TY - JOUR
ID - 92166
TI - Calculations of Dihedral Groups Using Circular Indexation
JO - Mathematics Interdisciplinary Research
JA - MIR
LA - en
SN - 2538-3639
AU - Dianat, Reza
AU - Mogharrab, Mojgan
AD - Department of Electrical Engineering, Persian Gulf University
AD - Department of Pure Mathematics, Persian Gulf University
Y1 - 2019
PY - 2019
VL - 4
IS - 1
SP - 37
EP - 49
KW - The group of symmetries of a regular polygon (dihedral group)
KW - Permutation
KW - periodic (circular) sequences
KW - the composition of functions
DO - 10.22052/mir.2019.176359.1124
N2 - 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.
UR - http://mir.kashanu.ac.ir/article_92166.html
L1 - http://mir.kashanu.ac.ir/article_92166_1ddcfd25f7118154422f34d7fffe929c.pdf
ER -