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.

Fractals & Chaos Recap for 12/6

You dived in deep to one of the questions we asked in our exploration of the Feigenbaum Plot: how many cycles of each type are there? 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. Moreover, not all cycles are necessarily bound by real numbers, instead moving to the realm of complex numbers. This results in many more 4-, 5-, 6-cycles and beyond.

Furthermore, 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.

Fractals & Chaos Recap for 11/5

We started in class today watching a Numberphile video about the Feigenbaum Plot and an interesting number that can be found in it that appears to have some surprising universality. The video does a great job of recapping what we’ve done over the past few days, so watch it if you’ve missed anything. You were also given an article to read about the plot and this constant.

The video also makes a point that the pattern we see in the Feigenbaum Plot is not unique to the Logistic function we’ve been iterating. In fact, any function that creates a bound area with the x-axis can exhibit such behavior. With the rest of the period, play around with Paul Fischer-York’s Bifurcation Diagram. Use the dropdown in the upper-right corner to examine diagrams for other functions. Some other functionality:

  • 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!)

Use this map to explore the Sharkovskii Ordering mentioned in the article above. Why does that ordering make sense given this picture?

Fractals and Chaos Recap for 12/4

We explored cycles a bit more in class today, noting that if {p,q} are the two values of a 2-cycle, then by definition (p) = q and (q) = p. Put another way, ((p)) = p and ((q)) = q. What’s important to note, though, is that the reverse is not true. If ((n)) = n, then n could be a member of a two cycle (n,m,n,m,n,m,…or it could just be a fixed point: (n,n,n,n,n,n,…)

There were two important takeaways from this algebraic definition of a 2-cycle. The first is that the local slope (Instantaneous Rate of Change) to all points of a 2-cycle, or indeed any cycle, is always the same. As a result, all points of a cycle will pass an IRC of -1 simultaneously, meaning all points of a cycle bifurcate simultaneously. This served as our proof of Question 2 posed early in our exploration of this pattern.

The second takeaway is 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!

Fractals & Chaos Lesson Recap for 12/3

We finally found our first 3-cycle, at a value of a = 3.83. Oddly, this was after chaos emerged, suggesting that sometimes, inexplicably, chaos can turn into order. This of course raised a slew of new questions:

  1. We also found a 5-cycle at a = 3.74 and a = 3.906. Is there a 3-cycle before a = 3.74? Or does the 3-cycle necessarily come after the 5-cycle?
  2. For that matter, why do some cycles happen more than once? A 6-cycle can be found at a = 3.63 and 3.845. A 4-cycle can be found at a = 3.5 and 3.96. Where are these coming from?
  3. What even is the order of cycles, anyway? It seems that powers of 2 come first, born out of bifurcations from our original pattern of fixed points.
  4. Finally, where do the cycles that emerge from chaos come from? Why do they emerge? How are they “born”?

To make exploring these equations easier, I’ve created a Logistic Orbit Iterator on Google Sheets. Make a copy of it and save it to your drive, and you can edit the value of a and the seeds to get a picture of the destinations of the orbit (or scroll down to the bottom and see how it behaves after 500 iterations).

Tomorrow, we’ll finally see the full picture of what we’ve been discussing.

Fractals & Chaos Lesson Recap for 11/30

We did some more research in the logistic map and the catalog of behavior we’ve been using for the past few days. We’ve answered one question from yesterday’s list, the first one: The first bifurcation happens at precisely a = 3.0. With a proof in class, we showed that this is because for a = 3.0, the slope of the curve at the value of the fixed point (which happens to precisely equal 2/3) is exactly -1. Any value of a less than 3.0 will have a slope at its fixed point that is shallower than -1, and as a result the fixed point is an attractor. Any value of a greater than 3.0 will have a slope at its fixed point that is steeper than -1, and as a result the fixed point is a repeller. So when we notice that there’s a 2-cycle at a = 3.1, this is because the fixed point is a repeller, pushing the orbit to the 2-cycle. This 2-cycle exists for all values of 3.0 < a < 3.1.

As a clue to further revelations, I also provided the answer to question 2: In fact, the 2 elements of the 2-cycle split simultaneously, meaning that the 2-cycle splits directly to a 4-cycle (and furthermore that many students’ hunt for the elusive 3-cycle within the interval 3.4 < a < 3.5 is fruitless!).

Nobody has found a 3-cycle yet, or indeed any cycle that isn’t a power of 2 (we found an 8-cycle at a = 3.55, a 16-cycle shortly thereafter, and a 32-cycle shortly after that).

