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?
We had our art show gala opening in class today, and took the opportunity to view each other’s Fractal Art, vote on which ones we liked the best in each category, and try our skills at matching templates to the completed fractals we saw on the walls. If you weren’t done casting your votes, please turn in your ballots on Monday. Remember also to read the section from our new books by Monday: pages 61-73. We’ll have a quick debrief of this reading then introduce the next method of calculating dimension of fractals, which will help to resolve the two remaining issues we have with the Hausdorff Dimension formulas!
We finished calculating dimension of various fractals, and observed that our Generalized Hausdorff Dimension formula still falls short when a fractal is not perfectly self-symmetric. There’s still work to be done.
We also got a textbook of sorts, the book Fractals: The Patterns of Chaos. We will occasionally have reading assignments out of this book, the first of which is pages 61-73, the section on fractal dimension. Please read this section by Monday.
Tomorrow, we will have our Fractal Art Show Gala Opening! Remember to bring your snacks!
We’ve finalized our fractal designs and everyone has submitted their works. We will have our Fractal Art Gala on Thursday, October 18. Please be sure to bring in some snacks to share!
For Wednesday, you should read Pollack’s Fractals, an article from Discover Magazine about the math underlying Jackson Pollack’s famous paintings. For some interesting follow-up reading, check out this article from the New York Times about the use of fractal analysis to examine the authenticity of supposed Pollack paintings and this article from the Science Daily blog suggesting that such an analysis is not scientifically valid.
We finished our discussion of the first dimension calculation practice sheet with an observation that the S^d = N definition for dimension has some weaknesses, primarily:
- What do you do with fractals with non-iterating stems (e.g., 2 and 13)?
- What if the fractal is not exactly self-similar (like 5)?
- What if it is self-similar, but at differing scales (like 15)?
We will continue to work on our definition of dimension to accommodate these issues. You have also been issued a challenge: use what you’ve learned from the S^d=N definition to design a fractal with a dimension of exactly 1.5. Think about this as you read Gould’s essay Size and Shape.
We spent the rest of the period working on our FractaSketch designs.
We started class with a writing prompt based on Haldane’s On Being the Right Size, a response to the meme “Would you rather fight one horse-sized duck or 100 duck-sized horses?” The horse-sized duck won the argument, as an application of the central takeaway from Haldane’s article is that artificially scaling a duck up to the size of a horse would increase its weight exponentially faster than its strength could accommodate, resulting in a duck incapable of walking around, let alone flight. We followed this up with a relevant Kurzgesagt (In a Nutshell) video: What Happens if we Throw and Elephant From a Skyscraper (as promised, the follow up — How to Make an Elephant Explode — can be found here).
In the second half of the period, we continued our discussion of the dimension of the various fractals found on our first dimension practice sheet. We found that one of the templates was a cleverly disguised version of the Sierpinski Carpet, and that another–Sierpinski’s Pyramid–has the interesting property of being simultaneously defined as having 1, 2, or 3 dimensions, depending on what type of dimension you use.
Tomorrow, we go back to the computer lab to work on our fractal designs. By Monday, read Size and Shape, paleontologist and evolutionary biologist Stephen Jay Gould‘s follow up to Haldane’s essay. If Gould’s name sounds familiar, it’s because he co-developed the idea of (and coined the term for) punctuated equilibrium, a theory of evolution.
We discussed the article Ants in Labyrinths, then did some more practice with finding dimension of a variety of fractals. Some of the designs were templates, but some of them were “completed” fractals, which prompted us to discuss how to recreate the template from these designs. We have also already observed that the S^d = N definition for dimension may not be sufficient!
We’ll have some more time to work in the lab tomorrow, and for Thursday you should read On Being the Right Size, an essay written by biologist JBS Haldane in 1926.
We spent the day working on our FractaSketch designs. By tomorrow, please read the passage Ants in Labyrinths, from Ivars Peterson’s The Mathematical Tourist. As usual, make a note of questions you have and passages you think are significant.
We practiced finding dimension with our new definition for a few additional fractals, including seeing a surprising result with the Dragon Curve, then spent the remaining time working on our fractal designs in lab.
This weekend, please read the passage Ants in Labyrinths, from Ivars Peterson’s The Mathematical Tourist. As usual, make a note of questions you have and passages you think are significant.