A spiral of numbers05
19

For the past few days, I've been studying the following question on StackExchange about a spiral of integers. We are looking for a closed formula for the coordinates of the \(n\) -th element in the following integer spiral, which expands from the origin outwards and further and further into infinity:

..  9 10 11 12
23  8  1  2 13
22  7  0  3 14
21  6  5  4 15
20 19 18 17 16

---

First we divide the natural numbers into the following groups:

$$G_1 = \{ 1,...,8 \}\\G_2 = \{ 9, ..., 24 \}\\G_3 = \{ 25, ... 48 \}\\...\\G_k = \left\{ (2k-1)^2, (2k+1)^2 - 1 \right\}$$

We call

$$a^2 = \left(\left \lfloor \left( \frac{\left \lfloor \sqrt{n} \right \rfloor + 1}{2} \right) \right \rfloor \cdot 2 - 1 \right)^2$$

the starting number in a group.

within a group there are four subgroups of equal size (for example for \(G_1\) this is \(g_1 = \{1,2 \}, g_2 = \{3,4 \}, g_3 = \{5,6 \}, g_4 = \{7,8\}\)). when \(n\) changes from one subgroup to the next, the direction on the spiral path (clockwise in our example) is changed at the same time. we obtain the current direction \( b \in \{0,1,2,3\} \) by dividing the position \(n-a^2\) within a group by the number of elements of a subgroup:

$$b = \left \lfloor \frac{ n - a^2 }{ \text{abs}\left( a \right) + 1 } \right \rfloor$$

Now we can determine the first element within a subgroup \(g_n\) by adding to the initial number \(a^2\) a multiple \(b\) of the number \(a+1\) of elements within a group:

$$c = a^2 + b \cdot (a+1)$$

Now we can determine by simple observation:

$$x_{right} = \left(n - c - \frac{ a + 1 }{2}+1\right),\, y_{right} = \left(\frac{ a + 1 }{2}\right) \\ x_{bottom} = \left(\frac{ a + 1 }{2}\right),\, y_{bottom} = (-1) \cdot \left( n - c - \frac{ a + 1 }{2}+1\right) \\ x_{left} = (-1) \cdot \left(n - c - \frac{ a + 1 }{2}+1\right),\, y_{left} = (-1) \cdot \left(\frac{ a + 1 }{2}\right) \\ x_{top} = (-1) \cdot \left(\frac{ a + 1 }{2}\right),\, y_{top} = \left( n - c - \frac{ a + 1 }{2}+1\right)$$

Now we want to represent \(f(n)\) in closed representation and without case distinctions. We use the sign function, the procedure I have described here. To get a pure formula depending on \(n\), we get:

That the whole thing really works, we see with the help of SVG.js in this little visualization:

See the Pen ulam spiral by David Vielhuber (@vielhuber) on CodePen.

The spiral is also known as Ulam's spiral and has an exciting relation to prime numbers.