Groups(3) - Measuring Symmetry

A mathematical tripod

Continuing with Crilly:

MEASURING SYMMETRY

In the case of our triskelion the basic symmetry operations are the (clockwise) rotations R through 120 degrees and S through 240 degrees. The transformation I is the one that rotates the triangle through 360 degrees or, alternatively, does nothing at all.

We can create a table based on the combinations of these rotations, in the same way we might create a multiplication table:

*	I	R	S
I	I	R	S
R	R	S	I
S	S	I	R

This table is like an ordinary multiplication table with numbers except we are "multiplying" symbols. According to the most widely used convention, the multiplication R * S means first rotate the triskelion clockwise through 240 degrees with S then by 120 degrees with R, the result being a rotation by 360 degrees, as if you did nothing at all. This can be expressed as R * S = I, the result found at the junction of the last but one row and the last column of the table.

The symmetry group of the triskelion is made up of I, R, and S and the multiplication table of how to combine them. Because the group contains three elements its size (or "order") is three. The table is also called a Cayley  table (named after the mathematician Arthur Cayley, distant cousin to Sir George Cayley a pioneer of flight).

Like the triskelion, the tripod without feet has rotational symmetry. But it also has mirror symmetry and therefore has a larger symmetry group. We'll call U, V, and W the reflections in the three mirror axes.

The larger symmetry group of the tripod, which is of order six, is composed of the six transformations I, R, S, U, V, and W and has the following multiplication table:

	I	R	S	U	V	W
I	I	R	S	U	V	W
R	R	S	I	V	W	U
S	S	I	R	W	U	V
U	U	W	V	I	S	R
V	V	U	W	R	I	S
W	W	V	U	S	R	I

An interesting transformation is achieved by combining two reflections in different axes, such as U * W (where the reflection W is applied first and is followed by the reflection U). This is actually a rotation of the tripod through 120 degrees, in symbols U * W = R. Combining the reflections the other way around W * U = S, gives a rotation through 240 degrees. In particular U * W does not equal W * U. This is a major difference between a multiplication for a group and an ordinary multiplication table with numbers.

A group in which the order of combining the elements is immaterial is called an abelian group, named after the Norwegian mathematician Niels Abel. The symmetry group of the tripod is the smallest group which is not abelian. 

ABSTRACT ALGEBRA

The trend in algebra in the 20th century has been towards abstract algebra, in which the group is defined by some basic rules known as axioms. With this viewpoint the symmetry group of the triangle becomes just one example of an abstract system. There are systems in algebra that are more basic than a group and require fewer axioms; other systems that are more complex require more axioms. However the concept of a group is just right and is the most important algebraic system of all. It is remarkable that from so few axioms such a large body of knowledge has emerged. The advantage of the abstract method is that general theorems can be deduced for all groups and applied, if need be, to specific ones.

A feature of group theory is that there may be small groups sitting inside bigger ones. The symmetry group of the triskelion of order three is a subgroup of the symmetry group of the tripod of order six. J.L. Lagrange proved a basic fact about subgroups. Lagrange's theorem states that the order of a subgroup must always divide exactly the order of the group. So we automatically know the symmetry group of the tripod has no subgroups of order four or five.

CLASSIFYING GROUPS

There has been an extensive programme to classify all the possible finite groups. There is no need to list them all because some groups are built up from basic ones, and it is the basic ones that are needed. The principle of classification is much the same as in chemistry where interest is focused on the basic chemical elements and not the compounds which can be made from them. The symmetry group of the tripod of six elements is a "compound" being built up from the group of rotations (of order three) and reflections (of order two).

Steve here. Crilly then goes on to describe the classification program, which is interesting from a Pure mathematical point of view but not for our purposes. If you wish to continue in that vein, you can buy the book and read about "the enormous theorem", "sporadic groups", the "baby monster" group and the "monster" group, and "the six pariahs", but none of these are going to help us understand gauge theory and The Standard Model of Particle Physics, our ultimate goal.

PREVIOUSLY:

Groups(1) - Gross Introduction

Groups(2) - Rotational Symmetry and Mirror Symmetry

UPDATE: Found a nice short video of Group Theory on youtube, below. The young man's enthusiasm is infectious, and although he doesn't go into the detail we discuss here, that's fine for an "Intro." One caveat: he has a pronounced accent (not unpleasant) but your parents may wonder why you're listening to a video presentation with expressions like "Cockster" and "shoe-full" by someone named Singing Banana. Not to worry, parents, he simply has an accent (except in his home country) and the words he's actually using are "Coexter" and "shuffle." Try not to be so paranoid, kthnx.



If you found that 2-D stuff fun, check out what happens when we consider 3 dimensions:

A tetrahedron can be placed in 12 distinct positions by rotation alone. These are illustrated above in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through the positions. The 12 rotations form the rotation (symmetry) group of the figure.

To see the multiplication table for the Tetrahedral Group, click here.