The Schoenberg curve is a fractal plane-filling curve, similar to the Peano, Hilbert, Moore, Lebesgue, and Wunderlich curves. The definition of the Schoenberg curve begins with a piecewise sawtooth-like function whose value falls between 0 and 1.
The curve is defined parametrically using sums of scaled copies of the original function. The first iteration is a one-half scale copy of the sawtooth function, beginning at the point (0, 0) and ending at (1/2, 1/2).
The second iteration adds a one-quarter scale copy to the previous copy, ending at (3/4, 3/4).
Each iteration spans more of the unit square. In its limit, the Schoenberg curve touches every point in the unit square.
See I. J. Schoenberg and C. de Boor (ed.), “On the Peano Curve of Lebesgue”, in I. J. Schoenberg: Selected Papers Vol. 1, Boston: Birkhäuser, 1988.
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