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 Image Title*: Upload a Math Image Each snowflake in the main image has the dihedral symmetry of a natual regular hexagon. The group formed by these symmetries is also called the dihedral group of degree 6. '''Order''' refers to the number of elements in the group, and '''degree''' refers to the number of the sides or the number of rotations. The order is twice the degree. In mathematics, a '''dihedral group''' is the group of symmetries of a regular polygon, including both '''rotations''' and '''reflections'''. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. Dihedral groups arise frequently in art and nature. Many of the decorative designs used on floor coverings, pottery, and buildings have one of the dihedral groups of symmetry. Chrysler’s logo has $D_5$ as a symmetry group, and that Mercedes-Benz has $D_3$. The ubiquitous five-pointed star has symmetry group $D_5$. ==Notation== There are two different kinds of notation for a dihedral group associated to a polygon with $n$ sides. In geometry, we usually call it $D_n$ or $Dih_n$, where $n$ indicates the number of the sides. In algebra, we call it $D_2n$, where $2n$ indicates the number of elements in the group. On this page, we will use the notation $D_n$ to describe a dihedral group. For $D_n$, we will call it ''the dihedral group of order $2n$'' or ''the group of symmetries of a regular $n$-gon.'' Below is an example of Dihedral symmetry of $D_3, D_4, D_5,$ and $D_6$. [[image:DG29.jpg|Example of Dihedral symmetry|thumb|300px|center]] ==Elements== The $n^\text{th}$ dihedral group is the symmetry group of the regular $n$-sided polygon. The group consists of $n$ reflections, $n-1$ rotations, and the {{EasyBalloon|Link=identity transformation|Balloon=Also called the identity element $I$ ($R_0, 1,$ or $e$) such that $IA=AI=A$ for every element $A \in G$.}}. Here is an example of $D_6$. This group contains 12 elements, which are all rotations and reflections. The very first one is the identity transformation. [[image:DG8.jpg|Image 1|thumb|center|800px]] If $n$ is odd each axis of symmetry connects the mid-point of one side to the opposite vertex. If $n$ is even there are $\frac{n}{2}$ axes of symmetry connecting the mid-points of opposite sides and $\frac{n}{2}$ 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. In [[#Image 1|Image 1]], through $S_0$ to $S_5$ are the axes of symmetries. All the reflections can be described as reflections of the identity through six axes of symmetries. =Definition= There are several different way to define a Dihedral Group. We will introduce three of them. We will use $R_0$ to represent the identity, $R_k, k=1,2,\cdots,n-1$, to represent the rotations, and $S_k, k=0,1, \cdots, n-1$, to represent the reflections. ==Complex Plane Presentation== For $n \geqslant 3$, the dihedral group $D_n$ is defined as the rigid motions of the plane preserving a regular $n$-gon, with the operation of composition. On complex plane, our model $n$-gon will be an $n$-gon centered at the origin, with vertices at {{EasyBalloon|Link=the n-th roots of unity|Balloon=For any x, $x^n=1$, we can get $x=\sqrt[n]{1}$. We call this x n-th root of unity.}}. $1$ is always an $n$-th root of unity, but $-1$ is such a root only if $n$ is even. In general, the roots of unity form a regular polygon with $n$ sides, and each vertex lies on the unit circle.
Image:DG27.jpg Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other {{#ev:youtube|TTlLRYbikDg}} ==In Music== The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords.[[#References|[3]]]
[[Image:DG17.jpg|frame| [[Dihedral Groups#Example of Dihedral Groups in Music|Example of Dihedral Groups in Music]]. Created by Benson, Dave J.[http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf]|thumb|center|500px]] Dihedral groups as a kind of special symmetric groups are studied in music. In music, we use the operations Transposition and Inversion, which are denoted as $T$ and $I$, to represente rotations and reflections in dihedral groups. Musicians usually study $D_{12}$, because 12 is the length of a normal cycle in music: C C D E E F F G G A B B, and then C again. Based on this 12 element cycle, $D_{12}$ is important in music theory. Musicians use Transpositions and Inversions (rotations and reflections) of a simple note to create other notes to complete a final composition. A transposition of a sequence $x$ of pitch classes by $n$ semitones is the sequence $T^n(x)$ in which each of the pitch classes in $x$ has been increased by $n$ semitones. So for example if :$x=3 0 8$, where the numbers denote pitches, then :$T^4(x)=T^4(3 0 8) = 7 4 0$. When doing the operation $T^n(x)$, add $n$ to each digit of $x$, and use arithmetic modulo 12 (clock arithmetic) when the resulting digit is over 12. For instance, in adding 4 to 8, the result is 12, but $12 \equiv 0 \pmod{12}.\,$ Turning to the next operation, inversion $I(x)$ of a sequence $x$ just replaces each pitch class by its negative (in clock arithmetic). So in the first example above with $x = 3 0 8$, we have :$I(x) = 9 0 4$. To do the operation $I(x)$, we need to do subtraction in clock arithmetic. For instance, if we want to get 12 from 3, we need to add 9. 0 is already 12, so we need to add 0.
