Posted by: Prof. Kate | September 17, 2008

## What is the mathematics of symmetry?

Symmetry is, loosely speaking, when an object remains the same under some transformation.  A heart shape, for example, is symmetrical in the sense that a right-left flip across a central vertical axis leaves the shape unchanged.  If you do this to other shapes, such as, say, the outline of North America, you’ll find the picture changes.

In mathematics, we refer to the transformation itself as the ‘symmetry’.  So a heart-shape has one reflectional symmetry.  A square has a 90-degree rotational symmetry because we can rotate it by 90 degrees and obtain a shape identical to the one we started with.  It also has a 180-degree rotational symmetry, and a 270-degree rotational symmetry.  We might be tempted to say it has a 360-degree rotational symmetry, and it does.  But rotating anything by 360 degrees gets you back where you started, so we think of rotation by 360 degrees as the same thing as “doing nothing”.  Incidentally, “doing nothing” is a symmetry, but it’s a boring one (so we call it “trivial.”)

Notice that two 90-degree rotations, performed in succession, give a 180-degree rotation.  Doing it four times in a row gets you back to 360, or “trivial” rotation.  If we call 90-degree rotation “r”, then we can write

$r^4 = id$

where “id” is short for “identity” or “trivial transformation”.

It’s almost like r is a number… a number whose fourth power is 1, or “identity”… almost as if these “transformations” form a sort of alternate number system.  Well, they do, and we call it the “group of symmetries” of the square (there are flips too, and other equations that flips and rotations satisfy).  So in this course one of the main things we’ll study is group theory.  A group is a collection of objects and a rule for putting them together (together with a few stipulations on how the rule behaves).  In this case, the objects are the “symmetries” which are a sort of transformation, and “putting them together” just means performing one and then the next.

We’ll see groups in all sorts of guises in this course.  The symmetries of the regular icosahedron and other shapes will show up again and again, in different areas of mathematics, and we’ll use this as a tour of the topics of discrete mathematics:  finite fields, linear algebra, graph theory, permutations, even probability.