Wed 14 Oct 2020, 16:36, Colin Wright wrote:
Embed Z^2 in R^2, and define the distance (0,0) to (x,y) to be |x|+|y|. Disks around a point are square diamonds.
(...) say that the distance from (0,0) to (1,1) is sqrt(2), and that we can only get somewhere by taking diagonal hops followed by taxi-cab journeys, and the *distance* is the minimum taken over all possible journeys. Disks around (0,0) are now octogons. I think.
I think you mean by octagon (or whatever shape of the disc) the convex hull in R^2 of the points in Z^2 having a given maximum distance from the origin, is that correct? Then the disc of radius 4 is not an octagon but a 12-gon with vertices at (+-4, 0), (0, +-4), (+-2, +-3), (+-3, +-2), the latter having distance 1+2 sqrt(2) = 3.828 from the origin. The distances of the grid points near the origin are as follows, in milli-units: [7071 6657 6243 5828 5414 5000 5414 5828 6243 6657 7071] [6657 5657 5243 4828 4414 4000 4414 4828 5243 5657 6657] [6243 5243 4243 3828 3414 3000 3414 3828 4243 5243 6243] [5828 4828 3828 2828 2414 2000 2414 2828 3828 4828 5828] [5414 4414 3414 2414 1414 1000 1414 2414 3414 4414 5414] [5000 4000 3000 2000 1000 0000 1000 2000 3000 4000 5000] [5414 4414 3414 2414 1414 1000 1414 2414 3414 4414 5414] [5828 4828 3828 2828 2414 2000 2414 2828 3828 4828 5828] [6243 5243 4243 3828 3414 3000 3414 3828 4243 5243 6243] [6657 5657 5243 4828 4414 4000 4414 4828 5243 5657 6657] [7071 6657 6243 5828 5414 5000 5414 5828 6243 6657 7071] - Maximilian