An easy starter problem is taking the endpoint to be any two knight moves from the origin. Sometimes it is better to move by three diagonals, and sometimes it is better to move by two horizontals or two verticals. The endpoint +(3,1) is also worth considering. Three diagonals is better than two Knight's moves, but not as good as one night move and one horizontal. Generally it seems best if (X,Y) is the total remaining distance to chose the move (x,y) for which x/y most closely matches X/Y, only resorting to horizontal and vertical at last. Is there a proof? I really love lattice walk problems, think that they are entirely practical, and with many branch points into other areas of mathematics and physics. Most of what I have seen has to do with counting and generating functions. The original article by Polya is an incredible landmark, and Alin Bostan's Habilitation is also great. Neither of these references deal with your question, but oh well, worth reading anyways. One critique, I'm not sure that I would say octagon, because it seems more like a square to me. Happy wanderings, --Brad On Wed, Oct 14, 2020 at 3:36 PM Colin Wright <math_fun@solipsys.co.uk> wrote:
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
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