Simon McGregor on Wed, 28 Jul 2010 08:43:18 -0700 (MST) |
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Re: [game-lang] Another quick tangential comment about integers and reals |
On Wed, Jul 28, 2010 at 2:21 PM, Joel Uckelman <uckelman@xxxxxxxxx> wrote: > Thus spake Simon McGregor: >> >> In regard to rationals and reals, imagine the following: a game is >> played by moving points on a board in R2, up to 1 unit's distance at a >> time (with the Euclidean metric). We can model this using the >> rationals, because rational arithmetic is adequate to determine >> whether two points with rational coordinates are within a rational >> distance of one another. >> >> However, suppose that the provably optimal move in some situation is >> to move your point as far as possible towards another point. Then (in >> general), no move in the rational model is optimal, although there's >> an optimal move in the continuous model. Uh-oh! >> > > I don't follow this last bit of reasoning. It seems to imply that there's > some real r which is nearer to a target t than any rational q is. Yup. Some real (r_1, r_2) which is nearer to a rational target (t_1, t_2) than any rational (q_1, q_2), given the constraints imposed by the movement range rule - that the Euclidean distances of (r_1, r_2) and (q_1, q_2) from a prior rational point (p_1, p_2) are a specified rational distance (1) or less. The need for irrationals comes from the square root sign in the Euclidean metric. Say my point A is at (0,0) and the target point T is at (1,1); my optimal move is to move A as close as possible to T; the rules prevent A's new position from being more than 1 unit away from its current position. * If I am allowed reals, the optimal move is "A to (sqrt 2, sqrt 2)". * If I am only allowed rationals, there is no optimal move, since there is always another rationals move which is closer to T within the allowable Euclidean distance of 1 unit. Does that make sense now? Simon _______________________________________________ game-lang mailing list game-lang@xxxxxxxxx http://lists.ellipsis.cx/mailman/listinfo/game-lang