geometry software

with Kali (Java) software from Jeff Weeks you can explore

frieze patterns - the 7 types

wallpaper patterns - the 17 types (some shown here)

and rosette (rotational symmetry) patterns

the options for the three types of symmetry sets are in easy to use menus 

illuminations NCTM: symmetry

cyclic figures: rotational - vary the original shape for different orders of symmetry

dihedral figures: line - change the start shape and see what happens with several mirror lines

shape tool : Pattern blocks - put shapes together and then reflect or rotate them