Escher s Tessellations: The Symmetry of Wallpaper Patterns II Symmetry II 1/38
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 2/38
Translations Symmetry II 3/38
Rotations Symmetry II 4/38
Reflections Symmetry II 5/38
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 6/38
This picture has reflectional symmetry. The vertical lines through the backbones of the beetles are reflection lines. Symmetry II 7/38
What symmetry can we find in this picture? Symmetry II 8/38
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 9/38
What about this picture? Symmetry II 10/38
Clicker Question Besides translational, what symmetry do you see? A Rotational only B Reflectional only C Rotational and reflectional Symmetry II 11/38
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 12/38
Glide Reflections Symmetry II 13/38
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 14/38
Symmetry II 15/38
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 16/38
Symmetry II 17/38
This picture has the same symmetry as the previous one, in that there are translational and glide reflectional symmetry and nothing else. Symmetry II 18/38
In each of these three pictures Escher used a glide reflection starting with a vertical reflection. Symmetry II 19/38
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 20/38
Different Combinations of Symmetry One can have rotational symmetry (180 ) along with glide reflectional symmetry. Symmetry II 21/38
One can also have rotational symmetry (120, one third turn) but no reflectional or glide reflectional symmetry. Symmetry II 22/38
Clicker Question What kind of symmetry does this picture have, besides translational? A Rotational only B Reflectional only C Rotational and reflectional D None Symmetry II 23/38
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 24/38
The following two picture indicates that performing a vertical reflection followed by a horizontal reflection results in a 180 degree rotation. Symmetry II 25/38
To each wallpaper pattern one can consider the collection of isometries which, when applied to the picture, superimposes it exactly onto itself. To understand a situation, mathematicians often look for some sort of structure on collections of objects rather than working just with the individual object. For example, the collection of numbers has the operation of addition, which takes two numbers and adds them, producing a third number. Symmetry II 26/38
The collection of isometries has the property that two isometries can be combined, or composed, by performing one, then the other, producing a third isometry. A glide reflection is an example of two isometries being composed. We get it by performing a reflection followed by a translation. As we just saw, if we compose a horizontal and a vertical reflection, we get a 180 degree rotation. Symmetry II 27/38
In arithmetic, addition satisfies: The associative property - e.g., 3 + (2 + 5) = (3 + 2) + 5. An identity 0 - e.g., 3 + 0 = 3. Additive inverses - e.g., 8 + ( 8) = 0. Symmetry II 28/38
Composing isometries also satisfy the same three properties. The analogue of 0 is the no motion, or identity, isometry. Also, each isometry has an inverse which, when performed after the original, results in no motion at all. For example, the inverse of a rotation by 90 degrees counterclockwise is a rotation by 90 degrees clockwise. A reflection is its own inverse. That is, performing a reflection twice accomplishes the same thing as no motion at all. Symmetry II 29/38
Group Theory Group theory studies collections of objects together with an operation which satisfies the same three properties mentioned above which addition satisfies. Group theory originated in the early 19th century through the work of Galois, who introduced the concept in order to study solutions of polynomial equations. Symmetry II 30/38
The collection of isometries associated to a wallpaper pattern is a group. There is one important difference between the group of isometries and the group of numbers with addition. The latter satisfies the commutative property (e.g., 2 + 3 = 3 + 2), while the former does not. To illustrate this, we consider a vertical reflection and a quarter turn (counterclockwise) rotation. Symmetry II 31/38
The series in blue results from doing a vertical reflection followed by a 90 degree rotation. The series in red results from performing the two isometries in the opposite order. Since the results are different, the order in which isometries are performed matters. Symmetry II 32/38
Clicker Question If you perform a 90 degree rotation counterclockwise followed by a reflection across a horizontal line on the figure below and to the left, which figure do you get? A B Symmetry II 33/38
Classification of Wallpaper Patterns It is through the study of groups of isometries that the classification of all possible types of symmetry of wallpaper patterns was made. It was discovered that there are exactly 17 different types of symmetry in wallpaper patterns, by seeing that there are 17 different groups of isometries. This was completed by Fedorov, Schoenflies, and Barlow at the end of the 19th century. Escher discovered the classification on his own. He drew pictures for 16 of the 17 symmetry types. Symmetry II 34/38
You can find a PDF file showing pictures of all 17 symmetry types, including Escher drawings for 16 of them, at the course website by clicking on Handouts, and then clicking on 17 Wallpapers.pdf. You can also find a webpage, also available from the Handouts link, titled Wallpaper Patterns. We will look at the 17 symmetry types now. Symmetry II 35/38
Crystallographers, interested in the chemical properties of crystals, studied their symmetry, and in the late 19th century classified the types of symmetry of crystals. They found that there are 230 different symmetry types. This is the 3-dimensional analogue of the classification of wallpaper patterns. Group theory has been used in encryption, coding theory, quantum mechanics, and crystallography, among other areas. Symmetry II 36/38
Next Week Next Tuesday we will continue our discussion of art by investigating fractals. On Thursday we will discuss encoding data in a way to be able to detect and correct errors. This is necessary for producing CDs and DVDs that can play without interruption when there is some dirt or a scratch on the disc. It is also necessary for producing hard drives that work even when there are small imperfections in the drive, which occur over time. Symmetry II 37/38
Assignment 2 due next Friday Variant 1: Draw a tessellation starting with a square (or rectangle) by following the instructions in the link Tessellations from Squares. Also, determine if the resulting tessellation has rotational, reflectional, and/or glide reflectional symmetry. Variant 2: Draw a tessellation starting with a triangle by following the instructions in the link Tessellations from Triangles Also, determine if the resulting tessellation has rotational, reflectional, and/or glide reflectional symmetry. Variant 3: Look over the pictures of the 17 symmetry types in 17 Wallpapers.pdf. Give a plausible reason why Escher did not draw a picture for the symmetry type on the third page of that handout (which is numbered page 59), when he drew (often many) pictures for the other 16 symmetry types? Give some rationale for your opinion. Symmetry II 38/38