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.

The inevitable Sweeney post

Everyone knows that ITV's classic gritty cop show, The Sweeney, was centred around two characters called Carter and Regan, between them a mere one letter away from the 39th and 40th presidents of the USA respectively. The actor who played their boss, DCI Haskins, is called Garfield Morgan (James Garfield was the 20th president). The 19th president was Rutherford B. Hayes, and large parts of the show were filmed in and around Hayes, in West London. We're told in one episode that Regan's mother came from Cleveland (c.f. Grover Cleveland, 24th president).

The show's title is of course rhyming slang (for Flying Squad), and the characters often use it. For example, in the episode "Crane Flies", George Carter describes one petty vandal as being "a bit William Howard in the 'ead, guv", meaning "daft" (after William Howard Taft, 27th president). And several times, Regan refers to desk-bound police officers as being "herberts", a clear reference to Herbert Hoover (31st president), who was notoriously desk-bound after a horse-riding injury. And now the Greater Manchester police have an Assistant Chief Constable Vincent Sweeney, responsible for Territorial Policing. I can hardly Adam'n'Eve it, guv!

(Of course, if you add up all the numbers of the presidents listed above... well, I needn't spell it out.)

Various Portlands

With somewhat belated thanks to the lovely Simone, I recently learned that many of the characters in the popular animated television series, "The Simpsons", are in fact named after streets of Portland, Oregon, where said Simone currently resides. Apparently this is very well known, but I've never been to that Portland and in fact have only visited the UK's Portland Bill the once, on a school field trip. Somehow, I had never heard of the FilmFair cartoon series, The Adventures of Portland Bill until today, though it does sound a) cracking, and b) exactly the kind of thing I lapped up when I was a kid. Back to the Simpsons' Portland though, and what people seem to have forgotten is where it got its names from in the first place. One of the cities founders, a Francis W. Pettygrove, chose in 1845 to name Portland after his hometown (also called Portland, in case you were wondering). He bought the claim to the land from a Mr Overton who had bought a land claim for half the area. The other half was owned by lawyer Asa Lovejoy, who later became mayor and chief justice, and is lovingly recorded for posterity via the character of the Reverend Lovejoy in the aforementioned Simpsons. Anyway, Lovejoy was a big fan of Charlotte Mary Yonge and in particular, her novel "Heartsease, Or, the Brother's Wife". When the street layout of Portland (Oregon) was being designed, Lovejoy went through his bookshelf and noted down suitable names from this and a handful of other books. So yes, Ned Flanders is named after Northeast Flanders St., but Northeast Flanders St. is only called Northeast Flanders St. because Yonge set part of "Heartsease" in a fictional Lancastrian village of Wrangerton, whose local vicar was called... Flanders. Indeed, in Heartsease there is a passing reference to a "Bishop Fox", "a north-country bishop" who was obsessed with the future, and particularly the Twentieth Century. No mention of Television though, clearly.

Just for the record, the opening lines from Heartsease are:
The sun shone slanting over a spacious park, the undulating ground here turning a broad lawn towards the beams that silvered every blade of grass; there, curving away in banks of velvet green; shadowed by the trees; gnarled old thorns in the holiday suit whence they take their name, giant's nosegays of horse-chestnuts, mighty elms and stalwart oaks, singly or in groups, the aristocracy of the place; while in the background rose wooded coverts, where every tint of early green blended in rich masses of varied foliage.
http://www.gutenberg.org/dirs/etext01/hrtss10.txt
They don't write 'em like they used to...