Those of you who own a four-legs will have noticed that they usually exhibit a range of different gaits depending on the occasion. Taking Mizuna our cat as an example - he walks (back-left, front-left, back-right, front-right, each leg a quarter of a cycle behind the previous), he trots (back-left and front-right together - then back-right and front-left together, each movement half a cycle behind the previous), he runs (back-left, back-right, front-left, front-right, or alternatively back-right, back-left, front-right, front-left). Mostly he sleeps, but that's not a gait.

There are two other forms of motion he employs occasionally. There's the bound (back legs together, front legs together) which is his gait-of-choice for high-speed ascent of the stairs (he trots on the way down) and the highly amusing pronk, in which he bounces on all four legs simultaneously - this is reserved for moments of great excitement such as being fed. Yes, pronk is a real word!

We bipeds, on the other hand, are rather short of choices when it comes to motion. We can walk (left, right...) or we can jump (legs together) and that is it. I don't count symmetry-breaking gaits here, such as the hop or the skip - they are a bit unusual, just like the quadruped's canter - and I count 'run' as being the same as walk, in that our two legs still move in anti-phase to each other.

The increased range of gaits available to the quadruped can be naively attributed to them having more legs, but perhaps a better description would be that they have increased symmetry. We bipeds have a single mirror plane, left-to-right when it comes to legs, but quadrupeds, with legs arranged in a rectangle, can be thought of as having a front-to-back mirror plane as well. (Yes, I know they don't really have a mirror plane here, but the legs approximately have.)

Group theory is a mathematical encapsulation of symmetry. We can use it in physics to simplify problems. A typical example is finding the modes of vibration of a molecule with a particular symmetry. It's often presented as rather abstract mathematics but when applied to physics it becomes beautifully and simply powerful. For example, our pronk, trot, bound, and another gait, the pace (left legs together, right legs together - not sure if any animal does this one), drop out of the analysis for a rectangular arrangement of legs. (The walk and run/gallop are a bit more subtle). Applied to the biped, we simply obtain the walk and the jump. The more symmetry you have, the greater the range of gaits you have.

In the limit of lots of symmetry indeed (the millipede, which approximately has complete translational symmetry along its length) there are a huge number of gait options. We can then start describing these in terms of waves, and, in particular, by wavelengths and frequencies of the action rippling down the body. This then has analogies with other physical systems, such as vibrational modes in solids (phonons), where different frequencies of sound wave travel at different velocities through the solid. Group theory isn't just abstract, as many textbooks would make out, it really is quite practical and fun.

So, next time you come across a four-legs, or a six-legs, have a careful look at how it moves.

P.S. I've drawn a bit from my memory of an Ian Stewart book here, though I can't quite remember which one. I'm not sure I still own it.