Fractals & Chaos Recap for 12/18

Today, we finished the Complex Paint Worksheet we’ve been working off of. In the left column for Part 2, we saw unambiguous numbers of spokes corresponding to the denominator of the fraction we were using for T in our Polar-Linear Form. Today, with T = 0.32, we saw something else: three clockwise spiral arms. Weird. When we slowed down the attraction from R = .9 to R = .99, we saw 25 spokes, which makes sense given 0.32 = 32/100 = 8/25, but where do those 3 spiral arms come from?

We saw similar results for T = 15/62 and T = 5/17:

  • For T = 15/62, we saw 4 clockwise spiral arms, which turned into 29 counter-clockwise spirals, before we finally saw the 62 spokes we expect.
  • For T = 5/17, we saw 3 clockwise, then 7 counter-clockwise spiral arms before finding the 17 spokes

What’s also odd is that when we used T = π, an obviously irrational quantity, we saw what looked like 7 spokes at R = 0.9. Slower values of R clearly indicated these were spiral arms, but then we saw the attraction pattern line up again with 113 “spokes.” When we used T = φ (the golden ratio), we never saw spokes, as we would expect from another irrational quantity, though we were amused to notice that the numbers of spiral arms we observed exactly matched the Fibonacci sequence.

Where are these spirals coming from in our rational rotation values? And where are these “spokes” coming from in our irrational ones? The answer to this will lead us to a surprising conclusion: Irrationality is not a strict “either/or;” there is a continuity to irrationality where some numbers are more irrational than others. Moreover, what we saw today will form the basis for what I think is a bold claim: Phi, the golden ratio, is the most irrational number.

Stay tuned!

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