We spent the period playing around with Fractal Zoomer, and have made a few observations:
- The central “bulb” of the Mandelbrot Set is a cardioid, and each other bulb off of that central body is a perfect circle.
- Each bulb has its own cycle of orbits. Some bulbs have the same size orbit as others, but with a different “order” (1,2,3,4,5,6 vs 1,3,5,2,4,6 for example)
- If you tile the plane with Julia Sets of a sufficient density, the collection of Julia Sets make a photo-mosaic of the Mandelbrot Set. This further reinforces that there is something important between the location of C within the Mandelbrot Set and the shape of the corresponding Julia Set.
- Along the central “spike” on the left side of the Mandelbrot Set, we can find baby Mandelbrot Sets strung along that string. The main body of these baby Mandelbrots show cycles instead of fixed points. Realizing that the horizontal axis on which the Mandelbrot set is centered is the real number axis, this suggests that there may be a meaningful connection between the Mandelbrot Set and Feigenbaum Plot…
Keep working with Fractal Zoomer and considering the questions posted yesterday. Please also read the section on Mandelbrot Set from pages 74-81 in your copy of Fractals: The Patterns of Chaos (and bring that book back on Monday).
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.”
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
We started watching a Nova documentary on The Strange New Science of Chaos. It’s from 1989, but it has held up well and serves as an excellent introduction to this strange new world of constrained randomness and sensitivity to initial conditions.
For Friday, there are three sections from Fractals: The Patterns of Chaos that I would like you to read:
- Pages 55-60 – Our Weather Today is Chaos
- Pages 93-97 – Chaos and Symmetry Hybrids
- Pages 99-106 – Chaos Sculpts Fractal Landscapes
We put up the random zig-zags that you created over the weekend and noticed that while the pattern tendered to wander away from the central line, it would eventually trend back as well. The pattern of central line crosses came in “bursts,” and the frequency of these bursts demonstrated fractal-like, self-similar qualities.
We returned to the classroom to examine another version of this, a Chaos Game app designed by Fractals & Chaos alum Istvan Burbank. We saw the impact that different constraints on random movement had on the pattern that movement created, including one very surprising result. We closed out the period investigating and playing with this app some more.
Remember: for tomorrow please read Chaos Hits Wall Street. This is required reading, and I will expect everybody to contribute to tomorrow’s discussion.
In class with Ms. Csaki today, you made your very own 3D models of 4-dimensional hypercubes! Take good care of it!
After you finished your model, Ms. Csaki should have given you a copy of the Discover Magazine article Chaos Hits Wall Street to read by Tuesday. While this is an old article (1993!) it addresses a lot of the topics we’ll be discussing in the second part of the course. We’ll see some updated takes on the theories presented in the next articles.
Finally, you should have also been given a piece of graph paper in class and told to come here to find out what to do with it. Here’s what you do:
- Fold it in half lengthwise (“hot dog” style).
- Unfold and put a dot on the left edge of your crease.
- Flip a coin (or use some other random procedure). If the coin lands heads, draw a diagonal that goes over and up one square. If the coin lands tails, draw a diagonal that goes over and down one square.
- Continue with this pattern, creating a zig-zag across your paper, until you reach the other side. Bring that in on Monday as well.
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.