In search of symmetry

Symmetry is something we all appreciate, whether in beautiful faces, stunning flowers or pleasing patterns.  And we have been exploring symmetry for thousands of years, as can be seen in the carved stone balls produced over 3000 years ago in the neolithic period in Scotland.  These are carefully decorated, most with evenly spaced carvings or knobs.  And most remarkably more than half have 4, 6 or 8 symmetrically placed knobs, showing a striking similarity to the highly symmetric platonic solids that would be discovered a thousand years later by the Greeks.

Three carved stone balls from the British Museum's collection.                   Image © Trustees of the British Museum

One way symmetry has been explored over the centuries by artists from all cultures is through decorative patterns and friezes, such as those decorating the Ashmolean museum.

Magic tricks and foot prints

Frieze patterns, such as the simple beaded pattern around the columns of the Ashmolean, are strips decorated with a repeating pattern.  As they are repeating patterns they all have translational symmetry – you can slide the pattern along until it matches itself exactly. 

Symmetries are just like magic tricks.  If you close your eyes and I perform one of these symmetry operations on the frieze pattern, it will appear unchanged when you open your eyes.  This beaded pattern holds many other symmetries apart from translation.  [What other symmetries can you see in this beaded pattern?] It would appear the same if I reflected the pattern horizontally or vertically, if I spun it through a half turn, or if I slide the pattern along and then reflected it horizontally (called a glide reflection).

The group of symmetries for this frieze pattern can be remembered by the simple name of "Spinning Jump".  You can see the pattern of footprints left behind by spinning between jumps has the same symmetries as the beaded pattern around the column.

If you jump out this pattern and leave behind the cut-out footprints, it makes it easy to demonstrate that it has the same symmetries as the beaded frieze.  Pick any of the symmetries, place four coloured cut-out footprints on the pattern, perform the symmetry transformation with these coloured footprints, and it will be clear that the footprint pattern will appear unchanged.

When symmetries fail

The spinning jump pattern has many symmetries but other frieze patterns, for example the "Sidle", are far less symmetrical. 

What symmetries does this pattern have?  Jump out the pattern, marking it with cut-out footprints.  Use coloured cut-out prints to explore which symmetries hold, and which don't, for this pattern.

Unlike the Spinning Jump, which had all possible symmetries, the only symmetries which hold for the Sidle are translation and vertical reflection.  Any other transformation will change the pattern.  [Can you spot this frieze pattern on the Ashmolean?]  This pattern can been seen in the egg and dart frieze running near the top of the building.

What other frieze patterns are there?

Some of the other frieze patterns you can see on the Ashmolean are the plait on the ceiling of the portico:

the plait running along the top of the walls:

and the spiralling pattern above the windows:

All three of these patterns have the same symmetries, a group of symmetries known as the "Spinning Hop".

What symmetries does this pattern have?  Use the footprint pattern and coloured cut-outs to explore the symmetries.

Apart from translation, the only other symmetry that holds for these patterns is rotation: if you spin the pattern by 180 degrees it will appear unchanged. 

The seven frieze groups

Mathematically we analyse frieze patterns by the combination of symmetries that can coexist within the frieze and how these symmetries interact with each other.  You might think there are endless possibilities of different combinations of symmetries, but actually there are only 7 possible ways that these symmetries can occur together in frieze patterns.  In addition to the Spinning Jump, Sidle and Spinning Hop groups we've already seen, there are four other frieze groups:

• the "Step"

• the "Spinning sidle"

• the "Hop"

• and the "Jump".

Artists have been playing with all seven of these patterns for thousands of years.  But it was only in the nineteenth century that mathematicians were able to prove that no other frieze groups existed and every possible frieze pattern could be described by one of these seven frieze groups.

Group theory

Mathematically symmetry is studied using group theory: each of the seven frieze groups above is an example of a mathematical group. One of the founders of group theory was the mathematician Everiste Galois.  He made significant contributions to mathematics during his short life before he died in a duel in 1932.  The night before the duel he passed the time writing letters and mathematical papers, which included him coining the word "group" for this type of mathematical object.

Group theory is used widely outside of mathematics too, in studying the symmetries of crystal structures and molecules in chemistry and in the Standard Model of particle physics currently being explored at the Large Hadron Collider.

We have only be able to spot three of the frieze groups decorating the Ashmolean.  Can you spot anymore?  And if you spot other examples of this beautiful piece of mathematics here, or in any city around the world, please let us know!

Demonstration

To make it easier for the group to see, hand around printouts of the image of the neolithic carved stone balls, the friezes of the Ashmolean  and the footprint patterns for the seven frieze groups (you can download this footprint bingo). 

We've included questions (in italics in the description above) that you can ask the group to get them involved.

To demonstrate the symmetries of these frieze patterns we jumped out the footprint patterns of the symmetry groups and marked the patterns with cut-outs.  As indicated in the captions of the footprint patterns above, an easy way to explore the symmetries is to use extra coloured footprints that you can move to see if a transformation leaves the pattern unchanged.