[[Image:DG16.jpg|frame| [[Dihedral Groups#Operation in Music|Operation in Music]]. Created by Crans, Alissa S. [http://myweb.lmu.edu/acrans/MusicalActions.PDF http://myweb.lmu.edu/acrans/MusicalActions.PDF]|thumb|center|700px]] :* An applet to explore dihedral groups: http://www.mathlearning.net/learningtools/Flash/Dihedral/dihedralExplorer.html [1] Wikipedia. (n.d.). ''Dihedral groups''. Retrieved from [http://en.wikipedia.org/wiki/Dihedral_group http://en.wikipedia.org/wiki/Dihedral_group] [2] de Cornulier, Yves. (n.d). ''Group Presentation.'' From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. [http://mathworld.wolfram.com/GroupPresentation.html http://mathworld.wolfram.com/GroupPresentation.html] [3] Wikipedia. (n.d.). ''Cayley table''. Retrieved from [http://en.wikipedia.org/wiki/Cayley_table http://en.wikipedia.org/wiki/Cayley_table] [4] Milson, Robert and Foregger, Thomas. ''dihedral group''. From PlanetMath.org. June, 12. 2007. Retrieved from [http://planetmath.org/encyclopedia/DihedralGroup.html http://planetmath.org/encyclopedia/DihedralGroup.html] [5] Gallian, Joseph A. ''Contemporary Abstract Algebra'' Seventh Edition. Belmont: Brooks/Cole, Cengage Learning. 2010. [6] Dahlke, Karl. (n.d). ''Groups, Dihedral and General Linear Groups.'' Retrieved from [http://www.mathreference.com/grp,dih.html http://www.mathreference.com/grp,dih.html] [7] Sharifi, Yaghoub. ''Subgroups of dihedral groups (1)&(2).'' Feb, 17, 2011. Retrieved from [http://ysharifi.wordpress.com/2011/02/17/subgroups-of-dihedral-groups-1/ http://ysharifi.wordpress.com/2011/02/17/subgroups-of-dihedral-groups-1/] [8] [[Dihedral Groups#References|[1]]]Scott, W. R. ''Group Theory.'' New York: Dover, 1987. [9] [[Dihedral Groups#References|[2]]]Hungerford, Thomas W. ''Graduate Texts in Mathematics - Algebra.'' New York: Springer, 1974. [10] [[Dihedral Groups#References|[3]]]Crans, Alissa S., Fiore, Thomas M. and Satyendra, Ramon. ''Musical Actions of Dihedral Groups.'' University of South Florida. Nov 3, 2007. Retrieved from [http://myweb.lmu.edu/acrans/MusicalActions.PDF http://myweb.lmu.edu/acrans/MusicalActions.PDF] [11] Benson, Dave J. ''Music: A Mathematical Offering.'' Cambridge University Press. Nov 2006. Retrieved from [http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf http://www.maths.abdn.ac.uk/~bensondj/html/music.pdf] [12] Rowland, Todd and Weisstein, Eric W. (n.d). ''Root of Unity.'' From MathWorld--A Wolfram Web Resource. Retrieved from [http://mathworld.wolfram.com/RootofUnity.html http://mathworld.wolfram.com/RootofUnity.html] [13] Conrad, Keith. (n.d). ''DIHEDRAL GROUPS.'' Retrieved from [http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/dihedral.pdf http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/dihedral.pdf] :#More information related to the other groups :#Add more about Dihedral Groups in 3D. I only talk about one property in 3D, but there must be some more. :#In the subgroups part, it is hard to explain only in words, so I use lots of notation, which is still not very clear. I hope can find a better way to illustrate it. :#Add more about applications :#Think about non-abelian in matrices which may relate to non-abelian in group theory. Yes, it is.