Editing Dihedral group (section)

===Elements===
[[File:Hexagon reflections.svg|thumb|The six axes of [[reflection symmetry|reflection]] of a regular hexagon]]
A regular polygon with <math>n</math> sides has <math>2n</math> different symmetries: <math>n</math> [[rotational symmetry|rotational symmetries]] and <math>n</math> [[reflection symmetry|reflection symmetries]]. Usually, we take <math>n \ge 3</math> here. The associated [[rotation]]s and [[reflection (mathematics)|reflections]] make up the dihedral group <math>\mathrm{D}_n</math>. If <math>n</math> is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If <math>n</math> is even, there are <math>n/2</math> axes of symmetry connecting the midpoints of opposite sides and <math>n/2</math> axes of symmetry connecting opposite vertices. In either case, there are <math>n</math> axes of symmetry and <math>2n</math> elements in the symmetry group.<ref>{{citation
 | last = Cameron | first = Peter Jephson
 | title = Introduction to Algebra
 | publisher = Oxford University Press
 | year = 1998
 | isbn = 9780198501954
 | page = 95
 | url = https://books.google.com/books?id=syYYl-NVM5IC&pg=PA95
}}</ref> Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.<ref>{{citation
 | last = Toth | first = Gabor
 | title = Glimpses of Algebra and Geometry
 | series = Undergraduate Texts in Mathematics
 | edition = 2nd
 | publisher = Springer
 | year = 2006
 | isbn = 9780387224558
 | page = 98
 | url = https://books.google.com/books?id=IRwBCAAAQBAJ&pg=PA98
}}</ref>

[[File:Dihedral8.png|550px|thumb|center|This picture shows the effect of the sixteen elements of <math>\mathrm{D}_8</math> on a [[stop sign]]. Here, the first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left.]]