Fractals & Chaos Recap for 1/16

We discussed an open problem related to the connectedness of the Mandelbrot Set today. It seems silly, but “connected” is actually a tough idea to define mathematically. We discussed the following ideas:

  • Not disconnected: That is, the set is not divided into pieces. We can’t draw a line “between” pieces of the set without crossing the set. The Mandelbrot set is not disconnected
  • “Simply” connected: Both the set and its complement (everything not a member of the set) are not disconnected. A circle is not simply connected, because it divides the plane into regions inside and outside of the circle, which are disconnected from each other by the circle. A disk — a circle that includes its interior — is simply connected, as anything not in the disk is outside the disk. The Mandelbrot Set is simply connected
  • “Path” connected: Any two points of the set can be joined by a path that stay in the set. We used the Topologist’s Sine Curve to define an example that is not “path” connected. The Mandelbrot Set, though, is path connected.
  • “Locally” connected: Around any point of the set, we can draw a circle small enough such that the set is not connected inside of it. We used the idea of a comb defined by the line segments y = 1/2^x from x: [0,1] as an example of a space not locally connected. Mathematicians are still unsure whether the Mandelbrot is Locally Connected or not, an open problem known as the MLC.

The last topic, which we will get into tomorrow, is a brief history lesson of the Julia Set that starts in Newton’s Method for Approximation.

For tomorrow, please read this article from the November, 1991 issue of Science News, published just after an important discovery was made about the border of the Mandelbrot Set, as well as this follow-up of sorts from 2017 in Scientific American

Fractals & Chaos Recap for 1/14

The central question I have asked of you is In what way are the shape of the Julia Set, the size of the cycle with it, and its location in the Mandelbrot Set related to each other? Today, we started to answer this question.

First, we observed that orbits along the horizontal x-axis of the Mandelbrot Set bifurcate in the same way that they do in the Feigenbaum plot, and for good reason. Furthermore, we noticed that elsewhere along the edge of the central cardioid of the Mandelbrot Set, we can find balls that yield cycles of any magnitude, and in general the size of the ball is inversely related to the magnitude of its cycle (bigger ball = smaller cycle).

When we look at the Julia Sets we get in each of these balls, we realize that each of them have a central area, centered at the origin, and each with an infinity of “pinch points” found along its edge. The number of areas pinched together at this point exactly matches the size of the orbit contained! Given a mystery Julia Set, then, we can determine the size of its contained orbit just by its shape!

Finally, we observed that every ball connected to the central cardioid has an infinity of balls attached to it, also of decreasing size. The fractal nature of the Mandelbrot Set manifests itself here: the largest sub-ball corresponds to a bifurcation, the next largest to trifurcations, and so on. Thus, the 2-cycle found in the San Marco Julia Set can bifurcate into a 4-cycle, trifurcate into a 6-cycle, or any other multiple of two. The 3-cycle found in the Douady Rabbit can bifurcate to a 6-cycle, trifurcate to a 9-cycle, and so on for any multiple of 3. And this pattern extends to the Julia Sets themselves: the 3-ball off of the 2-ball creates a Julia Set that is a San Marco of Douady Rabbits:

3 off 2

San Marco made up of Douady Rabbits — The 2-cycle of the San Marco trifurcates to create a 6-cycle

And the 2-ball off of the 3-ball creates a Julia Set that is a Douady Rabbit of San Marcos:

2 off 3

Douady Rabbit made up of San Marcos — The 3-cycle of the Douady Rabbit bifurcates to create a different 6-cycle

With this in mind, we can play the game: Find the Julia Set. For each Julia Set in the attached file, use the shape/structure of the Julia Set to predict where in the Mandelbrot Set it can be found, and as a result the magnitude of the cycle found within it. Then verify your predictions using Fractal Zoomer.


Fractals & Chaos Recap for 12/20

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, 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.

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 12/7

We closed out our work with the Feigenbaum Plot today and took a moment to reflect. With our work in the real numbers, we started with linear functions (mx+b) and quickly discovered nearly everything there was to find. Linear functions are easy and non-chaotic. When we moved to non-linear functions, things started getting interesting. We looked at a few examples of the form x^2 + c, but did a deep dive with the logistic map of ax(1-x). For a single parameter, we could make a cobweb diagram to show the behavior of that specific function, but the really interesting things happened when we made our catalog of behavior for all parameters, creating the Feigenbaum Plot.

