University of KashanMathematics Interdisciplinary Research2538-36394120190601Calculations of Dihedral Groups Using Circular Indexation37499216610.22052/mir.2019.176359.1124ENReza DianatDepartment of Electrical Engineering, Persian Gulf University0000-0003-1447-1383Mojgan MogharrabDepartment of Pure Mathematics, Persian Gulf UniversityJournal Article20190316In this work, a regular polygon with <em>n</em> sides is described by a periodic (circular) sequence with period <em>n</em>. 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 2π /<em>n</em> 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.https://mir.kashanu.ac.ir/article_92166_1ddcfd25f7118154422f34d7fffe929c.pdf