We started today with a debrief on the additional two continued fractions you were assigned to create yesterday:
- 7/38 = [0;5,2,3], creating successive approximations of 1/5, 2/11, and 7/38
- 71/360 = [0;5,14,5], creating successive approximations of 1/5, 14/71, and 71/360
We continued the conversation by observing that not only does the continued fraction give us rational approximations, it gives us a way of assessing how “good” these approximations are. We’ve already noticed in 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 (remember, each term adds on to the denominator). 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 continued fraction sequence for T = pi is [3;7,15,1,292,…]. The very first rational approximation we get for pi is 22/7, which has three digits of accuracy after only one iteration. This is a very good approximation of pi, one you probably used in middle school, which is why the 7 spiral arms we see in Complex Paint are so persistent. You’ll notice that the next term of the sequence, a_2 = 15, is followed immediately by a_3 = 1, the lowest value we could possibly add. This suggests that the second rational approximation, 333/106, is very bad. Even though it gives us 5 digits of accuracy, we don’t even see 106 of anything in Complex Paint. Instead, the third rational approximation, 355/113, produces seven digits of accuracy with a next term of 292, meaning we “lose” very little accuracy by stopping our continued fraction there. The pattern we get in complex paint is 113 spokes.
Finally, this also gives us a way of saying that PHI, the golden ratio, is the most irrational number. Its continued fraction sequence is [1;1,1,1,1,1,…], the worst possible sequence we could get. This is why in Complex Paint we never see spokes, and why we can see multiple spiral patterns within the same value of R. None of the rational approximations we create are “good” approximations.
This fact about Phi is also why the Golden Ratio comes up so much in nature: sprouting leaves or seeds in rotations around a central stem by a quantity of the golden ratio will guarantee that your seeds or leaves don’t line up. All leaves get some sun exposure, and you’ve maximized the quantity of seeds. See this Math is Fun page for a great explanation of this phenomenon, or Vi Hart’s series of videos on the topic starting here.
Over the break, please complete the back of the Complex Paint worksheet (noted “Linear Lab”). It’s a great way to reflect and summarize all the work we’ve done with complex linear functions.