Fractals & Chaos Recap for 12/15

Today we had the opportunity to play with 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 all 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. We started our analysis with the top half of a new worksheet, exploring and developing summary ideas of iterations of complex linear functions of the form Az+B. We saw that A is the major factor in the type of fixed point we get, while B only seems to affect where the fixed point is. This didn’t come as a significant surprise considering we made a similar observation about the slope and y-intercept of the linear functions we iterated in the real numbers.

We also finally developed a Polar Form for a linear function. Instead of always converting A from polar to rectangular in order to enter it into Complex Paint, we can instead use the form A = Rcos(2πT) + i*Rsin(2πT), where T represents the fraction of a full turn we are attempting to rotate our iterated points by (e.g., if we want a 180° rotation, T = 1/2; if we want a 45° rotation, T = 1/8). This eliminates the need to consider degree vs. radian mode for measuring angles.

Your homework over the weekend: continue the analysis we were doing in class on the Complex Paint worksheet, completing Part II up through T = 0.32. Do not go past that point just yet!

InCA Assignment for 12/13

Check out The Reveal for the New York Times’s graph we looked at this week. Also take a look at the comments, especially the shout-outs that students got for their catchy headlines. Next week, that could be you!


The third method of solving quadratics is to simply use the square root, but there’s a danger here: you still have two solutions. The + is built into the quadratic formula, but it still needs to be added here. And that falls to you, the solver!

Today’s lesson also included a brief revisit to word problems, this time resulting in a quadratic equation.

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Fractals & Chaos Recap for 12/12

After reviewing the rectangular/polar coordinate conversions we started yesterday, we revisited the original composition of transformations we looked at on Monday, noting that the Dilation x1/2, Rotation 45°, and Translation up 4 corresponds to an iteration of the function Az+B, where

  • A = [.5, 45°] = √(2)/4 + √(2)/4i
  • B = (0, 4) = 4i

Iterating this function allowed us to find that the coordinates of the attracting fixed point we identified earlier to be roughly positioned at (-2.605, 4.763) (or equivalent to the complex number -2.605+4.763i).

From there, we looked at a few problems from the back of the Complex Transformations Sheet, identifying again the specific geometric transformations that each complex linear function would produce and sketching the new location of a point that underwent that transformation. Your homework is to finish the back of that sheet

With the time we had left, we introduced a new piece of software: Complex Paint, a tool for more easily illustrating the transformations we have been identifying. We will work with this in more detail tomorrow.

InCA Assignment for 12/12

We had a homework quiz followed by a mastery quiz in class today. Your homework if to finish the mastery quiz if you did not complete it in class.

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Fractals & Chaos Recap for 12/11

Yesterday, we identified that adding complex numbers is equivalent to translating a point and multiplying complex numbers is equivalent to dilating and rotating that point. Furthermore, we noted that while the conventional rectangular coordinates for identifying a complex number are useful for identifying the direction and distance of a translation, to properly understand the dilation/rotation we need the polar coordinates.

Today, we quickly reviewed converting between rectangular and polar form at the start of class, then spent a considerable amount of time in class practicing this conversion and identifying the transformations they refer to. This work was on the Complex Numbers Transformations sheet and together we finished some of the front and about half of the back. Your homework is to finish the front.

InCA Assignment for 12/11

We did some more practice with solving quadratic equations today with a game of Trashketball.

Note: Tomorrow, we will have a HW quiz referring to assignments 3.9, 3.10, 3.11, and 3.12. We will also have a Mastery Quiz on solving quadratic equations.

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Fractals & Chaos Recap for 12/10

We saw yesterday that a sequence of geometric transformations can illustrate the same “fixed point attractor” behavior that we’ve seen when iterating a mathematical expression. We explained this by illustrating that such geometric transformations are analogous to iterating a linear function in the Complex number system. Specifically, adding two complex numbers is analogous to translating, and multiplying two complex numbers is analogous to a combined dilation/rotation.

The conventional rectangular form of a complex number a + bi tells us the horizontal and vertical components of the translation achieved by adding the 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.

We’ll dig in to this some more tomorrow, but for now please watch this video recapping polar vs rectangular coordinates (which the video refers to as “Cartesian coordinates”) and how to convert between the two.

InCA Assignment for 12/10

We didn’t actually finish our lesson yesterday, so today’s HW is the same as yesterday’s. We did some more work with the quadratic formula today, specifically with simplifying the radical expressions we sometimes get when we apply it.

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Fractals & Chaos Recap for 12/9

We wrapped up our conversation about numbers of cycles findable in the Feigenbaum Plot, noting again that for every attracting cycle we find, there is an accompanying “evil twin” repelling cycle. These “evil twin” repelling cycles serve a purpose. As the members of the attracting cycle bifurcate, their patterns branch out further and further. When this pattern of branching reaches one of the values of the repelling cycle – indeed it will reach all members of the repelling cycle simultaneously – the branching stops and the system dissolves into chaos once more. This explains why these spontaneously generating cycles appear in “windows” in the Feigenbaum Plot, with an abrupt end to chaos on the left, and an abrupt resuming of chaos on the right.

