A bit of (written) language for symmetries

Let's revisit the elements of the symmetry orbit of the triangle while replacing the three-coloring of their vertices with three-letter labeling using the letters  A, C, and T.  Let's first look at the single-reflections. 

We write (AC)  and say that

  label A goes to replace C and C goes to replace label A 

when referring to a transposition  of these two vertices, as another term indicating the reflection fixing the remaining vertex (T). Here is the diagram of the transposition (AC):

And here are the illustrations of the other two transpositions:

The double-reflections give rise to rotations:  Now we write (ACT) and say that

label A goes to replace label C, label C replaces T, and T replaces A.

Here is the illustration of the 2Pi/3 (clockwise) rotation (ACT):

The 4Pi/3 rotation in the same direction:

An additional rotation would bring us back to the original triangle.  This gives the identity transformation, and it is indicated by 1:

We now have language to write down the complete set of symmetries acting on the triangle:

{ 1, (ACT), (ATC), (AC), (AT), (TC) }. 

We also say that the symmetries were written using their cycles notation.

Last revised 02/23/11
These  postings are based on "Symmetry Studies An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)