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In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections.^{[1]} Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
See also: Dihedral symmetry in three dimensions.
Contents
Notation
There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted D_{n}, while in algebra the same group is denoted by D_{2n} to indicate the number of elements.
In this article, D_{n} (and sometimes Dih_{n}) refers to the symmetries of a regular polygon with n sides.
Definition
Elements
A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. The associated rotations and reflections make up the dihedral group D_{n}. If n is odd each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is even there are n/2 axes of symmetry connecting the midpoints of opposite sides and n/2 axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether and 2n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes. The following picture shows the effect of the sixteen elements of D_{8} on a stop sign:
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