Schoenberg Plane-Filling Curve

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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.

Contributed by: Robert Dickau (March 2011)
Open content licensed under CC BY-NC-SA



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.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.