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.