Escher s Tessellations: The Symmetry of Wallpaper Patterns II Symmetry II 1/27
Brief Review of the Last Class Last time we started to talk about the symmetry of wallpaper patterns. Recall that these are pictures with translational symmetry in two directions. Escher s tessellations are great examples. We discussed that there are certain movements of a picture (viewing it as a piece of an infinite picture) which, when made, superimpose the picture upon itself. The movements we discussed are called isometries. On Monday we discussed three types of isometries: translations, rotations, and reflections. Symmetry II 2/27
Translations Symmetry II 3/27
Rotations Symmetry II 4/27
Reflections Symmetry II 5/27
This picture has rotational symmetry. We can do a quarter turn rotation (90 ) and have the picture superimpose upon itself (if we ignore color). There are also half turns (180 ). There is no reflectional symmetry. Symmetry II 6/27
This picture has reflectional symmetry. The vertical lines through the backbones of the beetles are reflection lines. Symmetry II 7/27
What symmetry can we find in this picture? Symmetry II 8/27
Clicker Question What rotational symmetry is in this picture? A Quarter turn only B Half turn only C Quarter and half turn only D None E Something else Symmetry II 9/27
What about this picture? Symmetry II 10/27
Clicker Question Besides translational, what symmetry do you see? A Rotational only B Reflectional only C Rotational and reflectional Symmetry II 11/27
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 II 12/27
Homer Rotated The Homer on the right was obtained by rotating the Homer on the left. Symmetry II 13/27
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 II 14/27
Here is another reflection of Homer. Symmetry II 15/27
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 II 16/27
Clicker Question Was the Bart on the right obtained from the Bart on the left by A a rotation? B a reflection? Symmetry II 17/27
Combining Isometries Another way to build isometries is to perform two consecutively. One example is to do a reflection followed by a translation. This is important enough to be named. It is called a glide reflection. Symmetry II 18/27
Glide Reflections Symmetry II 19/27
If we perform two isometries consecutively, using any of the four types above, the end result will again be one of the four types. Thus, any isometry is one of the four types: translations, rotations, reflections, glide reflections. Escher made heavy use of glide reflections as we will illustrate with several pictures. There are some mathematical ideas behind glide reflections that Escher had to discover in order to draw pictures demonstrating glides. Note that in the pictures below, there are glide reflections, which are built from a reflection and a translation, in which neither the reflection nor the translation is a symmetry of the picture, only the combination. Symmetry II 20/27
Symmetry II 21/27
If you reflect the picture vertically and then shift an appropriate amount, the picture will superimpose upon itself. The resulting glide reflection is a symmetry of the picture, while the vertical reflection or the translation are not symmetries of the picture. The amount of shift in the glide reflection is shown in the next picture. We can view the reflection as being along the vertical line connecting the white horsemen s chins. The symmetry in the following pictures is probably the most common in Escher s tessellations. Symmetry II 22/27
Symmetry II 23/27
This picture has the same symmetry as the previous one, in that there are translational and glide reflectional symmetry and nothing else. Symmetry II 24/27
In each of these three pictures Escher used a glide reflection starting with a vertical reflection. Symmetry II 25/27
The amount of vertical shift in the glide is exactly half of the smallest vertical translation. This can be proven mathematically, and Escher had to discover this to make his drawings. Symmetry II 26/27
Next Time On Friday we will conclude our discussion of Escher s Tessellations and 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 II 27/27