Wallpaper Groups

Lance Drager

Discrete Groups of Geometric Transformations and Wallpaper What Makes a Discrete? Groups The Point Group Three Kinds of Groups of Geometric Patterns Classification of Rosette Groups (Discrete Groups of Isometries) The Lattice and the Point Group Frieze Groups Definition and Lattice Lance Drager Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Department of Mathematics and Statistics Groups Texas Tech University Classification Flowchart Lubbock, Texas

2010 Math Camp Wallpaper Groups Lance Drager Outline

Discrete Groups of Isometries What Makes a 1 Discrete Groups of Isometries Group Discrete? The Point Group What Makes a Group Discrete? Three Kinds of Groups Classification of The Point Group Rosette Groups The Lattice and Three Kinds of Groups the Point Group Classification of Rosette Groups Frieze Groups Definition and Lattice The Lattice and the Point Group Geometric Isomorphism Names for the Frieze Groups 2 Frieze Groups The 7 Frieze Groups Definition and Lattice Classification Flowchart Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart Wallpaper Groups Lance Drager Discrete Groups

Discrete Groups of Isometries • What Makes a Let G ⊆ E be a group of isometries. If p is a point, the Group Discrete? The Point Group orbit of p under G is Three Kinds of Groups Classification of Rosette Groups Orb(p) = {gp | g ∈ G}. The Lattice and the Point Group Frieze Groups Definition and Definition Lattice Geometric Isomorphism A group G of isometries is discrete if, for every orbit Γ of a Names for the Frieze Groups point under G, there is a δ > 0 so that The 7 Frieze Groups Classification Flowchart p, q ∈ Γ and p 6= q =⇒ dist(p, q) ≥ δ.

• D3, as the symmetries of an equilateral triangle, is discrete. The orthogonal group O is not discrete. Wallpaper Groups Lance Drager The Point Group

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Three Kinds of Definitions Groups Classification of Let G be a of isometries. We can define a group Rosette Groups The Lattice and the Point Group map Frieze Groups π : G → O:(A | a) 7→ A. Definition and Lattice Geometric The kernel N(G) of this map is the normal of G Isomorphism Names for the Frieze Groups consisting of all the translations in G. Naturally, it’s called the The 7 Frieze Groups subgroup of G. Classification Flowchart The subgroup K(G) = π(G) ⊆ O is called the Point Group of G. Note carefully that the point group K(G) of G is not a subgroup of G! Wallpaper Groups Lance Drager Three Kinds of Groups

Discrete Groups of Isometries What Makes a Group Discrete? • There are three possibilities for the translation subgroup The Point Group Three Kinds of N(G) of G. Groups Classification of 1 N(G) contains only the identity. These groups G are the Rosette Groups The Lattice and the Point Group symmetries of rosette patterns and are called rosette Frieze Groups groups. Definition and 2 All the translations in N(G) are collinear. These groups Lattice Geometric Isomorphism are the groups of frieze patterns and are called Names for the Frieze Groups frieze groups. The 7 Frieze Groups 3 N(G) contains noncollinear translations. These groups are Classification Flowchart the symmetries of wallpaper patterns and are called wallpaper groups. • We’ll describe the classification for each of these types of groups. Wallpaper Groups Lance Drager Classification of Rosette Groups

Discrete Groups of Isometries • Rosette groups are the symmetries of rosette patterns What Makes a Group Discrete? The Point Group Three Kinds of Groups Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Definition and Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart Wallpaper Groups Lance Drager Cyclic Groups

Discrete Groups of Isometries • A group is cyclic if it consists of the powers of a single What Makes a Group Discrete? element. The order of a group is the number of elements The Point Group Three Kinds of Groups in the group. Classification of Rosette Groups • An element g ∈ G has order n if n is the smallest number The Lattice and the Point Group so that g n = e. (If there’s no such n, then g has infinite Frieze Groups Definition and order). In this case the hgi has n elements, Lattice Geometric Isomorphism 2 n−1 Names for the hgi = {e, g, g ,..., g }. Frieze Groups The 7 Frieze Groups Classification • A cyclic group of order n is isomorphic to the group C Flowchart n consisting of the integers

