Symmetry

Our focus is on the use of Symmetry to study molecules in Chemistry and Physics.

Symmetry is a study of the geometry of an object. Symmetry makes it possible to do things such as rotate an object and the result after the rotation looks the same as before the rotation.

A triangle can be rotated 120 degrees and still look the same. Every point in the object goes through this rotation. It is convenient for us to just consider the vertices of the triangle.

A rectangle can be reflected through two axes of reflection: 1) up-down, 2) left-right. Every point goes through a reflection.

• Point Groups
• Representation Theory
• Group Representation
• Group Theory
• Homomorphism

Appendix A

Somewhat as a joke, we’ve come up with two different definitions of Symmetry:

noun definition

Symmetry is about symmetry elements that control the operations; a geometric object is given a name and a symbol and found in a group; symmetry operation just refers to actions like rotations, reflections and inversions.

verb definition

Symmetry is about symmetry operations that describe the action of the movement, the actions are given names and symbols and are found in groups; symmetry element just refers to geometric objects like an axis or a plane.

This came about because of the question: “hey, regarding symmetry operations and symmetry elements, which is the dog and which is the tail?”

After some discussion, a student thought the answer might be “neither”. Symmetry Element has a problem if you consider that both a proper rotation and an improper rotation refer to the same axis. We can’t have one name referring to two different things.

Regarding operation, that will get confusing when you get to improper rotation and one name is linked to a combination of two operations (you rotate then you reflect), rather than a single operation.

Here’s how we climbed out of the hole:

All the symbols are elements of a set and that set is a Group. We might thus go with “Symmetry Element” since the thing of interest to us (that thing being identity, proper rotation, improper roation, horizontal reflection, vertical reflection or inversion) will be thought of as an element when we start working with the group that contains it.