Editing Dihedral group (section)

==The dihedral group as symmetry group in 2D and rotation group in 3D==

An example of abstract group {{math|D{{sub|''n''}}}}, and a common way to visualize it, is the group of [[Euclidean plane isometry|Euclidean plane isometries]] which keep the origin fixed. These groups form one of the two series of discrete [[point groups in two dimensions]]. {{math|D{{sub|''n''}}}} consists of {{math|''n''}} [[rotation]]s of multiples of {{math|360°/''n''}} about the origin, and [[Reflection (mathematics)|reflection]]s across {{math|''n''}} lines through the origin, making angles of multiples of {{math|180°/''n''}} with each other. This is the [[symmetry group]] of a [[regular polygon]] with {{math|''n''}} sides (for {{math|''n'' ≥ 3}}; this extends to the cases {{math|''n'' {{=}} 1}} and {{math|''n'' {{=}} 2}} where we have a plane with respectively a point offset from the "center" of the "1-gon" and a "2-gon" or line segment).

{{math|D{{sub|''n''}}}} is [[generating set of a group|generated]] by a rotation {{math|r}} of [[order (group theory)|order]] {{math|''n''}} and a reflection {{math|s}} of order 2 such that

:<math>\mathrm{srs} = \mathrm{r}^{-1} \, </math>

In geometric terms: in the mirror a rotation looks like an inverse rotation.

In terms of [[complex numbers]]: multiplication by <math>e^{2\pi i \over n}</math> and [[complex conjugation]].

In matrix form, by setting

:<math>
  \mathrm{r}_1 = \begin{bmatrix}
    \cos{2\pi \over n} & -\sin{2\pi \over n} \\[4pt]
    \sin{2\pi \over n} &  \cos{2\pi \over n}
  \end{bmatrix}\qquad
  \mathrm{s}_0 = \begin{bmatrix}
    1 & 0 \\
    0 & -1
  \end{bmatrix}
</math>

and defining <math>\mathrm{r}_j = \mathrm{r}_1^j</math> and <math>\mathrm{s}_j = \mathrm{r}_j \, \mathrm{s}_0</math> for <math>j \in \{1,\ldots,n-1\}</math> we can write the product rules for D<sub>''n''</sub> as

:<math>\begin{align}
  \mathrm{r}_j \, \mathrm{r}_k &= \mathrm{r}_{(j+k) \text{ mod }n} \\
  \mathrm{r}_j \, \mathrm{s}_k &= \mathrm{s}_{(j+k) \text{ mod }n} \\
  \mathrm{s}_j \, \mathrm{r}_k &= \mathrm{s}_{(j-k) \text{ mod }n} \\
  \mathrm{s}_j \, \mathrm{s}_k &= \mathrm{r}_{(j-k) \text{ mod }n}
\end{align}</math>

(Compare [[coordinate rotations and reflections]].)

The dihedral group D<sub>2</sub> is generated by the rotation r of 180 degrees, and the reflection s across the ''x''-axis. The elements of D<sub>2</sub> can then be represented as {e,&nbsp;r,&nbsp;s,&nbsp;rs}, where e is the identity or null transformation and rs is the reflection across the ''y''-axis.

[[File:Dihedral4.png|thumb|500px|The four elements of D<sub>2</sub> (x-axis is vertical here)]]

D<sub>2</sub> is [[group isomorphism|isomorphic]] to the [[Klein four-group]].

For ''n'' > 2 the operations of rotation and reflection in general do not [[commutative|commute]] and D<sub>''n''</sub> is not [[abelian group|abelian]]; for example, in [[Dihedral group of order 8|D<sub>4</sub>]], a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees.

[[File:d8isNonAbelian.png|thumb|500px|D<sub>4</sub> is nonabelian (x-axis is vertical here).]]

Thus, beyond their obvious application to problems of [[symmetry]] in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.

The {{math|2''n''}} elements of {{math|D{{sub|''n''}}}} can be written as {{math|e}}, {{math|r}}, {{math|r{{sup|2}}}}, ... , {{math|r{{sup|''n''−1}}}}, {{math|s}}, {{math|r s}}, {{math|r{{sup|2}}s}}, ... ,  {{math|r{{sup|''n''−1}}s}}. The first {{math|''n''}} listed elements are rotations and the remaining {{math|''n''}} elements are axis-reflections (all of which have order&nbsp;2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection.

So far, we have considered {{math|D{{sub|''n''}}}} to be a [[subgroup]] of {{math|[[orthogonal group|O(2)]]}}, i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation {{math|D{{sub|''n''}}}} is also used for a subgroup of [[SO(3)]] which is also of abstract group type {{math|D{{sub|''n''}}}}: the [[symmetry group|proper symmetry group]] of a ''regular polygon embedded in three-dimensional space'' (if ''n'' ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore, it is also called a ''dihedron'' (Greek: solid with two faces), which explains the name ''dihedral group'' (in analogy to ''tetrahedral'', ''octahedral'' and ''icosahedral group'', referring to the proper symmetry groups of a regular [[tetrahedron]], [[octahedron]], and [[icosahedron]] respectively).

===Examples of 2D dihedral symmetry===
<gallery widths="180">
File:Imperial Seal of Japan.svg|2D D<sub>16</sub> symmetry &ndash; Imperial Seal of Japan, representing eightfold [[chrysanthemum]] with sixteen [[petal]]s.
File:Red Star of David.svg|2D D<sub>6</sub> symmetry &ndash; [[Magen David Adom|The Red Star of David]]
File:Naval Jack of the Republic of China.svg|2D D<sub>12</sub> symmetry — The Naval Jack of the Republic of China (White Sun)
File:Ashoka Chakra.svg|2D D<sub>24</sub> symmetry &ndash; [[Ashoka Chakra]], as depicted on the [[Flag of India|National flag of the Republic of India]]. 
</gallery>