The definition of the Schoenberg curve begins with a piecewise sawtooth-like function whose values lie between 0 and 1. The plane-filling curve is defined parametrically using sums of scaled copies of the original function. In the limit, the Schoenberg curve touches every point in the unit square.
Snapshot 1: first term of the summation, the upper-right corner of which is the point ; the line color changes from blue to red to help the eye track the progress of the curve from the lower left-corner to the upper-right corner
Snapshot 2: the Schoenberg curve differs from such plane-filling curves as the Peano curves, Hilbert and Moore curves, Lebesgue curve, and so forth, in that the approximations intersect themselves and indeed double back on themselves
Snapshot 3: greater iterations reach more points in the unit square
I. J. Schoenberg, "On the Peano Curve of Lebesgue," I. J. Schoenberg: Selected Papers, Vol. 1 (C. de Boor, ed.), Boston: Birkhäuser, 1988.