Does there exists a one-to-one and onto map $f:\mathbb{N}\rightarrow \mathbb{Z}[i]$ such that $|f(n+1)-f(n)|=\sqrt{5}$?
That is to get from $f(n)$ to $f(n+1)$ you have to move like a knight in chess: move two vertically and then one horizontally or two horizontally and then one vertically.
I know there is a lot written about when a knight tour is possible on a finite board. Check out the wiki and wolfram for some background. I can see how you might be able to do this in a type of infinite outward spiral if you could travel down a corridor of an arbitrary length. It seems that there is already some interest in that. But you would also need to make sure that you "end" your journey on the far side of the corridor in such a way that you could turn to travel down the next corridor.
Motivation: The key part of my question is the infinite part. I have recently been thinking about how we can traverse infinite spaces which is particularly relevant to mathematicians (as opposed to say computer scientists). For example when computing something over the entire set of Guassian integers. Let's say that some function $f$ does exist. Then $$\sum_{z\in \mathbb{Z}[i]} \alpha_z =\sum_{n=1}^\infty \alpha_{f(n)}$$
I am not sure that such a thing could practically be done. But my point is that it's valuable to know different ways we can traverse infinite spaces.
This does look pretty convincing: http://demonstrations.wolfram.com/AnInfiniteKnightsTour/
– Mason Jun 30 '18 at 22:51