We will follow the same path through the forest of Complex Numbers. We will start with linear functions of the form Az+B, and understand what we can find there. We’ll then move on to non-linear functions, specifically of the form z^2 + C, and look at the behavior of single, specific values of C. Eventually, we will move to a catalog view there and see what we find.

Today was our first step towards that goal, with a discussion of how arithmetic on complex numbers can mimic geometric transformations. Complex numbers have two coordinates, a real part and an imaginary part, so their operations provide a convenient way of movie around the coordinate plane. Specifically, adding complex numbers produces translations and multiplying complex numbers produces a dilation/rotation. Exactly how the dilation/rotation is understood requires another way of referring to these complex numbers: Polar Form. We will discuss this on Monday.

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 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 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 11/1

We started today with announcing the winners of the fractal art show competitions. Congratulations to all who won, and thanks to all for particpating!

We finished our conversation about the 4-dimensional hypercube, concluding that there are 32 segments and 24 squares to be found in the patterns of the table we created in class yesterday. Then we talked about how to sketch a hypercube, and looked at some animations of folding/unfolding cubes and hypercubes to understand how the “net” view of the hypercube would fold back into an actual hypercube shape.

Fractals & Chaos Recap for 10/30

After talking so much these past few weeks about fractional dimension—dimension values that fill in the gaps between 1, 2, and 3 dimensions, we turn our perspective in the other direction on the number line: towards the 4th dimension. It stands to reason that the trends we’ve observed could be extended past the 3rd dimension, but considering fractals, or even Euclidean shapes, is immensely challenging for us as 3-dimensional creatures.

What helps is to consider the perspective by analogy: we can better understand the fourth dimension by putting ourselves in the mindset of a 2-dimensional creature considering the third dimension. Fortunately, this is territory that has been well-covered.

In 1884, British teacher and theologian Edwin Abbott Abbott published Flatland: A Romance of Many Dimensions. It is told from the perspective of A. Square, a denizen of the titular 2-dimensional world, and starts off explaining aspects of their universe in great detail. The second part describes his first encounter with the 3-dimensional Spaceland, and the changes to his world-view as a result.

Abbott wrote the book partly as an exercise in geometry, but also partly as a satire on the regimented Victorian-era social hierarchy. As a result, there are some rather uncomfortable characterizations of women as lower-class citizens, among other shocking commentary.

You can read the full book here (I have paper copies if you’d prefer that), but for tomorrow please at least read this excerpt.

Fractals & Chaos Recap for 10/22

If you have a compass (the circle-drawing kind) please bring it with you to class over the next few days.

We discussed the assigned reading from Fractals: The Patterns of Chaos, then segued into a further discussion of dimension. As we are aware, we still have problems with the Hausdorff dimension formula for calculating dimension of fractals. It can’t handle fractals with stems (i.e., non-iterating segments that never disappear) and with fractals that are not exactly self-symmetric.

Today, we considered a football field, a circle, and a Koch Curve, and looked at how the size of the measuring stick we use to measure the length or perimeter of such things has an impact on the total amount of length we actually calculate. For a football field, the size of the stick makes no difference. We’ll be obtaining 100 yards worth of length even if we use a foot (S = 3) or an inch (S = 36) as our step size.

For a circle, this isn’t the case. Use a measuring stick the length of the diameter, and we can only make two steps before we end where we started. Use a stick the size of the radius (S = 2) and we can make 6 such steps (resulting in a measure of three diameters). Use a half-radius (S = 4), and we wind up with a total length of slightly more than 3 diameters. There is a limit to this, of course: pi*d, which is precisely the formula for the circumference of a circle.

For the Koch Curve, the story is very different. Use a step size the length of the original baseline, and we can make one step. Use a step size of 1/3 the baseline (S = 3), we can make 4 steps, giving a length of 4/3 the base. Use a step size of 1/9 the baseline (S = 9), and we can make 16 steps, for a total length of 16/9 the base. As we shrink the length of the ruler we use, the number of steps increases more quickly, and so the total length increases without bound.

We’ve seen suggestions at this idea before. In the second article we read (The Diversity of Life), we saw that reducing the scale of our perspective dramatically increases the amount of living space we can find. This idea is also found at the center of the coastline paradox, hinted at in the Ants in Labyrinths article.

We will be expanding on this in class tomorrow, including a discussion on what all this has to do with the dimension of what we’re measuring.