Escher s Tessellations: The Symmetry of Wallpaper Patterns Symmetry I 1/38
This week we will discuss certain types of art, 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 2D and 3D classification are very similar, and are easier to visualize in 2D. What can this mean, as there are no limit to the number of designs of wallpaper? The artist M. C. Escher created many interesting drawings of wallpaper patterns. We will use some of them to illustrate these ideas. Symmetry I 2/38
M. C. Escher Symmetry I 3/38
Maurits Cornelis Escher (1898-1972) 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. Here is a video about him and his art: The Mathematical Art of M.C. Escher Symmetry I 4/38
Let s now look at several of his tessellations. To simplify our discussion, we are going to ignore color in discussing symmetry even though Escher considered color important. Symmetry I 5/38
Horsemen Symmetry I 6/38
Lizards Symmetry I 7/38
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 8/38
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 9/38
Symmetry I 10/38
Here is another example; the four-sided figure below can be repeated over and over to fill out the picture. Symmetry I 11/38
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 12/38
There is more than one way to do this. One is to draw the square connecting the top fins of four fish. It may be hard to see that this works because we are seeing such a small part of the (infinite) picture. Symmetry I 13/38
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 s pictures. Symmetry I 14/38
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 15/38
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 16/38
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 17/38
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 18/38
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 19/38
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 20/38
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 21/38
Translations Symmetry I 22/38
Can we see translations in this picture? Symmetry I 23/38
Clicker Question Do you see translations of this picture in two different directions? A Yes B No Symmetry I 24/38
Rotations Symmetry I 25/38
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? Symmetry I 26/38
Clicker Question What rotations can you see? A Half turn only B Half and quarter turn C None Symmetry I 27/38
Reflections Symmetry I 28/38
This picture has reflectional symmetry. Where can you place a mirror and reflect the picture onto itself? There are multiple reflection lines. Symmetry I 29/38
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 30/38
Homer Rotated The Homer on the right was obtained by rotating the Homer on the left. Symmetry I 31/38
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 32/38
Here is another reflection of Homer. Symmetry I 33/38
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 34/38
Clicker 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 35/38
Solution B Since Bart s arm resting on the skateboard switches from right to left, this was accomplished by a reflection. Symmetry I 36/38
Here is another YouTube Video. This one shows lots of Escher s tessellations. Escher s Tessellations Symmetry I 37/38
Next Time On Thursday we will continue our discussion of symmetry, looking at a fourth type of isometry that Escher utilized frequently. We ll illustrate this isometry with several of his pictures. We will also discuss the classification of these pictures. We ll discuss briefly the broad mathematical ideas used to obtain the classification. We ll also see examples of all 17 symmetry types. Escher drew pictures representing 16 of the 17 symmetry types. We ll see these pictures. Symmetry I 38/38