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