Predicting Spirals I am currently in the process of analyzing a polyspiral, a spiral where each successive length drawn is increased at specified increment at the same angle.
*Please note the angles selected are the exterior angles or the angle by how much the turtle turns by.
144 degrees:

216 degrees:

versus 140 degrees
 
or 120 degrees:

Is there any way to predict the outcome (maybe the category or type of the spiral) of these spirals mathematically. 
Help would be much appreciated.

Let's say that you turn by an angle $\theta$ each time, and you find that after you've made $m$ such terms, you end up facing in the same direction you started at.  Then it must be the case that you've turned some integer number $n$ of full rotations, which means that
$$
360 n = \theta m
$$
(with $\theta$ measured in degrees.)  Moreover, the number of "points" on your design will be the smallest integer $m$ for which this is the case.  This implies that
$$
360 n = \theta m = \text{lcm} (\theta, 360),
$$
where "lcm" is the least common multiple of $\theta$ (in degrees) and 360.
So, taking your examples in turn:


*

*For $\theta = 144$, we have $5 \times 144 = 2 \times 360$.  So you get a five-pointed star. 

*For $\theta = 216$, we have $5 \times 216 = 3 \times 360$.  So you get a five-pointed star again.

*For $\theta = 140$, we have $18 \times 140 = 7 \times 360$.  So you get an 18-pointed star.

*For $\theta = 120$, we have $3 \times 120 = 1 \times 360$.  So you get a three-pointed "star" (otherwise known as a triangle).


Note that so long as you only use integer numbers of degrees, it must be the case that the number of "points" of the star will be a divisor of 360.
