by Jongsuk Keum
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.
C_{n }( Cyclic group) and D_{n}( 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.
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).
R_{0} | R_{1} | R_{2} | R_{3} | R_{4} | F_{0} | F_{1} | F_{2} | F_{3} | F_{4} | |
R_{0} | R_{0} | R_{1} | R_{2} | R_{3} | R_{4} | F_{0} | F_{1} | F_{2} | F_{3} | F_{4} |
R_{1} | R_{1} | R_{2} | R_{3} | R_{4} | R_{0} | F_{1} | F_{2} | F_{3} | F_{4} | F_{0} |
R_{2} | R_{2} | R_{3} | R_{4} | R_{0} | R_{1} | F_{2} | F_{3} | F_{4} | F_{0} | F_{1} |
R_{3} | R_{3} | R_{4} | R_{0} | R_{1} | R_{2} | F_{3} | F_{4} | F_{0} | F_{1} | F_{2} |
R_{4} | R_{4} | R_{0} | R_{1} | R_{2} | R_{3} | F_{4} | F_{0} | F_{1} | F_{2} | F_{3} |
F_{0} | F_{0} | F_{4} | F_{3} | F_{2} | F_{1} | R_{0} | R_{4} | R_{3} | R_{2} | R_{1} |
F_{1} | F_{1} | F_{0} | F_{4} | F_{3} | F_{2} | R_{1} | R_{0} | R_{4} | R_{3} | R_{2} |
F_{2} | F_{2} | F_{1} | F_{0} | F_{4} | F_{3} | R_{2} | R_{1} | R_{0} | R_{4} | R_{3} |
F_{3} | F_{3} | F_{2} | F_{1} | F_{0} | F_{4} | R_{3} | R_{2} | R_{1} | R_{0} | R_{4} |
F_{4} | F_{4} | F_{3} | F_{2} | F_{1} | F_{0} | R_{4} | R_{3} | R_{2} | R_{1} | R_{0} |
D_{5} has 10 symmetries, i.e., 5 reflections and 5 rotations;
R_{0 } -------- rotation by 0
R_{1} -------- rotation by 72
R_{2} -------- rotation by 144
R_{3} -------- rotation by 216
R_{4} -------- rotation by 288
F_{0} -------- reflection mirror 0
F_{1 } -------- reflection mirror 36
F_{2} -------- reflection mirror 72
F_{3} -------- reflection mirror 108
F_{4} -------- reflection mirror 144