Predicting Spirals

Question

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.

Answer 1

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.