Wednesday, 23 May 2007

Tubular balls

Mathematically, a "metric" is a function that defines the distance between pairs of points. The usual, everyday metric is the Euclidean metric, but there are plenty of others. The Euclidean metric is defined as the square root of the sum of the squares of the distances along each dimension, such as the north-south difference and east-west difference between two points. So if you draw a line between two points on a map, and form a right-angle triangle with the north/south and east/west grid lines, then the Euclidean distance is the length of the hypotenuse of the triangle. Other metrics include the so-called "taxi cab" or "Manhattan" metric, which adds together the north-south distance to the east-west distance, just like walking along the lines of grid.

In general, all metrics are non-negative (nothing is closer a point than the point itself), symmetric (going is the same as coming back) and have triangular inequality (meaning there's no shortcut to a straight line). If we take the p-th root of the sum of the differences, each raised to the power p, then we have the p-norm. So the Euclidean distance is a 2-norm and the Manhattan distance is the 1-norm. There's also an infinity norm, which if you don't mind thinking about taking the infinitieth-root of numbers raised to the power infinity is actually very simple: it's just the length of the single longest side of the triangle not counting the hypotenuse, in the map example.

And today I learned that travelling from King's Cross to Euston on the London Underground is 0.1 miles shorter on the Northern line than it is on the Victoria line. Let's use the notation that <A>L<B> means travelling from A to B on underground line L. So if I do <KC>N<E> then <E>V<KC> I end up back at King's Cross where I started, but travelled further on the way back (Victoria line) than I did on the way out (Northern line). By extension, if I do <KC>N<E>V<KC>N<...>V<KC> (i.e. repeat the looping journey) enough times, then I should end up back where I started, but will have travelled a negative distance. Thus that corner of North London obeys none of the three laws of metrics listed above.

Given that the Parisian underground is the "Metro", which sounds a lot like "metric", one can only assume that their system is more rigorously Euclidean, and no such shortcuts are allowed with typical Napoleonic efficiency.

1 comment:

Jeff Boyce said...
This comment has been removed by a blog administrator.