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My bad, I'd just ^F'd with a purely syntactic search on a few terms. If you can arrange your space from 0,..1 and identify 1 with 0 to make it circular, obviously (consider 1/3, 2/3, etc.) advancing by any amount in ℚ will fail to explore a lot of the space. The flip side of this is that φ, or 1/φ, are the least-ℚ-like regular things in ℝ: if we consider good approximations to be places in the continued fraction where there's a large entry, that never occurs. (their continued fraction representation is repeated 1's) This comes up with "how do sunflowers know about φ?", the answer to which seems to be: they don't, they're just creating new seeds in the least-correlated place to all the existing seeds, and because of the property above, that tends to result in apparently-spiraling placements which can be fitted with fibonaccis or φs. Does that make sense? |