Day 7 : Symmetry Patterns in the Plane and Symmetry Group

by Jongsuk Keum

Students explore symmetry patterns in the plane using GSP, and make  figures using GSP.

Symmetry patterns in the plane

Every pattern P in the plane has one of the following types:

Asymmetric pattern: The identity is the only symmetry of P.

Bilateral pattern: The only symmetries of P are the identity and reflection with mirror L.
Example: A human's picture is almost bilateral symmetry.

Radial pattern(Wheel pattern): The only possible symmetries of P are the identity, rotations with center C, and reflections with mirror through C.
Cn ( Cyclic group) and Dn( Dihedaral group) are only possible infinite classes.

Strip pattern:  There are seven types of strip patterns.

Wallpaper pattern: There are seventeen types of wallpaper patterns. 

Symmetry Group of a Plane Pattern P

The set of symmetries of a pattern P is a group because the symmetries of a plane pattern P have the following four group axioms:

1. Identity: The identity isometry is a symmetry of P.

2. Product ( closed ): If the isometries S and T are symmetries of P, then their product isometry ST is a symmetry of P.

3. Inverse: If the isometry S is a symmetry of P, then the inverse isometry S-1 is a symmetry of P.

4. Associativity: If the isometries S, T, U are symmetries of P, then (ST)U = S(TU). 

Multiplication table for D5

R0 R1 R2 R3 R4 F0 F1 F2 F3 F4
R0 R0 R1 R2 R3 R4 F0 F1 F2 F3 F4
R1 R1 R2 R3 R4 R0 F1 F2 F3 F4 F0
R2 R2 R3 R4 R0 R1 F2 F3 F4 F0 F1
R3 R3 R4 R0 R1 R2 F3 F4 F0 F1 F2
R4 R4 R0 R1 R2 R3 F4 F0 F1 F2 F3
F0 F0 F4 F3 F2 F1 R0 R4 R3 R2 R1
F1 F1 F0 F4 F3 F2 R1 R0 R4 R3 R2
F2 F2 F1 F0 F4 F3 R2 R1 R0 R4 R3
F3 F3 F2 F1 F0 F4 R3 R2 R1 R0 R4
F4 F4 F3 F2 F1 F0 R4 R3 R2 R1 R0

D5  has 10 symmetries, i.e., 5 reflections and 5 rotations;

R -------- rotation by 0
R1  -------- rotation by 72
R2  -------- rotation by 144
R3  -------- rotation by 216
R4  -------- rotation by 288
F0  -------- reflection  mirror 0
F --------  reflection  mirror 36
F2  -------- reflection  mirror 72
F3  -------- reflection  mirror 108
F4  -------- reflection  mirror 144

Making a figure with D5 symmetry using GSP

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