Escher s Tessellations: The Symmetry of Wallpaper Patterns 27 January 2014 Symmetry I 27 January 2014 1/30
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 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 27 January 2014 2/30
M. C. Escher Symmetry I 27 January 2014 3/30
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. We are going to ignore color in discussing symmetry, while Escher considered color important. This will simplify our discussion. Let s look at some of his art. Symmetry I 27 January 2014 4/30
Horsemen Symmetry I 27 January 2014 5/30
Lizards Symmetry I 27 January 2014 6/30
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 27 January 2014 7/30
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 27 January 2014 8/30
Symmetry I 27 January 2014 9/30
Here is another example; the four-sided figure below can be repeated over and over to fill out the picture. Symmetry I 27 January 2014 10/30
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 27 January 2014 11/30
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 we are seeing such a small part of the (infinite) picture. Symmetry I 27 January 2014 12/30
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 27 January 2014 13/30
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 27 January 2014 14/30
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 27 January 2014 15/30
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 27 January 2014 16/30
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 27 January 2014 17/30
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 27 January 2014 18/30
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 27 January 2014 19/30
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 27 January 2014 20/30
Translations Symmetry I 27 January 2014 21/30
Can we see translations in this picture? Symmetry I 27 January 2014 22/30
Clicker Question Do you see translations of this picture in two different directions? A Yes B No Symmetry I 27 January 2014 23/30
Rotations Symmetry I 27 January 2014 24/30
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 27 January 2014 25/30
Clicker Question What rotations can you see? A Half turn only B Half and quarter turn C None Symmetry I 27 January 2014 26/30
Reflections Symmetry I 27 January 2014 27/30
This picture has reflectional symmetry. Where can you place a mirror and reflect the picture onto itself? There are multiple reflection lines. Symmetry I 27 January 2014 28/30
Next Time On Wednesday we will continue our discussion of symmetry, compare and contrast rotations and reflections, and look at a fourth type of isometry that Escher utilized frequently. We ll illustrate this isometry with several of his pictures. Symmetry I 27 January 2014 29/30