Riding a fixed-wheel bike is another British peculiarity... Een baan-fiets op de weg! exclaim Dutch tourists. It seems to be quite common for trike riders to ride fixed when they're on two wheels; here's Kevin McLellan's TransCycle page with the same combination.
A fixed-wheel bike is one which has no free-wheel and no gear-change. It's extremely efficient, either for use in flat areas (such as Cambridgeshire) or hilly areas (such as the Bath area)... but not both without changing the gear. (A fixed-wheel hub has threads for a gear at each end of the axle, so to change gear, you take the wheel out and re-install it the other way around.)
A fixed-wheeler is far simpler than any other kind of bike; lighter (mine weighs 18lbs at its lightest, which is pretty light for bikes of that era -- I built it in the early 80s), more efficient, very low-maintenance, and very natural to ride (once you've learnt to handle it) -- you slow down by resisting the pedal motion (this can throw you over the handlebars if you don't know what you're doing) and hardly ever have to use the brake. (The law only requires one brake for fixed-wheelers.) You can also pedal much harder, as the chain does not jump when pulled hard like a derailleur does, hence the use of fixed-wheel for sprint racing.
The fixed-wheeler is a much older and purer design than the typical modern bike, and has not changed much in a long time, being already more efficient in its niche... a bit like the difference between sharks (chondroichthyes) which evolved to a stable state a long time ago, while bony fish (osteoichthyes) are still fiddling around trying to get things right...
Another page with fixed wheels is Kevin McLellan's. I was delighted to find a bike shop on the web that stocks fixed equipment (alas, a long way from me!): it is the Harris Cyclery, who have a page of fixed-wheel kit and a very good article enthusing about fixed-wheel riding.