Basic principle is to plot the 'seeds' at a step wise increasing intervals in polar coordinates where the angle is equivalent to the number of steps multiplied by a 'seed factor' and distance from the origin is proportional to the square root of the number of steps:

- x is 'seed factor'
- n is number of steps

This results for a value of 1/12 of the seed factor in the following steps: Only the fractional part of the 'seed factor' x has any significance: any integer part of x will result in full rotations.

To make this a bit more colorful I added color and the color will shift in a number of 'color steps', the interactive version: jvdm.info/NumberTheory/spiral.html. In order to better understand the patterns experimented with some numbers. Rational numbers can be represented as an integer numerator and a non-zero integer denominator. The number of arms the spiral has for rational numbers is the greatest common divisor of the numerator and denominator.