Calculations of Dihedral Groups Using Circular Indexation

Abstract

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and generates a processed circular output sequence‎. ‎The system can be described by a permutation function‎. ‎Permutation functions can be written in a simple form using circular indexation‎. ‎The operation between the symmetries of the polygon is reduced to the composition of permutation functions‎, ‎which in turn is easily implemented using periodic sequences‎. ‎It is also shown that each symmetry is effectively a pure rotation or a pure flip‎. ‎It is also explained how to synthesize each symmetry using two generating symmetries‎: ‎time-reversal (flipping around a fixed symmetry axis) and unit-delay (rotation around the center-point by $2pi‎ /‎n$ radians clockwise)‎. ‎The group of the symmetries of a polygon is called a dihedral group and it has applications in different engineering fields including image processing‎, ‎error correction codes in telecommunication engineering‎, ‎remote sensing, and radar‎.