11) Wallpaper Patterns

Question 1

What is a lattice

Answer 1

Question 2

What is the rank of a lattice

Answer 2

The rank of a lattice is k

Question 3

What is the rank of

Answer 3

Question 4

What are the symmetries of a lattice

Answer 4

Question 5

What rotation is always a symmetry of a lattice Λ, and why

Answer 5

Question 6

What translation and linear transformation are symmetries of a lattice Λ

Answer 6

Question 7

Describe the proof of which translations and linear transformations are symmetries of a lattice

Answer 7

Question 8

What is a short generator pair for Λ

Answer 8

Question 9

When are the elements of a short geneator pair greater than 0

Answer 9

Question 10

What are the five types of of lattice Λ

Answer 10

Question 11

When are the symmetry group for two lattices isomorphic

Answer 11

Question 12

What homomorphism is defined for isometries of the form f(x) =Ax+b in E(2)

Answer 12

Question 13

What is the subgroup of translations and the point group of a group G≤E(2)

Answer 13

Question 14

What are the possible point groups of a lattice

Answer 14

Question 15

What is a wallpaper group

Answer 15

Question 16

How many wall paper groups are there

Answer 16

There are precisely 17 types of wallpaper groups

Question 17

What is the quotient of a group G acting on a set X

Answer 17

The quotient is the set X/G ={Orb(x) | x ∈ X}, where each element of X/G is a subset of X, namely an orbit

Question 18

What is a fundamental domain

Answer 18

Let G be a group acting on a set X. A fundamental domain is a subset Y ⊆ X such that for all x ∈ X there exists a unique y ∈ Y such that y ∈ Orb(x)

Question 19

What is the relationship between a fundamental domain and the quotient space in group actions

Answer 19

Question 20

Describe the proof that π : Y → X/G is a bijection

Answer 20

Question 21

What are fundamental domains for the action of a lattice Λ on R^2 by translations

Answer 21