Ling Ling Chang

Math 107B

Midterm Project

FRIEZE GROUPS

The frieze groups are a collection of infinite symmetric groups of periodic designs in a plane. All of the frieze groups consist of translations. All frieze groups can be categorized by 7 different groups that have special symmetries. These symmetries include reflection along a horizontal axis, reflection along a vertical axis, a glide-reflection, and a rotation of 180 degrees. The following is an example of each of the frieze groups where the blue line represents the horizontal axis of reflection, the green line represents the vertical axis of reflection, the purple lines represent the glide-reflection, and the red dot represents the point where a 180 degrees rotation occurs.

Definition of a Group (from Gallian's book) : Leg G be a nonempty set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a,b) of elements of G an element in G denoted by ab. We say G is a group under this operation if the following three properties are satisfied.

1. Associativity. The operation is associative; that is, (ab)c = a(bc) for all a,b,c in G.

2. Identity. There is an element e (called the identity ) in G, such that ae=ea=a for all a in G.

3. Inverses. For each element a in G, there is an element b in G (called an inverse of a) such that ab=ba=e .

PROGRAMS

FRIEZE GROUP 1

FRIEZE GROUP 2

FRIEZE GROUP 3

FRIEZE GROUP 4

FRIEZE GROUP 5

FRIEZE GROUP 6

FRIEZE GROUP 7