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Wallpaper Groups Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Geometric Transformations and Wallpaper Groups Wallpaper Groups Lance Drager Department of Mathematics and Statistics Texas Tech University Lubbock, Texas 2010 Math Camp Wallpaper Groups Outline Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 1 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Wallpaper Groups Outline Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 1 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Wallpaper Groups Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Introduction to Wallpaper Groups • A Wallpaper Group is a discrete group of isometries of the plane that contains noncollinear translations. These are the symmetries of wallpaper patterns. Wallpaper Groups Outline Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 1 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Wallpaper Groups The Lattice Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify • Let G be a wallpaper group. • Choose a 6= 0 in the lattice L of G so that kak is a small as possible. • Choose b 6= 0 that is skew to a so that kbk is as small as possible. • kak ≤ kbk. Theorem The lattice L of G is L = {ma + nb | m, n ∈ Z}. Wallpaper Groups Lance Drager Picture of a Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a A Lattice Wallpaper Groups The Trace Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 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 2 × 2. 1 2 If A and B are matrices, show tr(AB) = tr(BA). Just use brute force. If P is an invertible matrix, show tr(P −1 AP) = tr(A). Wallpaper Groups The Crystallograhic Restriction 1 Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 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 P = [a | b], m p , M= n q so RP = PM. Wallpaper Groups Lance Drager The Crystallograhic Restriction 2 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 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. Wallpaper Groups Lance Drager Crystallographic Restiction 3 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Proof Continued. From Trig, we can figure out the corresponding angles: cos(θ) = −1 cos(θ) = −1/2 cos(θ) = 0 cos(θ) = 1/2 cos(θ) = 1 =⇒ =⇒ =⇒ =⇒ =⇒ θ θ θ θ θ = 180◦ = 120◦ = 90◦ = 60◦ =0 =⇒ =⇒ =⇒ =⇒ =⇒ o(R) = 2, o(R) = 3, o(R) = 4, o(R) = 6, R = I. Wallpaper Groups Picture of the Angles Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify • Here’s the picture of the angles. y (0, 1) √ (−1/2, 3/2) √ (1/2, 3/2) 90◦ 120◦ 60◦ (−1, 0) 180 ◦ 0 ◦ (1, 0) x Wallpaper Groups Outline Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 1 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Wallpaper Groups Classification of Lattices 1 Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify • The lattices of wallpaper groups can be divided into 5 classes. • Since a ± b are skew to a, kbk ≤ ka + bk, ka − bk. • We can arrange ka − bk ≤ ka + bk by replacing b by −b, if necessary. Then kak ≤ kbk ≤ ka − bk ≤ ka + bk. • We get 8 cases by considering = or < for each of the ≤’s above. Wallpaper Groups Classification of Lattices 2 Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Case 1 2 3 4 5 6 7 8 Inequality kak = kbk = ka − bk = ka + bk kak = kbk = ka − bk < ka + bk kak = kbk < ka − bk = ka + bk kak = kbk < ka − bk < ka + bk kak < kbk = ka − bk = ka + bk kak < kbk = ka − bk < ka + bk kak < kbk < ka − bk = ka + bk kak < kbk < ka − bk < ka + bk • Let’s look at the pictures. Lattice Impossible Hexagonal Square Centered Rect. Impossible Centered Rect. Rectangular Oblique Wallpaper Groups Lance Drager Case 8, Oblique Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 8, Oblique Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 7, Rectangular Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 7, Rectangular Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 3, Square Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 2, Hexagonal Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 2, Hexagonal Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 2, Hexagonal Lattice Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 6, Centered Rectangular Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Case 6, Centered Rectangular Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify b a Wallpaper Groups Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Case 4, Another Centered Rectangular b a Wallpaper Groups Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Case 4, Another Centered Rectangular b a Wallpaper Groups Outline Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 1 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Wallpaper Groups Names for the Wallpaper Groups Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 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. Wallpaper Groups Outline Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 1 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Wallpaper Groups Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Flowchart for Classification Wallpaper Groups Outline Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify 1 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Wallpaper Groups Lance Drager Patterns to Classify Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify • Classify the following patterns. Don’t look at the answers until you’ve given it a try! Wallpaper Groups Wallpaper Problem 1 Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Go to Answer Wallpaper Groups Lance Drager Wallpaper Problem 2 Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Go to Answer Wallpaper Groups Wallpaper Problem 3 Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Go to Answer Wallpaper Groups Wallpaper Problem 4 Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Go to Answer Wallpaper Groups Wallpaper Problem 5 Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify Go to Answer Wallpaper Groups Lance Drager Wallpaper Groups Introduction The Structure of Wallpaper Groups Classification of Lattices Names for the Wallpaper Groups How to Classify Wallpaper Groups Patterns to Classify End of Lectures Wallpaper Groups Answer to Wallpaper Problem 1 Lance Drager ∗632 Back to Problem Wallpaper Groups Lance Drager Answer to Wallpaper Problem 2 Back to Problem 3∗3 Wallpaper Groups Answer to Wallpaper Problem 3 Lance Drager ∗2222 Back to Problem Wallpaper Groups Answer to Wallpaper Problem 4 Lance Drager 22∗ Back to Problem Wallpaper Groups Lance Drager Answer to Wallpaper Problem 5 Back to Problem ∗333