Your homework this weekend is to turn your attention to the chaos past a = 3.6 and to address question 3. Is there truly only chaos there? Or is there something else, lurking in the shadows…

Fractals & Chaos Recap for 11/29

We did some more work collecting information for the Catalog of Behavior for the Logistic Map function we introduced on Monday, getting a clearer idea on the changes in behavior as a increases towards 4.0. The patterns we observed raised a few questions:

  1. Somewhere within the interval 3.0 < a < 3.1, the single fixed point attractor “bifurcates” (i.e., splits) into a 2 cycle. Where exactly does this happen?
  2. Later, the 2-cycle becomes a 4-cycle. Do all points of the cycle bifurcate simultaneously, or can one split before another? In other words, is there a 3-cycle within the interval 3.4 < a < 3.5, or does the 2-cycle split directly to a 4-cycle?
  3. Once chaos appears past a = 3.6, is that it? Or is there anything more to be found there?

Continue your explorations of this function to try and answer these questions. Use the applets posted yesterday. I have a special prize for the first person who can find a 3-cycle…


Fractals and Chaos Recap for 11/28

We continued our discussion of the logistic map from yesterday by iterating the function for various values of the growth parameter a. We observed that if a is too low, the population will die out and the destination of the orbit is zero. Once a passes 1, the population will eventually settle at some proportion of the maximum population; for example at a = 2.6, the orbit of iterations settles on a value of approximately 0.6154 (precisely, this is 8/13 of the possible maximum population). For a = 3.2, we observed a two-cycle of {0.5130,0.7995}, suggesting that the population here will year by year fluctuate between 51% and 80% of its possible maximum population. For a = 3.6, we observed two bands of chaos bound within (0.32,0.6) and (0.79,0.9), suggesting that the population never dies out, but never settles at a stable value (or set of values).

Tonight, please continue to explore values of a and the destinations of orbits within this graph. Use the apps at to help, as well as this Logistic Function Cobweb Diagram I made in Desmos. Keep track of your observations on the Catalog of Behavior you got in class. By tomorrow, you should at least have an observation for every tenth value of a between 2.2 and 4.0.

Fractals & Chaos Recap for 11/27

We took some time today to finish looking at the functions in question 5 of the Iterated Functions sheet, exploring different cases of what happens with the slope of a curve immediately at its fixed point is precisely 1.

From there, we moved to a different family of functions defined by the graph y = ax(1-x). This function is called a logistic map, and it represents the growth of a capped population where x represents the ratio of the current population to a maximum sustainable population (as defined by the carrying capacity), and a is known as a fecundity rate. Naturally, both x and y are bound within the interval (0,1), as a population that reaches 0 is extinct and a population that reaches 1 will necessarily exceed its carrying capacity and become doomed. Your homework tonight is to consider this equation and those limits and to derive the boundaries for a.

Fractals and Chaos Recap for 11/26

We discussed the results of 4g, 4h, and 4i from the Iterated Functions sheet and observed how the discussed the ancient divide and average method of approximating irrational square roots relates to 4i.

We then revisited an observation we made before break: the graph of a function “near” a fixed point will look linear if you magnify it enough, even if the function itself is not linear. As a result, we can use the rules we derived for classifying fixed points based on the slope of a linear function to help us classify fixed points now. For example, if the “nearby” slope around a fixed point is positive and greater than that of y=x (i.e., is greater than 1), then the fixed point will be a “direct attractor.” This raised the question: What if the nearby slope is exactly equal to 1? This occurs when the graph of f(x) is tangent to the graph of y = x at the location of the fixed point.

It turns out that the fixed point has no unique classification, as behavior of seeds to the left of the fixed point differs from the behavior of seeds to the right. A fixed point could be directly attracting on one side, but directly repelling on the other!

Your homework tonight is to finish the Iterated Functions sheet by exploring the equations of part 5. All five pairs of functions have a fixed point at 0, all of which having a nearby slope of 1. Use the Geogebra apps on to make a graph and test the behavior of seeds on both sides of the fixed point and write down your observations on the Iterated Functions Supplement. All five examples of two equations each: the positive and negative version. Make sure you look at both!

Fractals & Chaos Recap for 11/19

We observed some patterns that emerge in cobweb diagrams for certain types of fixed points. For an attractor, the pattern of steps in the cobweb diagram will be drawn towards the fixed point; for a repeller they move away. Furthermore, if the fixed point is direct, the pattern of steps will look like actual steps as they move towards or away from the fixed point, trapped between the two lines defined by the function we are iterating and y = x. If the fixed point is alternating, on the other hand, the pattern will spiral around the fixed point with each step, producing a picture that most clearly gives the cobweb diagram its name.

