The animation shows an approximation to Peano's curve. The horizontal and vertical movements do belong to the curve, while the diagonal movements are interpolations, to keep the file size manegeable. The speeds of the point in the different segments are straight from the definition of the curve, at the cost of making parts of the movie quite sluggish.
Here is the path followed by the point:
The movie started as a Mathematica animation made with the Real Analysis package, and was converted to QuickTime with Macintosh system resources.
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