# The Science of Symmetry I

Symmetry is found everywhere, in nature and artificial arrangements. This is the first of a two-part blog post about symmetry and chemistry. This week we will be talking about the basics of symmetry and how they relate to molecules. Usually, we talk about beauty when concerning symmetry, however Mathematicians have a different use for the word.

Symmetry in Shapes

Symmetry is an intrinsic property of many shapes. We can take a square and a hexagon as below:

For the purposes of the diagram I used reflectional symmetry as it’s more convenient to draw, although reflectional and rotational symmetries are the same with regular polygons. To achieve a reflection across the dotted lines you can rotate the hexagon any number of 60 degrees. The square would be the same, apart from it being rotated any number of 90 degrees. The rotation that needs to be applied several times can be simply calculated by taking 360°, the angle around a point, and dividing it by the number of sides the shape has.
We can also take shapes that aren’t regular and find symmetries of them. The rotational symmetry of irregular shapes is very rare, but not impossible. Note that we are excluding the trivial symmetry of 360°. The order of symmetry is the number of times the shape matches itself and if we had a star shape, technically not a regular polygon, then the order of symmetry would be the number of points.

For a concrete example, let us look at the Triskellion on the flag of the Isle of Mann. The flag has an order of symmetry of 3.

Symmetry in molecules

If we look back to our hexagon, we can relate this to cyclohexane, a molecule shaped like a hexagon in a skeletal formula. Although, before we run into this, let me do a brief overview of how I’m going to draw the molecules for the remainder of this blog post, in skeletal formulae.

Skeletal Formulae

Skeletal formulae are a way to abbreviate the drawing of organic compounds in chemistry. Briefly, we use line segments at 60° angles, and the vertices symbolise the carbons. We also assume that any ‘free’ spaces on the carbons are hydrogens.

The advantage to us is that it makes molecules look like shapes. It makes it simpler when considering their symmetry. Look at the examples below, see how they are becoming lines and points:

However, it gets a lot more complicated, but it’s useful for organic chemists to name compounds.

For our purposes, we just want the mapping from molecules to shapes. Finally, we can write the molecule benzene like this (a hexagon!):

Now we have looked at how we are drawing our molecules. Looking back at the symmetry of shapes, we can map them to molecules, meaning that molecules have symmetries. If we start with regular polygons, we know that there exist similar molecules to cyclohexane but for all the regular polygons (we are talking about them in a totally 2D context).

There are many other types of molecules that we can show have symmetry, infinitely many in fact. We can take a couple of simple examples:

The first has vertical reflective symmetry, the reflective line being from the apex of the pentagon downwards. The second has rotational symmetry similar to the flag of the Isle of Mann, again of order 3.

Before we go and talk about other bits of symmetry in chemistry, let us take the rather particular case of benzene. We have two ways of drawing the molecule. We have Devil Benzene Kekule Structure, and we have the Thiele Structure. One is more symmetrical than the other.

The Kekule has double bonds in the ring, denoted by the double lines. This produces the fact that you must rotate it by 120° rather than the 60° that Thiele needs. However, the Thiele structure is closer to the actual structure of benzene, especially in terms of rotational symmetry, as the electrons in its delocalised ring have an equal probability of being everywhere in that ring.

Now we know that we can have symmetric molecules and how they relate to the ordinary symmetries we know and love. Next week we get into the meaty bits of the post; groups, symmetry and isomerism! So, until then, have a good week and happy mathsing!