Annotation 2019-12-06 074512.png

We wrapped up today with a recap of the work we’ve done so far, occupying our time exclusively within the Real numbers. The last portion of the course will now branch into the Complex numbers, effectively repeating the analyses we’ve done with iterating functions into the complex plane. Our first stop will be to understand how to represent geometric transformation with operations on complex numbers. We modeled this in class by showing how the repetition of a series of geometric transformations can result in the same “fixed point attracting” behavior as we saw with the real numbers.

InCA Assignment for 12/9

Each week, starting today, we will be participating in the New York Times’s What’s Going On in This Graph activity. Click the link to see today’s graph, and go to Desmos to enter some of your thoughts.


We did some more work with the quadratic formula today, specifically with simplifying the radical expressions we sometimes get when we apply it.

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AP Statistics Assignment for week of 12/9

You will spend this week working on your projects. You should strive to be done with data collection by Thursday the 12th, and you’ll have up to Monday the 16th to work on your presentations and written report. Refer to the contents of the midterm project Drive folder for details

We’ll spend the 17th-20th presenting on your findings, and your reports (one per group!) will be due on the 20th.

Fractals & Chaos Recap for 12/6

We have observed that cycles are born in the Feigenbaum Plot in one of two ways: bifurcations of lower cycles and spontaneously out of chaos. Yesterday we understood how these cycles are spontaneously born and how this phenomenon coupled with the self-similarity of the Feigenbaum Plot suggests an order to cycles. What it also gives us is a way of counting how many cycles of each type there are.

We know there are two fixed points on the original f (x) = ax(1 – x). One is attracting for a < 3, the other (at zero) is always repelling.

For 2-cycles, we look at f (f (x)). We see as many as four fixed points on this graph. But two are already members of 1-cycles and must be eliminated. This leaves only two points eligible to be members of 2-cycles, meaning we only have one 2-cycle, which we see bifurcating at a = 3.

To count 3-cycles, we observe that f (f (f (x))) has as many as eight fixed points. A 3-cycle pattern n, ___, ____, n could be explained two ways: as a 1-cycle (n, n, n, n) or as a real 3-cycle (n, o, p, n). So again, we remove the two 1-cycle points and are left with six points eligible to be members of 3-cycles, suggesting two 3-cycles. We see one in the Feigenbaum plot at a ≈ 3.83, but that’s an attracting fixed point. Where’s the other one? It turns out that for every cycle born out of chaos, there is an “evil twin” repelling cycle born as well. As a result, there are actually two 3-cycles born at a ≈ 3.83: one attracting and one repelling.

To count 4-cycles, we observe that f(f(f(f(x)))) has as many as 16 fixed points. But if we see a pattern of n, ___, ____, ____, n, we could explain this by:

  • A 1-cycle: n, n, n, n, n
  • A 2-cycle: n, o, n, o, n
  • A real 4-cycle: n, o, p, q, n

So we eliminate the two 1-cycle points and the two 2-cycle points. This leaves 12 points eligible to be members of 4-cycles, suggesting three 4-cycles. One is the bifurcating cycle we clearly see in the Feigenbaum plot, the second and third are an attracting/repelling 4-cycle pair found at a ≈ 3.96.

To count 5-cycles, we start with the 32 fixed points, eliminate the two that are members of 1-cycles, and observe that the 30 remaining points must create six 5-cycles. One attracting/repelling pair we see at a ≈ 3.74, but the other two are harder to find (one is at a ≈ 3.906, the other pair at a ≈ 3.99028).

Your homework for this weekend: continue this train of thought and identify how many 6-, 7-, and 8-cycles there are. If you’re feeling ambitious, try to locate all of them!

InCA Assignment for 12/6

More work with solving quadratic equations today, this time with equations that did not factor. Our solution? The quadratic formula.

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AP Statistics Assignment for 12/6

By the end of the day today, please write a response to the following statement. Write from your own perspective, not necessarily from the perspective of a stance to which you may have been assigned.

It is necessary to give terminally ill patients a placebo instead of potentially life-saving drugs in the name of proper experimental design.

Log in to this Google Form and record your response there.

Fractals & Chaos Recap for 12/5

We observed yesterday that cycles appear to be “born” in one of two ways: bifurcations of “lower” cycles, and spontaneously arising from chaos. We’ve already shown why cycles bifurcate, so we started today with an explanation about how cycles spontaneously emerge from chaos.

