Sets {fractal set}| can have members related by fractal steps. All initial values used in Newton's method lead to square root. Following fractal paths, initial value lying between two roots can lead to any value. Feigenbaum functions on complex plane reach all frequencies {Julia set}. Julia sets can generalize. Take complex number, square it, add square to original complex number, square result, add square to original complex number, and so on. If result does not diverge from complex-plane origin, result has bound, and complex number is in set {Mandelbrot set}. If real or imaginary part becomes greater than 2 or smaller than -2, number diverges. Mandelbrot set is boundary between the regions {fractal basin boundary}. Mandelbrot set is not precisely self-similar.
Sets {Levy set}, such as rectifiable or connected curves, can have dimension between zero and one and density between zero and one. Points have dimension zero. Lines have dimension one. Lower dimension means more clustering.
Remove line-segment middle third, then remove middle thirds of both remaining line segments, and repeat indefinitely {triadic set}.
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Date Modified: 2022.0225