UC SCI 111, vM 
Notebook 14

Needs["Graphics`Colors`"];
Off[General::spell1];
<<  Graphics`ImplicitPlot`

Dihedral groups.

The permutation group associated with the equilateral triangle.

On different occasions we have already studied symmetries of equilateral n-gons. In the theory it is showed that the equilateral triangle  has six different symmetries. The group of these symmetries is often called the dihedral group on 3 points, or just (Subscript[D, 3]).

Recall that the matrix that describes a rotation over an angle α around the origin is given by
	
	(Subscript[R, α]) = (({{cos α, -sin α }, {sin  α, cos  α}}))

while the matrix that describes a reflexion in the x-axis is given by

	S = (({{1, 0}, {0, -1}}))

Now we fix a numbering of the vertices of the triangle like in the picture. Every time we let a combination of (Subscript[R, α]) and S act on this triangle, the vertices get permuted. The procedure showaction[n_ ,m_ ]performs this action and shows the results. Here n is the number of vertices and m the matrix that acts on the polygon.

\(showaction[\ n_\ , m_\ ]\  := \ 
      Module[
{eps, \ bit, \ ff, \ points, \ text, \ 
          imagepoints, \ imagetext, \ i}, 
eps\  = \ 
          1.1; 
bit = { .2, 0}; 
ff = 
          TextStyle\  -> {\ 
              FontSize -> \ 16}; 
points\  = \ 
          Table[\ {Cos[\ \((2\ Pi\ i\ )\)/n], \ Sin[\((2\ Pi\ i)\)/n]}, \ {i, 
              0, n}]; 
text\  = \ 
          Table[\ Text[i, eps\ \ points[\([i]\)]\ , \ ff\ ], \ {i, 
              n}]; 
imagepoints\  = \ 
          Table[\ m . points[\([i]\)], {i, 
              n}]; 
imagetext\  = \ {}; 
For[\ i = 1, \ 
          i\  ≤ \ n, 
imagetext\  = 
            Join[\ imagetext, \ {Black, 
Text[\ " -> ", \ eps\ imagepoints[\([i]\)] +  .5\ bit, 
                  ff], 
Blue, 
Text[i, 
                  eps\ imagepoints[\([i]\)] + bit, 
                  ff]}]; 
\(i++\);
]; 
Show[\ 
          Graphics[\ 
            Join[{\ Line[\ points], \ Red}, text, {Blue}, 
              imagetext, {PointSize[0.03], Black, Point[{0, 0}]}]\ , \ 
            AspectRatio\  -> \ 
              Automatic], 
PlotRange\  -> \ 
            All]
]\)

The following example illustrates what happens when we reflect the original triangle in the horizontal x-axis.

s = {{1, 0}, {0, -1}};
showaction[ 3,  s];

Exercise 1.
a) What is the value of α in this example?
b) Write down all six symmetries of the triangle in terms of (Subscript[R, α]) and S.
c) Describe how (Subscript[D, 3]) acts on the set {1,2,3}.
d) Write down the six permutations of {1,2,3} that correspond with the symmetries.
e) Are there any other permutations of {1,2,3}? 
f) Explain why we have found an isomorphism between a symmetry group and a permutation group.