0, 1, 2,..., n − 1

with addition mod n (clock arithmetic). Wallpaper Groups Lance Drager Classification of Rosette Groups

Discrete Groups of Isometries Theorem (Classification of Rosette Groups) What Makes a Group Discrete? The Point Group A rosette group G is either a cyclic group Cn generated by a Three Kinds of Groups rotation or is one of the dihedral groups D . Classification of n Rosette Groups The Lattice and the Point Group Outline of Proof. Frieze Groups Definition and 1 Lattice There is smallest rotation r in G, which has finite order n. Geometric Isomorphism This takes care of all the rotations, so the rotation Names for the Frieze Groups subgroup is hri ≈ Cn. The 7 Frieze Groups Classification 2 If there is no reflection, we’re done. Otherwise there is a Flowchart reflection s = S(θ) with a smallest θ.

3 If there are no rotations, G = hsi, which is D1 4 If r has order n, we have the reflections s, rs,... r n−1s and no others. The group is Dn. Show me the proof Wallpaper Groups Lance Drager The Lattice and the Point Group

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Definition Three Kinds of Groups If G is a discrete group of isometries, the lattice L of G is the Classification of Rosette Groups orbit of the origin under the translation subgroup N(G) of G. The Lattice and the Point Group To put it another way, Frieze Groups Definition and Lattice 2 Geometric L = {v ∈ R | (I | v) ∈ G} Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification • Observe that if a, b ∈ L then ma + nb ∈ L for all integers Flowchart m and n. This is because (I | a) and (I | b) are in G, so (I | a)m = (I | ma) and (I | b)n = (I | nb) are in G. Thus ma + nb = (I | ma)(I | nb)0 is in L. Wallpaper Groups Lance Drager The Point Group Preserves the

Discrete Groups of Lattice Isometries What Makes a Group Discrete? The Point Group Three Kinds of Theorem Groups Classification of Rosette Groups Let L be the lattice of G and let K = K(G) be the point group The Lattice and the Point Group of G. Then KL ⊆ L, i.e., if A ∈ K and v ∈ L then Av ∈ L. Frieze Groups Definition and Lattice Proof. Geometric Isomorphism We must have (I | v) ∈ G and (A | a) ∈ G for some a. Then Names for the Frieze Groups The 7 Frieze −1 −1 −1 Groups (A | a)(I | v)(A | a) = (A | a + Av)(A | − A a) Classification Flowchart = (AA−1 | a + Av + A(−A−1a)) = (I | Av).

Thus, Av ∈ L. Wallpaper Groups Lance Drager Frieze Groups 1

Discrete Groups of Isometries • A discrete group of isometries G is a frieze group if all the What Makes a Group Discrete? vectors in the lattice are collinear. These are the symmetry The Point Group Three Kinds of groups of frieze patterns. Groups Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Definition and Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart Wallpaper Groups Lance Drager Frieze Groups 2

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Three Kinds of Groups • A wallpaper border. Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Definition and Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart Wallpaper Groups Lance Drager The Lattice

Discrete Groups of Isometries Theorem What Makes a Group Discrete? Let G be a frieze group and let L be its lattice. Choose a ∈ L The Point Group Three Kinds of Groups so that 0 < kak is as small as possible. Such an a exists Classification of Rosette Groups because the group is discrete. Then The Lattice and the Point Group Frieze Groups L = {na | n ∈ Z}. Definition and Lattice Geometric Isomorphism Names for the Frieze Groups Proof. The 7 Frieze Groups Classification Suppose not. Then there is a c ∈ L so than c 6= na for an Flowchart n ∈ Z. But must have c = sa for some scalar s, so s ∈/ Z. Then we have s = k + r where k is an integer and 0 < r < 1. But then c = ka + ra. Since c, ka ∈ L, this implies ra = c − ka ∈ L. But 0 < krak = |r| kak < kak, which contradicts the choice of a. Wallpaper Groups Lance Drager Geometric Isomorphism

