Fractals & Chaos Recap for 12/19

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.

Fractals & Chaos Lesson Recap for 12/14

We did some more work with Complex Paint today, in the hopes of “solving” Az+B. To make this easier, we derived the “Polar-Linear” form of this formula, where instead of referring to A in rectangular coordinates, we refer to it in a modified form of polar coordinates, where we use R and T, the fraction of a full turn that the angle of polar form refers to. This allowed us to make the following observations:

  • B has no impact on the type of fixed point we get, just its location
  • If R < 1, the fixed point is an attractor
  • If R = 1, the fixed point is neutral
  • If R > 1, the fixed point is a repeller
  • The denominator of T (in lowest terms) tells us how many spokes we get in a pattern.

The only thing we haven’t figured out yet is where spirals come from, and why spirals turn into spokes when we use values of R closer to 1. We’ll explore that on Monday.

Fractals & Chaos Recap for 12/13

We then introduced our next Very Important Program: Complex Paint. This software was created by former F&C students Devon Loehr and Connor Simpson, updating software originally available only on MacOS9, and will be invaluable to us going forward. We’ll discuss the features of the program as they become relevant, but after downloading/unzipping the folder listed in the Google Drive folder linked, read the README file for help on how to use the software.

For now, we are using “Linear” mode, and working off of Complex Paint Worksheet 1. We’ve observed that in the linear complex expression Az+B, the value of A can affect the type of fixed point we get. For example, the angle of A written in polar form seems to govern the number of spokes we get, and the attracting/repelling/neutral nature of the fixed point is tied to the value of r. The value of B seems to just change its location. We’ve also noticed some unusual differences in the patterns we see: sometimes the path of attraction towards a fixed point forms straight-line spokes, but sometimes it forms spirals instead. Odd…

We’ll have to look at this some more going forward!

Fractals & Chaos Recap for 12/11

We spent some time reviewing answers to the Complex Numbers Transformations sheet from yesterday, and used the principles to find the precise value of the fixed point you were finding graphically last week with the transformations Dilation x1/2, Rotation 45°, Translation Up 4.

We then introduced our next Very Important Program: Complex Paint. This software was created by former F&C students Devon Loehr and Connor Simpson, and will be invaluable to us going forward. We’ll discuss the features of the program as they become relevant, but after downloading/unzipping the folder listed in the Google Drive folder linked, read the README file for help on how to use the software.

Your homework tonight is to finish the back of the Complex Numbers Transformations sheet.

Fractals & Chaos Lesson Recap for 12/10

We reviewed that the values of a and b of the Rectangular Form tell us the horizontal and vertical components, respectively, of the translation achieved by adding a complex number, but this form is not useful for when the number is being used as a multiplication factor. For that, we need the number’s Polar Form. Once finding it, the value of r tells us the dilation factor and the value of θ tell us the rotation factor.

To better understand this requires practice, and so we spend a considerable amount of time in class today converting between rectangular and polar forms for a single complex number and identifying the transformations they refer to. This work was on the Complex Numbers Transformations sheet, and your homework is to finish the front as well as questions 3, 4, and 5 on the back.