Peano curve (nonfiction)
In geometry, the Peano curve, discovered by Giuseppe Peano in 1890, is the earliest known space-filling curve.
Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective.
Peano was motivated by an earlier result of Georg Cantor that these two sets have the same cardinality. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve.
Peano's ground-breaking article contained no illustrations of his construction, which is defined in terms of ternary expansions and a mirroring operator. But the graphical construction was perfectly clear to him—he made an ornamental tiling showing a picture of the curve in his home in Turin. Peano's article also ends by observing that the technique can be obviously extended to other odd bases besides base 3. His choice to avoid any appeal to graphical visualization was, no doubt, motivated by a desire for a well-founded, completely rigorous proof owing nothing to pictures. At that time (the beginning of the foundation of general topology), graphical arguments were still included in proofs, yet were becoming a hindrance to understanding often counterintuitive results.
A year later, David Hilbert published in the same journal a variation of Peano's construction. Hilbert's article was the first to include a picture helping to visualize the construction technique, essentially the same as illustrated here. The analytic form of the Hilbert curve, however, is more complicated than Peano's.
In the News
Papa Giuseppe is best father ever, says Peano curve.
Fiction cross-reference
Nonfiction cross-reference
External links
- Peano curve @ Wikipedia
- Space-filling curve @ Wikipedia