Geometric Transformations and Wallpaper. Groups. Lance Drager Math Camp

Transcription

1 How to Geometric Transformations and Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 2010 Math Camp

2 Outline How to 1 How to

3 Outline How to 1 How to

4 How to to A Group is a discrete group of isometries of the plane that contains noncollinear translations. These are the symmetries of wallpaper patterns.

5 Outline How to 1 How to

6 How to The Lattice Let G be a wallpaper group. Choose a 0 in the lattice L of G so that a is a small as possible. Choose b 0 that is skew to a so that b is as small as possible. a b. Theorem The lattice L of G is L = {ma + nb m, n Z}.

7 Picture of a Lattice How to b a A Lattice

8 How to The Trace Definition Here s some standard linear algebra. If A is the matrix [ ] a b c d we define tr(a), the trace of A, by tr(a) = a + d Exercises All matrices here are If A and B are matrices, show tr(ab) = tr(ba). Just use brute force. 2 If P is an invertible matrix, show tr(p 1 AP) = tr(a).

9 How to The Crystallograhic Restriction 1 Theorem (The Crystallographic Restriction) The possible orders for a rotation in a wallpaper group are 2, 3, 4 and 6. Proof. Let R be a rotation in the point group K(G). Then RL L. This means that Ra must be in the lattice, so Ra = ma + nb for integers m, n. Similarily Rb = pa + qb for some p, q Z. These equations can be written in matrix form as [ ] m p R[a b] = [Ra Rb] = [a b]. n q Let [ ] m p P = [a b], M =, so RP = PM. n q

10 How to The Crystallograhic Restriction 2 Proof Continued. Since a and b are not colinear, P = [a b] is invertible. so R = PMP 1. But then tr(r) = tr(pmp 1 ) = tr(m) = m + q Z, i.e., the trace of R is an integer. But [ ] cos(θ) sin(θ) R = R(θ) = = tr(r) = 2 cos(θ) sin(θ) cos(θ) Thus, cos(θ) must be a half-integer. Since 1 cos(θ) 1, the possiblities are 1, 1/2, 0, 1/2, 1.

11 How to Crystallographic Restiction 3 Proof Continued. From Trig, we can figure out the corresponding angles: cos(θ) = 1 = θ = 180 = o(r) = 2, cos(θ) = 1/2 = θ = 120 = o(r) = 3, cos(θ) = 0 = θ = 90 = o(r) = 4, cos(θ) = 1/2 = θ = 60 = o(r) = 6, cos(θ) = 1 = θ = 0 = R = I.

12 How to Picture of the Angles Here s the picture of the angles. y (0, 1) ( 1/2, 3/2) (1/2, 3/2) ( 1, 0) (1, 0) x

13 Outline How to 1 How to

14 How to 1 The lattices of wallpaper groups can be divided into 5 classes. Since a ± b are skew to a, b a + b, a b. We can arrange a b a + b by replacing b by b, if necessary. Then a b a b a + b. We get 8 cases by considering = or < for each of the s above.

15 How to 2 Case Inequality Lattice 1 a = b = a b = a + b Impossible 2 a = b = a b < a + b Hexagonal 3 a = b < a b = a + b Square 4 a = b < a b < a + b Centered Rect. 5 a < b = a b = a + b Impossible 6 a < b = a b < a + b Centered Rect. 7 a < b < a b = a + b Rectangular 8 a < b < a b < a + b Oblique Let s look at the pictures.

16 How to Case 8, Oblique Lattice b a

17 How to Case 8, Oblique Lattice b a

18 How to Case 7, Rectangular Lattice b a

19 How to Case 7, Rectangular Lattice b a

20 Case 3, Square Lattice How to b a

21 How to Case 2, Hexagonal Lattice b a

22 How to Case 2, Hexagonal Lattice b a

23 How to Case 2, Hexagonal Lattice b a

24 How to Case 6, Centered Rectangular b a

25 How to Case 6, Centered Rectangular b a

26 How to Case 4, Another Centered Rectangular b a

27 How to Case 4, Another Centered Rectangular b a

28 Outline How to 1 How to

29 How to Theorem The are 17 geometric isomophism classes of wallpaper groups. Conway notation 2, 3, 4, 6 indicate a class of rotocenters of the given order. indicates a mirror. Digits after the star mean the rotocenters are on a mirror line. x indicates a glide reflection (that does not come from a pure reflection). 1 means there are only translations in the group. Crystallographic Notation (abbreviations from a longer, logical system) p or c mean a primative or centered cell in the lattice. 2, 3, 4, 6 denotes the highest order of a rotation in the group. m for mirror, stands for reflection, g stands for a glide.

30 Outline How to 1 How to

31 Flowchart for Classification How to

32 Outline How to 1 How to

33 How to the following patterns. Don t look at the answers until you ve given it a try!

34 Problem 1 How to Go to Answer

35 Problem 2 How to Go to Answer

36 Problem 3 How to Go to Answer

37 Problem 4 How to Go to Answer

38 Problem 5 How to Go to Answer

39 End of Lectures How to

40 Answer to Problem Back to Problem

41 Answer to Problem 2 Back to Problem 3 3

42 Answer to Problem Back to Problem

43 Answer to Problem 4 Back to Problem 22

44 Answer to Problem 5 Back to Problem 333