If you restrict the hops to integer multiples of the allowed moves, then the set of points at a given distance from (0, 0) is going to be finite and lie on integer grid points. If you want to associate a polygon with it, you could define it as the convex hull of all points at the given distance from (0, 0). Consider the case where diagonal moves are allowed, but not knight moves. What is the set of points that are distance 4 from (0, 0)? They are: (4, 0) (0, 4) (-4, 0) (0, -4) And that's it. There are no diagonal moves involved in any of these points, since any diagonal move would result in a non-integral distance. The convex hull is a diamond. For a distance of sqrt(2), we get: (1, 1) (-1, 1) (-1, -1) (1, -1) In this case the convex hull is a square. On the other hand, if we want a distance of 1 + sqrt(2), then we get: (2, 1) (1, 2) (-1, 2) (-2, 1) (-2, -1) (-1, -2) (1, -2) (2, -1) So in this case the convex hull is an octogon. On the other hand, if the definition allowed arbitrary multiples of the allowed moves, then any point could be reached by: a*(1, 1) + b*(-1, 1) + c*(1, 0) + d*(0, 1) where a, b, c, and d are real rather than restricted to integers. (And only one of a, b is non-zero, and only one of c, d is non-zero, since we only care about minimal paths). Any given minimal path would then consist of a 45 degree diagonal segment, of any length, and a horizinal or vertical segment, again of any length. Determining the path is simple: Choose a diagonal segment that places you on a horizontal or vertical line with the point, then complete the path with a horizonal or vertical segment. "Circles" centered at (0, 0) would then be continuous curves consisting of the set of points at some given distance from (0, 0). I haven't looked looked at it closely, but I think you'd end up with an octogon in all cases (where the radius is > 0). Adding knight's moves to the mix complicates it a little, but I think the same basic approach would work. In all cases, the vertices of the polygon correspond to points where a particular class of line segment appears or disappears from the minimal path. Tom Colin Wright writes:
I've been asked a question to which I don't know the answer. Although this is not unusual, I'm wondering if there is an answer at all. If there is one, the audience here is likely to know of it.
Let's start with the well-known to give a context.
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.
We now depart into vaguely familiar territory, but I'm finding it hard to pin down exactly what the question is, so I'm going to ask for some leniency, and for some creativity in finding the right question that's close to what I might be asking.
When we use the Taxi-Cab metric above we are saying that the distance from (0,0) to (1,1) is 2. So let's short-circuit that and 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. Are they regular? I suspect not.
And then add that we are permitted to make a knight's move of distance sqrt(5). So journeys now consist of diagonal steps, knight's move, and taxi-cab journeys, and the distance from (a,b) to (c,d) is the minimum over all such journeys.
Disks around (0,0) are now 16-gons, but are not regular.
Obviously we can continue, but the question is:
* Have these specific n-gons been studied?
* Do these specific n-gons have names?
* Do they have any interesting properties?
Thanks for reading, I look forward to your thoughts, which I'll pass on to my interlocutor.
Colin -- The power of accurate observations is commonly called cynicism by those who haven't got it. -- George Bernard Shaw