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Definition Three Kinds of Groups Classification of Two discrete groups of isometries G1 and G2 are geometrically Rosette Groups The Lattice and isomorphic if the is a group isomorphism φ: G1 → G2 that the Point Group Frieze Groups preserves the geometric type of the elements, i.e., φ(T ) always Definition and Lattice has the same type as T . Geometric Isomorphism Names for the • We classify frieze (and wallpaper groups) up to geometric Frieze Groups The 7 Frieze Groups isomorphism Classification −1 Flowchart • If T ∈ E, the conjugation map ψT : E → E: Q 7→ TQT preserves type. Hence, for any subgroup G, ψT (G) and G are geometrically isomorphic. Wallpaper Groups Lance Drager Lattice Direction and Classification

Discrete Groups of • If G is a frieze group, we can conjugate G by a rotation, Isometries What Makes a to get a geometrically isomorphic group where a points in Group Discrete? The Point Group the positive x-direction. Thus, the lattice L looks like this. Three Kinds of Groups Classification of y Rosette Groups The Lattice and a the Point Group Frieze Groups x Definition and Lattice Geometric Isomorphism • There are only four possible elements of the point group. Names for the Frieze Groups The 7 Frieze 1 The identity Groups Classification 2 Vertical reflection, i.e., reflection through the y-axis Flowchart 3 Horizontal reflection, i.e, reflection through the x-axis. 4 A half-turn (180◦) rotation. Theorem There are exactly seven geometric isomorphism classes of frieze groups. Wallpaper Groups Lance Drager Names for the Frieze Groups

Discrete Groups of Isometries What Makes a Group Discrete? • A system for naming these groups has been standardized The Point Group Three Kinds of by crystallographers. Another (better) naming system has Groups Classification of been invented by Conway, who also invented English Rosette Groups The Lattice and the Point Group names for these classes. Frieze Groups • The crystallographic names consist of two symbols. Definition and Lattice 1 First Symbol Geometric Isomorphism Names for the 1 No vertical reflection. Frieze Groups The 7 Frieze m A vertical reflection. Groups Classification 2 Second Symbol Flowchart 1 No other symmetry. m Horizontal reflection. g Glide reflection (horizontal). 2 Half-turn rotation. Wallpaper Groups Lance Drager Conway Names

Discrete Groups of Isometries • In the Conway system two rotations are of the same class What Makes a Group Discrete? if their rotocenters differ by a motion in the group. Two The Point Group Three Kinds of reflections are of the same class if their mirror lines differ Groups Classification of by a motion in the group. Rosette Groups The Lattice and the Point Group • Conway Names Frieze Groups ∞ We think of the translations as “rotation” Definition and Lattice Geometric about a center infinitely far away. There are Isomorphism Names for the two rotocenters, one in the up direction, and Frieze Groups The 7 Frieze Groups one in the down direction. These “rotations” Classification Flowchart have order ∞. 2 A class of half-turn rotations. ∗ Shows the presence of a reflection. If a 2 or ∞ comes after the ∗, then the rotocenters are at the intersection of mirror lines. x Indicates the presence of a glide reflection. Wallpaper Groups Lance Drager The 7 Frieze Groups 1

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Three Kinds of Groups Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Hop, 11, ∞∞ Definition and Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart

Jump, 1m, ∞∗ Wallpaper Groups Lance Drager The 7 Frieze Groups 2

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Three Kinds of Groups Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Definition and Sidle, m1, ∗∞∞ Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart

Step, 1g, ∞x Wallpaper Groups Lance Drager The 7 Frieze Groups 3

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Three Kinds of Groups Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Definition and Spinhop, 12, 22∞ Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart

Spinjump, mm, ∗22∞ Wallpaper Groups Lance Drager The 7 Frieze Groups 4

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Three Kinds of Groups Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Definition and Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart Spinsidle, mg, 2 ∗ ∞ Wallpaper Groups Lance Drager How to Classify Frieze Groups

Discrete Groups of Isometries What Makes a Group Discrete? The Point Group Three Kinds of Groups Classification of Rosette Groups The Lattice and the Point Group Frieze Groups Definition and Lattice Geometric Isomorphism Names for the Frieze Groups The 7 Frieze Groups Classification Flowchart Wallpaper Groups Lance Drager Proof of the Classification of

Details for The Rosette Groups 1 Classification of Rosette Groups • Let G be a rosette group. Let J be the subgroup of G consisting of rotations. • Suppose J 6= {I }. • There is a smallest θ in the range 0 < θ < 360 so that r = R(θ) ∈ G, since otherwise the group is not discrete. • Claim: There is an n ∈ N so that nθ = 360. • Suppose not. Then there is an integer k so that kθ < 360 < (k + 1)θ, so 360 = kθ + φ, where 0 < φ < θ. • This means I = R(360) = R(kθ + φ) = R(θ)k R(φ). • But then R(φ) = R(θ)−k ∈ G, where 0 < φ < θ, which contradicts the definition of θ. Wallpaper Groups Lance Drager Proof of the Classification of

Details for The Rosette Groups 2 Classification of Rosette Groups • Claim: J = {I , r, r 2,..., r n−1}. • Suppose not. Then there is some R(φ) ∈ J, where 0 < φ < 360 and φ is not an integer multiple of θ. As above, we can find an integer k so that kθ < φ < (k + 1)θ, so φ = kθ + ψ, where 0 < ψ < θ. • But then R(φ) = R(kθ + ψ) = r k R(ψ) is in G, so R(ψ) = r −k R(φ) ∈ G. Since 0 < ψ < θ, this contradicts the definition of θ. • Conclusion: J is a finite cyclic group. • Consider Reflections in G. If there a no reflections, we’re done, so suppose there is at least one reflection. There is a smallest φ is the range 0 ≤ φ < 360 so that S(φ) is in G. Otherwise the group is not discrete. Let s = S(φ). Wallpaper Groups Lance Drager Proof of the Classification of

Details for The Rosette Groups 3 Classification of Rosette Groups

• Case 1: J = {I }.

• Claim: G = {I , s}, which is D1. • Suppose not. There must be another reflection S(ψ) in G, φ < ψ < 360. But then G contains S(ψ)S(φ) = R(ψ − φ), which is a rotation that is not the identity, contradicting our assumption on J. • Case 2: J = {I , r, r 2,..., r n−1}. • G must contain the reflections s, rs,..., r n−1s. These are the reflections r k s = R(θ)k S(φ) = S(kθ + φ), where k ∈ {0, 1,..., n − 1}. Wallpaper Groups Lance Drager Proof of the Classification of

Details for The Rosette Groups 4 Classification of Rosette Groups • Claim: There are no other reflections in G. • Suppose not. There is some S(ψ) ∈ G where φ < ψ < 360, but ψ 6= kθ + φ for any k ∈ {0, 1,..., n − 1}. • But then G contains S(ψ)S(φ) = R(ψ − φ). Since 0 < ψ − φ < 360 we must have ψ − φ = kθ for some k ∈ {1, 2,..., n − 1}. This is a contradiction. • We now know that G = {I , r, r 2,..., r n−1, s, rs,..., r n−1s}. • For any angles α and β, S(α)R(β)S(α) = R(−β) = R(β)−1. −1 n−1 n−1 • Thus, srs = r = r , so sr = r s. Thus, G = Dn. • This finishes Case 2. • The proof is complete! Back to the Theorem