Formula for Nautilus shell spiral

Question

I am programming a game and i can't find out the formula to spawn villages for newly created users.

This is how i want it to be. I want it to have a formula, where i give ID and it returns x, y coordinates. Its like the Nautilus shell spiral in x/y coordinates. I dont know how to explain, just giving an example.

Please in an easy-to-read way where i could make the formula with javascript.

ID   x   y
1   15   15
2   20   15
3   25   15
4   25   20
5   25   25
6   20   25
7   15   25
8   10   25
9   10   20
10  10   15
11  10   10
12  15   10
13  20   10
14  25   10
15  30   10
16  30   15
17  30   20

Answer 1

Parametric equations of a ''logarithmic spiral'' (classical name for the nautile shell).

$$\cases{x=0.1*exp(a*t)*cos(t)\\y=0.1*exp(a*t)*sin(t)}$$

with, for example $a=0.2$, for $t=0...(8*\pi)$ with a small step for $t$.

In this way you get 4 turns (4 is the number of times one take a 2\pi turn); take $(10*\pi)$ if you want one more turn, etc.