Escher s Tessellations: The Symmetry of Wallpaper Patterns II 1 February 2012 Symmetry II 1 February 2012 1/32
Brief Review of Monday s 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. Symmetry II 1 February 2012 2/32
Brief Review of Monday s 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. Symmetry II 1 February 2012 2/32
Brief Review of Monday s 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 1 February 2012 2/32
Translations Symmetry II 1 February 2012 3/32
Rotations Symmetry II 1 February 2012 4/32
Reflections Symmetry II 1 February 2012 5/32
Symmetry II 1 February 2012 6/32
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 1 February 2012 6/32
Symmetry II 1 February 2012 7/32
This picture has reflectional symmetry. The vertical lines through the backbones of the beetles are reflection lines. Symmetry II 1 February 2012 7/32
What symmetry can we find in this picture? Symmetry II 1 February 2012 8/32
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 1 February 2012 9/32
What about this picture? Symmetry II 1 February 2012 10/32
Clicker Question Besides translational, what symmetry do you see? A Rotational only B Reflectional only C Rotational and reflectional Symmetry II 1 February 2012 11/32
Performing More Than One Isometry 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 1 February 2012 12/32
Glide Reflections Symmetry II 1 February 2012 13/32
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. Symmetry II 1 February 2012 14/32
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. Symmetry II 1 February 2012 14/32
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 1 February 2012 14/32
Symmetry II 1 February 2012 15/32
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. Symmetry II 1 February 2012 16/32
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. Symmetry II 1 February 2012 16/32
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 1 February 2012 16/32
Symmetry II 1 February 2012 17/32
Symmetry II 1 February 2012 18/32
This picture has the same symmetry as the previous one, in that there are translational and glide reflectional symmetry and nothing else. Symmetry II 1 February 2012 18/32
Symmetry II 1 February 2012 19/32
In each of these three pictures Escher used a glide reflection starting with a vertical reflection. Symmetry II 1 February 2012 19/32
Symmetry II 1 February 2012 20/32
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 1 February 2012 20/32
There were lots of other things Escher had to discover in order to make his drawings. Let s look at the following picture. This picture has rotational symmetry. Symmetry II 1 February 2012 21/32
There is a pattern of rotation centers; that is, points about which we can rotate and have the picture superimpose upon itself. Symmetry II 1 February 2012 22/32
The green points are centers of 90 degree rotations and the magenta points are centers of 180 degree rotations. It is no coincidence that the magenta points are halfway along the sides of the square. Symmetry II 1 February 2012 23/32
No matter how you draw the picture, if it has the same symmetry as this picture, it must have the same pattern of rotation points. Escher had to discover this. Symmetry II 1 February 2012 24/32
Recall this picture. We ll draw a diagram to represent translations in two directions and reflection lines. Symmetry II 1 February 2012 25/32
The yellow lines represent smallest possible translations. The red lines are reflection lines. Corners of the yellow square, and the center are rotation centers for quarter turn rotations. Midpoints of the yellow lines are half turn rotation centers. Symmetry II 1 February 2012 26/32
Different Combinations of Symmetry Symmetry II 1 February 2012 27/32
Different Combinations of Symmetry One can have rotational symmetry (180 ) along with glide reflectional symmetry. Symmetry II 1 February 2012 27/32
Symmetry II 1 February 2012 28/32
One can also have rotational symmetry (120, one third turn) but no reflectional or glide reflectional symmetry. Symmetry II 1 February 2012 28/32
Symmetry II 1 February 2012 29/32
It is also possible to have rotational symmetry and reflectional (rather than glide reflectional) symmetry. Escher drew this picture with reflectional symmetry in two perpendicular directions. Doing so forces the picture to have 180 degree rotational symmetry. Symmetry II 1 February 2012 29/32
The following two picture indicates that performing a vertical reflection followed by a horizontal reflection results in a 180 degree rotation. Symmetry II 1 February 2012 30/32
Quiz Question What type of symmetry, besides translational, does this picture have? A Only rotational symmetry B Only reflectional symmetry C Rotational and reflectional Symmetry II 1 February 2012 31/32
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 1 February 2012 32/32