometries The notion of isometry is a formalization of the high school notion of congruence. Two geometric shapes are congruent if one can be moved to be exactly superimposed upon the other. Symmetry I 30 January /32
31 Isometries The notion of isometry is a formalization of the high school notion of congruence. Two geometric shapes are congruent if one can be moved to be exactly superimposed upon the other. Symmetry I 30 January /32
32 More formally, two shapes are congruent if there is an isometry which moves one exactly onto the other. Symmetry I 30 January /32
33 More formally, two shapes are congruent if there is an isometry which moves one exactly onto the other. There are three basic types of isometries of the plane: translations, rotations, and reflections. Symmetry I 30 January /32
34 Translations Symmetry I 30 January /32
35 Can we see translations in this picture? Symmetry I 30 January /32
36 Rotations Symmetry I 30 January /32
37 This picture has rotational symmetry. About what points can you rotate, and by how much of a full turn, and rotate the picture onto itself? There are ways to do a quarter turn, and ways to do a half turn. Symmetry I 30 January /32
38 Reflections Symmetry I 30 January /32
39 This picture has reflectional symmetry. Where can you place a mirror and reflect the picture onto itself? There are multiple reflection lines. Symmetry I 30 January /32
40 Rotations versus Reflections Sometimes it is difficult to distinguish between rotations and reflections. One way to distinguish them is that reflections switch orientation; that is, right and left are switched. Rotations do not switch orientation. Symmetry I 30 January /32
41 Rotations versus Reflections Sometimes it is difficult to distinguish between rotations and reflections. One way to distinguish them is that reflections switch orientation; that is, right and left are switched. Rotations do not switch orientation. Think about looking into a mirror. If you hold something in your right hand, in the mirror it looks like you are holding it in your left hand. Symmetry I 30 January /32
42 Homer Rotated The Homer on the right was obtained by rotating the Homer on the left. Symmetry I 30 January /32
43 Homer Reflected The Homer on the right was obtained by reflecting the Homer on the left. The program I used also made Homer look upside down. Symmetry I 30 January /32
44 Here is another reflection of Homer. Symmetry I 30 January /32
45 In the original and rotated images, Homer is holding the donut in his right hand. In each of the reflected images, he is holding the donut in his left hand. Original and Rotation Original and Reflection Symmetry I 30 January /32
46 Quiz Question Was the Bart on the right obtained from the Bart on the left by A a rotation? B a reflection? C a translation? D none of the above? Symmetry I 30 January /32
47 Next Time On Wednesday we will continue our discussion of symmetry and look at a fourth type of isometry that Escher utilized a lot. We ll illustrate this isometry with several of his pictures. Symmetry I 30 January /32
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