We looked briefly at Pascal’s Triangle today, and some of the neat patterns that can be found there. I hinted at some hidden fractals that could be found by removing numbers from the triangle, so your homework is to fill in circles in this smaller version that would represent removing every even number from the triangle (remember, we observed that two filled in circles create a filled in one, two empty circles create a filled in one, and an empty and filled circle create an empty one).
We wrapped up class by playing with FractaSketch some more (linked at left). Before everybody left, I also handed out the next assigned reading for the course: this Science News article from 1997 (Fractal past, Fractal future) and this supplementary article from a 1997 issue of Popular Science about the Heartsongs album mentioned in the first one.
We finished our discussion of the ideas inspired by the Jurassic Park excerpt, including looking at a few theories of using fractals to predict financial markets (see the silver and bitcoin articles here if you’d like to read them more closely).
From there, we discussed last night’s assigned reading and used it to form some properties about fractals. In particular:
- They demonstrate self-symmetry or self-similarity (each part could be viewed as a scaled-down version of the whole thing)
- They are Non-Euclidean (for a Euclidean curve, no matter how wiggly it is, zooming in far enough eventually makes it look linear, but for a fractal zooming in just reveals the same level of detail).
- They have fractional dimension (unlike one-dimensional lines, two-dimensional squares, or three-dimensional cubes, fractals live in a space between and could have a non-integer dimension)
This last idea is, of course, pretty wild, and if you feel skeptical about it, you should. Hold on to that skepticism! Let me convince you.
We finished the day making a brief list of ideas of fractals, including snowflakes, ferns, feathers, trees, and river deltas. Your homework is twofold:
- Continue to think of examples of fractals in the world around you, and
- Read over the excerpt from Edward O Wilson’s book The Diversity of Life: Living Labyrinths. As before, make a note of 2-3 passages that seems significant or questions you have.
We started out today with a discussion about course expectations. This is a pass/fail course but I still expect you to take it seriously. I will often ask you to read an article or do some math work at home, and I expect that work will be done the next day and ready to be discussed. I expect everyone to actively participate in class discussions and engage in the work we do during class time. This is not a study hall, so please don’t bring other work to do during this class or I will ask you to leave.
After a discussion of some of the pre-existing notions on what fractals are, we read an excerpt from Michael Crichton’s Jurassic Park and discussed some of the big ideas to be found there (you can see a clip of this scene from Steven Spielberg’s 1993 film version here).
For your first reading assignment, I’m asking you to read the article distributed in class: Fractals: Magical Fun or Revolutionary Science, from the March 21, 1987 issue of Science News. Take some notes as you read, jotting down the 2 or 3 major points of the article. Pay particular attention to how this article defines the term “fractal”.