Reflections and reflection orbits

Reflections are very useful mechanisms to express symmetry relations among objects, images, ideas, narratives, poems, lyrics, melodies, movements, arguments, molecules, and many more.

The squares shown below illustrate the mechanism of line reflections.

Select any one of the squares. Can you then identify the square that is its image by a vertical line of reflection? Next, identify the other square that is its image by an horizontal line of reflection.  Now observe that the remaining square is the image of the originally selected one by:

• Iterating the two reflections (in any order) or;
• Directly rotating it (by 180 degrees).

Thus, it seems that here the iteration of two line reflections (with 90 degrees of separation) produces the same effect of a (planar 180 degree) rotation of the original square.

Let's indicate these symmetry operations with the letters:

v to indicate a reflection relative to a vertical line;

h to indicate a reflection relative to a horizontal  line;

o to indicate a double-reflection or the 180-degree central rotation.

You will observe that it really does not matter which square you select as your starting point: after applying  the reflections and double-reflections to that initial square you will always end up with the same set. This set is called an orbit of symmetry. We say that the orbit has the symmetry of those reflections and their iterations. To include the starting square in the orbit we add a transformation that just leaves any object as is. We call it the identity transformation, and write it as 1.

Together, then, the set of transformations, that we write as

D_2 = {1, v, h, o},

generates the reflection orbits. Shortly, we say that the resulting orbits have the symmetry of D_2.

In the Flag Experiment each row of flags is an orbit with the symmetry of D_2.

Last revised 02/02/2011
06/06/06
These  postings are based on "Symmetry Studies An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)