From there, we took our first look at non-linear functions, realizing quickly that in addition to having more than one fixed point, such functions can have different types of fixed points. One could be a direct repeller, while another an alternating attractor. The cobweb diagram remains our best way to observe these differences, and we used the Geogebra-based applications found here to create them.

We looked at 4a (y = x^2) together, classifying the fixed point at 1 as a direct repeller and the fixed point at 0 as a direct attractor. We furthermore observed that -1 is a “pre-image” to the fixed point at 1, and therefore we have different behavior for different seeds. Your homework tonight is to continue and look at 4b-4f, using the cobweb diagram app linked above and the Iterated Functions Supplement to keep track of your results.

Snow Day Items of Note

Happy snow day, all! I hope you are able to stay warm, dry, and safe today. It was a mess out there this morning, and it still is in many places, so if you can avoid going out you probably should!

There will be no make up assignments over the weekend for any of my classes, with the exception of AP Statistics. For you, I would like you to watch a pair of videos that will give you a solid introduction to our next chapter about observational studies and experiments. They are

  • The Question of Causation – A historical story describing how researchers untangled the relationship between smoking and lung cancer
  • Designing Experiments – A discussion of the difference between observational studies and experiments, with examples about marine life on the remote Line Islands and a medical study about osteoarthritis treatments.

Have a good weekend!

Fractals & Chaos Recap for 11/15

We continued our work with the Iterated Functions sheet, making some observations about how we might predict the classification of fixed point we get based on the parameters of the function we are iterating (see a friend in class for the exact notes on this if you missed them!)

We also introduced a new way of visualizing the behavior of functions: the cobweb diagram. In this style of graph, we draw the function we are iterating on the same plot as the line y=x. We pick a seed and move vertically to the graph of y=f(x), then horizontally to the line y=x, resulting in the output we just got becoming the input for the next iteration. We then move vertically again to y=f(x) and horizontally again to y=x, continuing until we get a view of the behavior of the function.

Your homework tonight then is to finish question 1 (parts f and g) and to do parts a, b, and c of question 3. Draw both the time diagram and cobweb diagram for each of these functions, and try to make some observations about how the patterns of the two representations align with each other.

Fractals & Chaos Recap for 11/14

We’ve started down the road towards understanding the mathematics of chaos by iterating linear functions. Imagine a recursively-defined sequence, where each term of the sequence is defined based on the previous value. That’s what we’re doing here: we start with a seed, then plug that seed into a formula. Each output becomes the next step’s input, and we seek to understand the long-term behavior of formulas.

We are primarily working with this classwork sheet: Iterated Functions. We’ve done parts a, b, and c for part 1, looking at the behavior of seeds both numerically and graphically, creating a plot of steps vs. value called a Time Diagram. In all three examples, we found a fixed point: a value of a seed that is constant through the formula (plugging that value in gets that same value out). Some of the fixed points were attractors, and some were repellers. We discussed how iterating the function repeatedly is a fine way to find an attracting fixed point, but obviously won’t work to find a repeller. The only option there is to solve the equation f(x) = x.

Tonight for homework, please do parts d and e of question 1, then all of question 2. You should work on identifying and classifying the fixed point (as an attractor or repeller).

Fractals & Chaos Recap for 11/13

We kicked things off today with a video on Musical Fractals in jazz music that I came across this weekend. It is pretty awesome, and if you can get through the somewhat tricky bit in the middle, the pay off at the end is really cool (there’s another interesting video that Vox just released about the song central to this video – Coltrane’s Giant Steps – that I recommend checking out as well for an interesting lesson in music theory and the circle of fifths).

After this, we shared some of our planetary designs, and then finished off the Nova Documentary from last week.

Tomorrow, we start our work on understanding the mathematics of chaos. Don’t miss it!

Fractals & Chaos Recap for 11/9

We discussed the reading from Fractals: The Patterns of Chaos and looked at some references to the “Butterfly Effect” in popular culture. We spent most of the rest of the period playing with the Solar System simulator online.

For Monday, please read pages 49-54 from your book, the section on “The Fractals and Chaos of Outer Space.”

Fractals & Chaos Lesson Recap for 11/8

We discussed some key terms that we observed in the first part of the Nova documentary (specifically: fixed point, attractor vs repeller, and strange attractor) and we modeled some of these terms in action with some magnet pendulums. At the close of the period, we also looked briefly at a Solar System simulator found here.

For tomorrow, please read the three sections mentioned on yesterday’s post from Fractals: The Patterns of Chaos