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Symmetries of Geometric Patterns (Discrete Groups of Isometries)

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Symmetries of Geometric Patterns (Discrete Groups of Isometries) Wallpaper Groups Lance Drager Discrete Groups of Geometric Transformations and Wallpaper Isometries What Makes a Group Discrete? Groups The Point Group Three Kinds of Groups Symmetries 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 discrete group 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 subgroup of G Isomorphism Names for the Frieze Groups consisting of all the translations in G. Naturally, it’s called the The 7 Frieze Groups translation 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 symmetry 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 cyclic group 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.
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