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Dihedral Groups

When you click on Dihedral Groups, the adjacent box displays with D selected.
Enter the number of elements.

Dn is the set of symmetries of a regular polygon with n sides under the operation of function composition. A symmetry of a figure is a rigid (preserves distance) mapping of the plane that maps the figure back onto itself.  

When you open a Dn table, the table, a key, and a polygon display. The symmetries of a regular polygon can be tracked by where they map each vertex, which is indicated in the key that displays with the table.  

 
 

 

In the original position of the polygon, vertex 1 is always displayed
in the 1 o'clock position. In the key (13)(45) means:

      Vertex 1 is mapped to 3 and 3 is mapped to 1.
      Vertex 4 is mapped to 5 and 5 is mapped to 4.
      The other vertices are fixed.

Rotations

Dn has n rotations which are labeled as R0, R1, R2, . . . , Rn-1.
R1 is a rotation of 360/n degrees. It maps vertex 1 to 2, vertex 2 to 3, etc.
R2 is a rotation of 2(360/n) degrees, etc.

Rotations

Dn has n reflections (flips) which are labeled as F1, F2, . . . , Fn.
F1 is the reflection that leaves vertex 1 fixed.

If n is odd, the reflections leave only 1 vertex fixed,
so Fj is always the reflection that leaves vertex j fixed.

If n is even, some reflections leave 2 vertices fixed and others leave no vertix fixed.
If j <= n/2, Fj is the reflection that leaves vertex j fixed, along with its diagonal mate.
If j > n, Fj is the reflection that maps j-1 to j.

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If you click on a symmetry listed above the polygon, you can watch the polygon do the rotation or reflection, and then see how the vertices are arranged. If you click on R1, you will see that 1 is now in the 2-position, so 1 is mapped to 2, etc.

When you click on a symmetry the second time, GU computes the outcome of the composition of those two symmetries.

If you would like to see the Key notation display in the table, go to the Edit Menu and click on Change Detail. To go back to the condensed form, click on Change Detail again.