Referring to our previous proof, graphing the function y = ((x)) can be a way of finding new cycles. Fixed points on the graph of y((x)) that are not common with the graph of y = (x) will be the parameters of our 2-cycle. By extension, graphing y = f (((x))), or any number of nested iterations, will give us a tool of finding new cycles. More crucially, this also shows us why cycles spontaneously appear from chaos. Explore the graph here. For a = 3.84, the “wiggliness” of the graph of y = f (((x))) is enough for the fingers of the graph to touch the line y = x. But for a < 3.84, it isn’t. The moment that a becomes large enough for those fingers to touch the line y = x is the moment that a 3-cycle is born.

The graph above shows that it is not possible to get a 3-cycle before a = 3.84, meaning the 3-cycle “window” we see in the Feigenbaum Plot is the first time we get a 3-cycle (addressing one of the other questions we asked yesterday). It also gives a clue to the order of cycles. We’ve already noticed that the Feigenbaum Plot exhibits fractal-like self-similar behavior, and the 6-cycle we observed at a = 3.63 could almost be viewed as two groups of three. If we consider that the 3-cycle at a = 3.84 is “born” from the original fixed point trend we observed for a < 3.0, then we could argue that the 6-cycle is actually two conjoined 3-cycles, each born from the first bifurcation at a = 3.0. This would suggest that there is a 12-cycle for an even lower value of a, born from the second bifurcations 4-cycle (and indeed there is, at a = 3.5821).

The 5-cycle we see at a = 3.74 then is mirrored with a 10-cycle at a = 3.6053, and a 20-cycle at a = 3.5775. This pattern could continue forever, to find any cycle, of any length.

This argument forms the basis for the Sharkovskii order we saw in yesterday’s article. The 3 cycle is the very last cycle to be born out of the chaos of this trend. The 5-cycle is the second-to-last, and the 7-cycle and every other odd-numbered cycle comes before those. But before we get to any odd-numbered cycle, we first would find the 6-cycle (2 x 3). Before that, the 10 cycle (2 x 5); before that, the 14 -cycle (2 x 5), and so on. But before any of those, we find the 12-cycle (4 x 3); before that the 20-cycle (4 x 5); before that the 28-cycle (4 x 7). And so on, reading the Feigenbaum plot right-to-left, until we find our “un-bifurcating” powers of two cycles, stitching back together to 16, to 8, to 4, to 2, and then finally back to 1.

Your homework: read pages 138-145 in Fractals: The Patterns of Chaos, about “Visualizing Chaos”

If you’d like to learn a bit more about the Universality of the Feigenbaum Plot, check out this long article from a class similar to ours at Georgia Tech.

InCA Assignment for 12/5

Part 1 of this unit was on creating graphs of parabolas and other polynomials, identifying their key features. Part 2 is more algebraic, focused on solving quadratic equations by hand and connecting the equations to the graphs more deeplyl. Today was a review of factoring and how that method of algebraic manipulation allows us to solve these equations.

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AP Statistics Assignment for 12/5

Your project proposals are due at the start of class (access the full project details here). When you submit the proposal, please include all of your group members’ names in the filename (LastName.LastName.LastName.MidYearProjectProposal) before sharing it with me.

Today, we watched a Vsauce video about the classic trolley problem. The video deals with ethics, both of the problem itself and with creating a realistic experimental scenario to test how people would react in the situation posed. Tomorrow, we will have an organized and structured conversation about ethics ourselves. Please read over the Ethics Discussion Guidelines for details. At the start of class tomorrow, you will be assigned one of two stances to argue:

  • For some experiments, it is okay to lie to subjects in the name of proper experimental design and to preserve the integrity of what is being tested.
  • It is never okay to lie to subjects in an experiment, regardless of what is being tested.

Tomorrow, you’ll be asked to refer to the Vsauce video and the articles below to inform your arguments. Please read these articles, as well as take notes or even print them out so you can refer to them on Friday.

Fractals & Chaos Recap for 12/4

We spent the first portion of class today exploring the Feigenbaum Plot today by asking a few supplemental questions to question 3 (is there anything to be found after chaos):

  • Which comes first, a 3 cycle (so far found at a = 3.84) or the 5 cycle (found at a = 3.74 but also a = 3.906)?
  • Why do some cycles happen more than once (see the 5-cycle above, or the two 6-cycles at a = 3.63 and a = 3.845 or two 4-cycles at a = 3.5 and a = 3.961)?
  • In general:
    • Where do cycles come from? How are they “born”?
    • Is there some order to cycles?

Instructions on using Paul Fischer-York’s Bifurcation Diagram

  • Use the Darkness slider to make the image darker and easier to see.
  • Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.
  • Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)

After some time to explore, we made a few observations about the general questions. The first is that cycles are born in two ways:

  1. Bifurcations of “lower” cycles, or
  2. Spontaneously arising from chaos

This suggests a certain ordering of cycles: the 2-cycle comes first, then the 4-cycle, the 8-cycle, the 16-cycle, and so on for powers of 2. Eventually, these cycles become so large that the system becomes chaotic. But this only explains cycles that are powers of 2. What about any other? For this, please read this article about the Feigenbaum plot, its constant, and the ordering of cycles.