Escher s Tessellations: The Symmetry of Wallpaper Patterns III. Symmetry III

Transcription

1 Escher s Tessellations: The Symmetry of Wallpaper Patterns III Symmetry III 1/20

2 In the past two classes we saw many examples of Escher s tessellations, and different combinations of symmetry. We will talk briefly about the broad mathematical structures used to classify wallpaper patterns, and we ll see pictorial examples of the different symmetry types which are possible. Symmetry III 2/20

3 Different Combinations of Symmetry One can have rotational symmetry (180 ) along with glide reflectional symmetry. Symmetry III 3/20

4 One can also have rotational symmetry (120, one third turn) but no reflectional or glide reflectional symmetry. Symmetry III 4/20

5 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 III 5/20

6 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 III 6/20

7 The following two picture indicates that performing a vertical reflection followed by a horizontal reflection results in a 180 degree rotation. Symmetry III 7/20

8 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 III 8/20

9 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 III 9/20

10 In arithmetic, addition satisfies: The associative property - e.g., 3 + (2 + 5) = (3 + 2) + 5. An identity 0 - e.g., = 3. Additive inverses - e.g., 8 + ( 8) = 0. Symmetry III 10/20

11 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 III 11/20

12 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 III 12/20

13 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., = 3 + 2), while the former does not. To illustrate this, we consider a vertical reflection and a quarter turn (counterclockwise) rotation. Symmetry III 13/20

14 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 III 14/20

15 Clicker Question If you perform a 90 degree rotation counterclockwise followed by a reflection across a horizontal line on the following figure, which figure do you get? A B Symmetry III 15/20

16 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 III 16/20

17 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 III 17/20

18 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 III 18/20

19 Next Week We will begin a three week discussion of probability. Next week will focus on a little history of the subject and some of the principal ideas. In particular, we ll conduct some probability experiments to help understand the meaning of probability and to see some common misconceptions. In the second week we ll discuss poker and several aspects of how probability comes up in poker. In the final week we ll discuss various games of the New Mexico Lottery and see how likely it is that you d win when playing those games. Symmetry III 19/20

20 Assignment 3 due next Friday Do one of the following three activities (assignment on the website): 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 III 20/20