Tiling a Flat Torus Stephen Hetterich, Corey Manack April 7, 2013 Important: wallpaper spans the entire plane! What is a wallpaper? A wallpaper, A, is a subset of R2; (C), whose translation subgroup LA is a rank 2 discrete subgroup of R2. What is a wallpaper? A wallpaper, A, is a subset of R2; (C), whose translation subgroup LA is a rank 2 discrete subgroup of R2. Important: wallpaper spans the entire plane! Wallpaper: Not a wallpaper: What is a wallpaper? Examples: Not a wallpaper: What is a wallpaper? Examples: Wallpaper: What is a wallpaper? Examples: Wallpaper: Not a wallpaper: Let FA be the set of all isometries of A. FA is called a wallpaper group. LA ⊂ FA Two wallpapers are equivalent if their corresponding wallpaper groups are isomorphic. 17 different wallpaper groups in total. What is a wallpaper? Examples: Now, assume A is a wallpaper. FA is called a wallpaper group. LA ⊂ FA Two wallpapers are equivalent if their corresponding wallpaper groups are isomorphic. 17 different wallpaper groups in total. What is a wallpaper? Examples: Now, assume A is a wallpaper. Let FA be the set of all isometries of A. LA ⊂ FA Two wallpapers are equivalent if their corresponding wallpaper groups are isomorphic. 17 different wallpaper groups in total. What is a wallpaper? Examples: Now, assume A is a wallpaper. Let FA be the set of all isometries of A. FA is called a wallpaper group. 17 different wallpaper groups in total. What is a wallpaper? Examples: Now, assume A is a wallpaper. Let FA be the set of all isometries of A. FA is called a wallpaper group. LA ⊂ FA Two wallpapers are equivalent if their corresponding wallpaper groups are isomorphic. What is a wallpaper? Examples: Now, assume A is a wallpaper. Let FA be the set of all isometries of A. FA is called a wallpaper group. LA ⊂ FA Two wallpapers are equivalent if their corresponding wallpaper groups are isomorphic. 17 different wallpaper groups in total. Wallpaper groups Wallpaper groups Wallpaper groups In other words, L is the set of all integer linear combinations of two linearly independent vectors ~ ~ l1; l2. Given a wallpaper A, LA can be identified with some lattice L (for example, identify a translational ~ ~ symmetry in the direction of l1 with l1 itself). We will use LA and L interchangeably. What is a lattice? A lattice L is a (rank 2) discrete additive subgroup of R2 (or C). Given a wallpaper A, LA can be identified with some lattice L (for example, identify a translational ~ ~ symmetry in the direction of l1 with l1 itself). We will use LA and L interchangeably. What is a lattice? A lattice L is a (rank 2) discrete additive subgroup of R2 (or C). In other words, L is the set of all integer linear combinations of two linearly independent vectors ~ ~ l1; l2. We will use LA and L interchangeably. What is a lattice? A lattice L is a (rank 2) discrete additive subgroup of R2 (or C). In other words, L is the set of all integer linear combinations of two linearly independent vectors ~ ~ l1; l2. Given a wallpaper A, LA can be identified with some lattice L (for example, identify a translational ~ ~ symmetry in the direction of l1 with l1 itself). What is a lattice? A lattice L is a (rank 2) discrete additive subgroup of R2 (or C). In other words, L is the set of all integer linear combinations of two linearly independent vectors ~ ~ l1; l2. Given a wallpaper A, LA can be identified with some lattice L (for example, identify a translational ~ ~ symmetry in the direction of l1 with l1 itself). We will use LA and L interchangeably. Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Several descriptions of T Description 1 - The lattice Λ itself. So we can make calculations. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 So we can look at pretty pictures. What is a flat torus? Several descriptions of T Description 1 - The lattice Λ itself. So we can make calculations. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 So we can look at pretty pictures. What is a flat torus? Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Description 1 - The lattice Λ itself. So we can make calculations. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 So we can look at pretty pictures. What is a flat torus? Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Several descriptions of T So we can make calculations. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 So we can look at pretty pictures. What is a flat torus? Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Several descriptions of T Description 1 - The lattice Λ itself. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 So we can look at pretty pictures. What is a flat torus? Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Several descriptions of T Description 1 - The lattice Λ itself. So we can make calculations. Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 So we can look at pretty pictures. What is a flat torus? Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Several descriptions of T Description 1 - The lattice Λ itself. So we can make calculations. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) So we can look at pretty pictures. What is a flat torus? Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Several descriptions of T Description 1 - The lattice Λ itself. So we can make calculations. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 What is a flat torus? Definition: T = R2=Λ, meaning T is the quotient of R2 by some lattice Λ =< v~1; v~2 >,. Several descriptions of T Description 1 - The lattice Λ itself. So we can make calculations. Description 2 - Schematically: A parallelogram with opposite edges identified (donut shape, cut and laid flat.) Description 3: The fundamental domain F: the convex hull of v~1; v~2; v~1 + v~2 So we can look at pretty pictures. A wallpaper tiles T if translations of F \ A by Λ recover A. (A \ F) + Λ = A Think of Λ as a set of translational symmetries of the plane. BIG Thm. Up to symmetry, A tiles T iff Λ is a rank 2 subgroup of LA. Nutshell. Given a wallpaper A, and a torus T = R2=Λ, we wish to know if the wallpaper “fits nicely” on the surface of the flat torus. (A \ F) + Λ = A Think of Λ as a set of translational symmetries of the plane. BIG Thm. Up to symmetry, A tiles T iff Λ is a rank 2 subgroup of LA. Nutshell. Given a wallpaper A, and a torus T = R2=Λ, we wish to know if the wallpaper “fits nicely” on the surface of the flat torus. A wallpaper tiles T if translations of F \ A by Λ recover A. BIG Thm. Up to symmetry, A tiles T iff Λ is a rank 2 subgroup of LA. Nutshell. Given a wallpaper A, and a torus T = R2=Λ, we wish to know if the wallpaper “fits nicely” on the surface of the flat torus. A wallpaper tiles T if translations of F \ A by Λ recover A. (A \ F) + Λ = A Think of Λ as a set of translational symmetries of the plane. Nutshell. Given a wallpaper A, and a torus T = R2=Λ, we wish to know if the wallpaper “fits nicely” on the surface of the flat torus. A wallpaper tiles T if translations of F \ A by Λ recover A. (A \ F) + Λ = A Think of Λ as a set of translational symmetries of the plane. BIG Thm. Up to symmetry, A tiles T iff Λ is a rank 2 subgroup of LA. Look at p2 vs p6m p2 tiles square T , but p6m does not tile T . Nutshell Look at p2 vs p6m p2 tiles square T , but p6m does not tile T . Nutshell Nutshell Look at p2 vs p6m p2 tiles square T , but p6m does not tile T . Answer: BIG Thm. Whenever Λ ⊂ LA. Counting: Each wallpaper A has a lattice LA =< l1; l2 > The parallelogram P spanned by l1; l2 captures the smallest piece of wallpaper that can be translated (via LA) to cover the entire plane.