On a two dimensional grid is there a formula I can use to spiral coordinates in an outward pattern?

Question

I don't know exactly how to describe it, but in a programmatic way I want to spiral outward from coordinates 0,0 to infinity (for practical purposes, though really I only need to go to about +/-100,000:+/-100,000)

So if this is my grid:

[-1, 1][ 0, 1][ 1, 1][ 2, 1]
[-1, 0][ 0, 0][ 1, 0][ 2, 0]
[-1,-1][ 0,-1][ 1,-1][ 2,-1]

I want to spiral in an order sort of like:

[7][8][9][10]
[6][1][2][11]
[5][4][3][12]
etc...

Is there a formula or method of doing this?

Answer 1

Here’s a recipe for finding the coordinates of your position after $n$ steps along the spiral.

It’s simpler to number the positions on the spiral starting at $0$: position $0$ is $\langle 0,0\rangle$, the origin, position $1$ is $\langle 1,0\rangle$, position $2$ is $\langle 1,-1\rangle$, and so on. Using $R,D,L$, and $U$ to indicate steps Right, Down, Left, and Up, respectively, we see the following pattern:

$$RD\,|LLUU\,\|\,RRRDDD\,|LLLLUUUU\,\|\,RRRRRDDDDD\,|LLLLLLUUUUUU\;\|\dots\;,$$

or with exponents to denote repetition, $R^1D^1|L^2U^2\|R^3D^3|L^4U^4\|R^5D^5|L^6U^6\|\dots\;$. I’ll call each $RDLU$ group a block; the first block is the initial $RDLLUU$, and I’ve displayed the first three full blocks above.

Clearly the first $m$ blocks comprise a total of $2\sum_{k=1}^{2m}k=2m(2m+1)$ steps. It’s also not hard to see that the $k$-th block is $R^{2k+1}D^{2k+1}L^{2k+2}U^{2k+2}$, so that the net effect of the block is to move you one step up and to the left. Since the starting position after $0$ blocks is $\langle 0,0\rangle$, the starting position after $k$ full blocks is $\langle -k,k\rangle$.

Suppose that you’ve taken $n$ steps. There is a unique even integer $2k$ such that $$2k(2k+1)<n\le(2k+2)(2k+3)\;;$$ at this point you’ve gone through $k$ blocks plus an additional $n-2k(2k+1)$ steps. After some straightforward but slightly tedious algebra we find that you’re at

$$\begin{cases} \langle n-4k^2-3k,k\rangle,&\text{if }2k(2k+1)<n\le(2k+1)^2\\ \langle k+1,4k^2+5k+1-n\rangle,&\text{if }(2k+1)^2<n\le 2(k+1)(2k+1)\\ \langle 4k^2+7k+3-n,-k-1\rangle,&\text{if }2(k+1)(2k+1)<n\le4(k+1)^2\\ \langle -k-1,n-4k^2-9k-5\rangle,&\text{if }4(k+1)^2<n\le2(k+1)(2k+3)\;. \end{cases}$$

To find $k$ easily, let $m=\lfloor\sqrt n\rfloor$. If $m$ is odd, $k=\frac12(m-1)$. If $m$ is even, and $n\ge m(m+1)$, then $k=\frac{m}2$; otherwise, $k=\frac{m}2-1$.

Answer 2

Here is some code that finds the $n$-th point in the spiral. Unfortunately it spirals the other way but perhaps it helps anyway.

function spiral(n)
        k=ceil((sqrt(n)-1)/2)
        t=2*k+1
        m=t^2 
        t=t-1
        if n>=m-t then return k-(m-n),-k        else m=m-t end
        if n>=m-t then return -k,-k+(m-n)       else m=m-t end
        if n>=m-t then return -k+(m-n),k else return k,k-(m-n-t) end
end

See http://upload.wikimedia.org/wikipedia/commons/1/1d/Ulam_spiral_howto_all_numbers.svg.