Fractals & Chaos Recap for 9/9

We added to our list of fractals and discussed some observations from Wilson’s excerpt. We observed that dimension is tied to measurement, with lengths being one-dimensional, areas being two-dimensional, and volumes being three-dimensional. This suggests that any notion of fractional dimension must correspond to measurements that don’t fit within those three categories.

Furthermore, the article illustrated an interesting point (one that we will come back to soon): shrinking our scale of measurement increases the quantity of measurement we can find, disproportionate to the shrinking scale. The amount of space found at progressively smaller scales increases rapidly, and with it biodiversity.

After a brief distraction with a mouse, we introduced our first mathematical fractal: the Cantor Ternary Set, developed in 1883 by German mathematician Georg Cantor to challenge contemporary ideas of infinity and length. In particular, the set has three seemingly contradictory properties:

  • Every member of the set is a limit point
  • It has an uncountably infinite number of points
  • Yet its Length is zero

We will explore these ideas in the next few days.

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