Abstract: This paper explored the orbit trap methods for generating images with cyclic or dihedral symmetry from a dynamical system's point of view, and illustrated the mathematical principles of the construction of the symmetric orbit traps. Constructing the function with Cn or Dn symmetry, tracking the series of points corresponding to the function and introducing a new coloring algorithm derived from colormap technique, visually fascinating images with Cn or Dn symmetry can be created.

Key words: dihedral symmetry, chaos, dynamical system, cyclic symmetry, dihedral symmetry, chaos, dynamical system