Multiplication tables

When I introduced the reflection symmetries, it was remarked that, for example, a double reflection is equivalent to a rotation, or that iterating the same reflection gives the identity symmetry. When we start with a set of symmetries and assemble all of their possible iterations, or compositions,  the resulting table is usually referred to as its multiplication table (in analogy to what we have learned in grade school... and perhaps selectively erased from memory...).

Here are the multiplication tables for the sets {1,v}, {1,h}, and {1,o}.

For example, we write vv=1 to indicate that the composition of a vertical reflection with itself gives the identity symmetry. Similarly, 1h=h1=h, or oo=1, and so on.

And similarly, here is the multiplication table for the set  D_2={1,v,h,o}.

We then observe a few important properties of these tables, where we write G to indicate any of the sets introduced above:

• The composition of any two symmetries in G remains in G;
• The element 1 in G is such that 1w=w1 for all w in G;
• To each w in G there is a w' in G such that ww'=w'w=1.

The element w' corresponding to w is called the inverse of w. In the above tables, clearly each element is its own inverse. The operation of composition of reflections is also associative, that is,

 u(vw) = (uv)w,

 so that I did not include that requirement in the list above.

When a multiplication table can be defined on a (finite) set G then we say that G, together with the operation defined in the table, is a (finite) group.

If G is group and uv = vu  for all elements in G, we then say that G is a commutative group. Inspecting the multiplication table above we can state that G={1,v,h,o} is a commutative group. 

The sets {1,v}, {1,h}, and {1,o} are subsets of  D_2={1,v,h,o} inheriting its multiplication table, so that they might be referred to as subgroups of D_2.

The subgroups {1,v}, {1,h}, and {1,o} are exactly the same, algebraically,  because their multiplication tables can be made to coincide. They are only realized differently. As such we say that these subgroups are isomorphic. 

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Unavoidable symmetry

Here are two bits of symmetry appreciation:

Can you see (with the eyes of imagination) the double reflections generating the icon (of the Publican and the Pharisee) below?

As such, the images of the Publican and the Pharisee form two distinct symmetry orbits generated by double reflections, introduced earlier here.

Now enjoy one a poem of J.L. Borges, where he (as often) plays with reflections, mirrors, and more:

We are the time. We are the famous
Jorge Luis Borges 

We are the time. We are the famous
metaphor from Heraclitus the Obscure.
    We are the water, not the hard diamond,
    the one that is lost, not the one that stands still.
We are the river and we are that greek
that looks himself into the river. His reflection
changes into the waters of the changing mirror,
into the crystal that changes like the fire.
   We are the vain predetermined river,
   in his travel to his sea.
The shadows have surrounded him.
Everything said goodbye to us, everything goes away.
   Memory does not stamp his own coin.
However, there is something that stays
however, there is something that bemoans. 


And here, perhaps Borges brings us an image of a single-point orbit when the colors of its element are erased:


"Denying temporal succession, denying the self, denying the astronomical universe, are apparent desperations and secret consolations. Our destiny is not frightful by being unreal; it is frightful because it is irreversible and iron-clad. Time is the substance I am made of. Time is a river, which sweeps me along, but I am the river; it is a tiger, which destroys me. But I am the tiger, it is a fire, which consumes me, but I am the fire. The world, unfortunately, is real. I, unfortunately, am Borges"

Jorge Luis Borges. Essay: A New Refutation of Time in Labyrinths. Ed Donald A Yates and James E Irby. Penguin Books 1987.

Revisions 02/25/2011
06/06/11
These  postings are based on "Symmetry Studies An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

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)

Symmetry orbits of the square

Here is the symmetry orbit of the four-color square.

Following the reasoning introduced with the symmetries of the three-color triangle, develop by analogy the corresponding narrative for this orbit.

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

Symmetry orbits of the triangle

These six images

illustrate the symmetry orbit of the three-color triangle. Note that:

• Any two triangles in different rows are single-reflection images of each other;
• Any two triangles in the same row are double-reflection images of each other;
• Any two triangles in the same row are rotated images of each other.

As a consequence, the symmetry orbit of the triangle is generated by single and double reflections, or, equivalently, by reflections and rotations. 

The separation angle between adjacent reflection axes in the triangle is 2Pi/6 and is referred to as its dihedral angle. The resulting rotations are in angles that are multiple of twice the dihedral angle.

The rotations in one row are in the opposite direction relative to the rotations in the other row. This may hint us to connecting symmetry orbits with space orientation: For example, the arms of  a transparent clock hanging in a transparent wall rotate in opposite directions when viewed from opposing sides of its wall. Conversely, your position relative to the wall may be determined or label by the direction of rotating arms.
More to come on this.

Any one of the six triangles generates all the remaining ones after the reflections and iterated reflections are applied to that initial triangle. Therefore, the initial choice, or the generating element, is arbitrary. This property of arbitrariness is present in any symmetry orbit and will be explored later on in future postings.

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)

More and less populated orbits

 Here are three orbits,

all with the reflection symmetries of  D_2={1,h,v,o}.

The gray orbit has four elements, the  green orbit has two elements, and the pink one has a single element. Can you identify what attribute affected the number of elements in each orbit? Since the pink orbit has a single square, we are allowed to say that it has the symmetries of D_2 as well.

Conversely, the pink square suggests the action of D_2. The pink square is a sort of an elementary framework. By coloring it in different ways we may eventually get different orbits. In Chemistry, molecules are different colorings of elementary frameworks.  

So, now it may become more apparent why symmetry is what you see at a glance  ... the elements in the gray orbit, from afar, seem like the pink square, and elusively borrow its symmetries .... until you look close enough!

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