Escher s Tessellations: The Symmetry of Wallpaper Patterns. 30 January PDF Free Download

Escher s Tessellations: The Symmetry of Wallpaper Patterns 30 January 2012 Symmetry I 30 January 2012 1/32

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 2012 2/32

M. C. Escher Symmetry I 30 January 2012 3/32

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. Let s look at some of his art. Symmetry I 30 January 2012 4/32

Horsemen Symmetry I 30 January 2012 5/32

Lizards Symmetry I 30 January 2012 6/32

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 2012 7/32

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 2012 7/32

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 2012 8/32

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 2012 8/32

Symmetry I 30 January 2012 9/32

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 2012 10/32

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 2012 11/32

Here is another example; the four-sided figure can be repeated over and over to fill out the picture. Symmetry I 30 January 2012 12/32

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 2012 13/32

First Example Symmetry I 30 January 2012 14/32

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 2012 14/32

Second Example Symmetry I 30 January 2012 15/32

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 2012 15/32

Third Example Symmetry I 30 January 2012 16/32

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 2012 16/32

Fourth Example Symmetry I 30 January 2012 17/32

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 2012 17/32

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 2012 18/32

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 2012 18/32

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 2012 19/32

More formally, two shapes are congruent if there is an isometry which moves one exactly onto the other. Symmetry I 30 January 2012 20/32

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 2012 20/32

Translations Symmetry I 30 January 2012 21/32

Can we see translations in this picture? Symmetry I 30 January 2012 22/32

Rotations Symmetry I 30 January 2012 22/32

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 2012 23/32

Reflections Symmetry I 30 January 2012 24/32

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 2012 25/32

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 2012 26/32

Homer Rotated The Homer on the right was obtained by rotating the Homer on the left. Symmetry I 30 January 2012 27/32

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 2012 28/32

Here is another reflection of Homer. Symmetry I 30 January 2012 29/32

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 2012 30/32

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 2012 31/32

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 2012 32/32

Escher s Tessellations: The Symmetry of Wallpaper Patterns. 30 January 2012