Simon Tatham has produced fractals derived from Newton-Raphson iteration which are quite interesting to look at.
The main idea here is that if you want to find a root of some function f, then you start from a guess a0; then compute a1 = a0 - f(a0)/f'(a0); geometrically this corresponds to replacing the graph of the function with its tangent line at (a0, f(a)0) and finding the root of the tangent line. Then starting from a1 we find a2 and so on. If you're already close to a root you'll get closer. But if you're far away from a root unexpected things can happen; the set of all starting points a0 for which the sequence (a0, a1, a@, a3, ...) converges to a given root of f is fractal.
(I've mentioned this before, and so has John Armstrong, but Tatham's pictures are better.)
Showing posts with label Simon Tatham. Show all posts
Showing posts with label Simon Tatham. Show all posts
19 January 2009
29 August 2008
The Olympic poset
During the Olympics (yeah, I know, you've all forgotten about the Olympics), there was much argument about whether the right thing to do, if we want to determine which country "won the Olympics", is to count the country that got the most gold medals (China), the most medals (the US), or something else -- say a points system that allocates three points for a gold, two for a silver, and one for a bronze.
Simon Tatham has prepared a Hasse diagram of the medal table. The main idea is:
Alternatively, we could say country A does strictly better than country B if and only if A gets more points than B under all weighting schemes in which we assign x points for a gold, y points for a silver, and z for a bronze, with x ≥ y ≥ z ≥ 0. This seems like it's equivalent to Tatham's order, but I haven't thought that hard about it.
One could extend this to include the population of a country; the natural order there would be that A did strictly better than B if the medal score of B can be transformed into that of A by a sequence of medal additions, medal upgrades, and population reductions. The idea here, of course, is that if two countries win the same assortment of medals, the one with lower population did better. But going there is dangerous; do we then take into account GDP? Popularity of sports in general in the country? The fact that the particular set of sports in the Olympics is more popular in some countries than others?
Simon Tatham has prepared a Hasse diagram of the medal table. The main idea is:
So we want to say that one country has done strictly better than another if the medal score of the latter can be transformed into the former by a sequence of medal additions and medal upgrades.This gives a partial order on the countries.
Alternatively, we could say country A does strictly better than country B if and only if A gets more points than B under all weighting schemes in which we assign x points for a gold, y points for a silver, and z for a bronze, with x ≥ y ≥ z ≥ 0. This seems like it's equivalent to Tatham's order, but I haven't thought that hard about it.
One could extend this to include the population of a country; the natural order there would be that A did strictly better than B if the medal score of B can be transformed into that of A by a sequence of medal additions, medal upgrades, and population reductions. The idea here, of course, is that if two countries win the same assortment of medals, the one with lower population did better. But going there is dangerous; do we then take into account GDP? Popularity of sports in general in the country? The fact that the particular set of sports in the Olympics is more popular in some countries than others?
Subscribe to:
Posts (Atom)
