We’ve been reflecting on the question of why the pattern of attraction for some values of R for the same value of T will produce spiral patterns and why others (specifically slower values of R closer to 1) will produce spokes. A clue was found in discussing T = 0.32, with the note that 0.32 is close to, but not quite equal to, 1/3. So every rotation of 0.32 is close to a perfect 1/3 rotation. But since 0.32 < 1/3, every 3rd step is slightly short of a full 1/3 rotation, so we wind up with a clockwise spiral pattern (T = 0.35 on the other hand would be slightly ahead of 1/3 with every three steps, so it creates a counterclockwise spiral pattern).
But why 3 specifically? 319/1000 is closer to 0.32 than 1/3 is. So is 6/19. Why don’t we see 1000 spiral arms, or 19? The answer to this has to do with a novel way of breaking down real numbers into a sort of skeletal structure: a continued fraction. Each step of a continued fraction produces rational approximations of a real number of progressively improved accuracy. It allows us to claim that some rational numbers are actually “more rational” than others and, even more surprisingly, some irrational numbers are “more irrational” than others. But most immediately, the denominators of these approximations correspond exactly to the numbers of spirals we see.
We confirmed this with T = 15/62 and T = 5/17. The continued fraction sequence for T = 15/62 is [0;4,7,2], producing a first rational approximation of 1/4 and then 7/29. The number 15/62 < 1/4, producing the four CW spirals we see in Complex Paint, and the number 15/62 > 7/29, producing the 29 CCW spirals. The sequence for T = 5/17 = [0;3,2,2], giving approximations of 1/3 (too big -> 3 CW spirals) and 2/7 (too small -> 7 spirals).
Not only does the continued fraction give us rational approximations, it gives us a way of assessing how “good” these approximations are. We notice in the complex paint that the 4 spiral arms of 15/62 last for several steps of R (it’s not until R = 0.99 that we even see the 29 spiral arms), whereas T = 5/17 only shows 3 spiral arms for R = 0.9, and then only barely (we can also somewhat see the 7 spiral arms at this stage). This tells us that 1/4 is a better approximation of 15/62 than 1/3 is of 5/17 (which makes sense, since 15/62 is a lot closer to 0.25 than 5/17 is to 0.3333….).
So, a “good” rational approximation can be illustrated by the following characteristics in Complex Paint:
- It is easy to see, and not hidden by another pattern
- It is straight or spoke-like
- It lasts through several magnifications (R = 0.9->0.95->0.99->0.995->…)
We can see this numerically as well: the third term of the sequence for 15/62 (a_2 = 7) is bigger than the second term (a_1 = 3), suggesting that we’re not adding a lot when we add on the third term of the continued fraction. On the other hand, the third term of 5/17 (a_2 = 2) is less than the second term (a_1 = 3), suggesting that we’ve left off a lot by stopping at just two terms.
All of this leads to why we see 7 “spokes” which very slowly turn into spirals for T = pi. The rational approximation 22/7 is a very common, and accurate to three digits, approximation of pi. The 7 “spokes” we see at low values of R slowly tighten as we move R closer to 1, until we get 113 “spokes.” This suggests there is another, even more accurate rational approximation of pi with a denominator of 113. Your homework tonight is to complete the continued fraction sequence for pi and find that approximation.