On Sun, May 26, 2013 at 4:31 AM, Bill Gosper <billgosper@gmail.com> wrote:
Wikipedia: Peano's curve is dense in the unit square<https://en.wikipedia.org/wiki/Unit_square>, and was used by Peano to construct a continuous function<https://en.wikipedia.org/wiki/Continuous_function>from the unit interval <https://en.wikipedia.org/wiki/Unit_interval> to the unit square, motivated by an earlier result of Georg Cantor<https://en.wikipedia.org/wiki/Georg_Cantor>that these two sets have the same cardinality <https://en.wikipedia.org/wiki/Cardinality>. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve.[2]<https://en.wikipedia.org/wiki/Peano_curve#cite_note-2>
Dense, hell. It COVERS the unit square. Not "to the unit square". ONTO the unit square. I was a victim of "some authors", for years misattributing Hilbert's spacefill to Peano.
"The Peano curve itself is the limit<https://en.wikipedia.org/wiki/Limit_%28mathematics%29>of the curves through the sequences of square centers, as *i* goes to infinity."
I.e., the unit square? How doe one take this limit? Total bull.
"In 1890, Peano <http://en.wikipedia.org/wiki/Peano> discovered a densely self-intersecting curve, now called the Peano curve<http://en.wikipedia.org/wiki/Peano_curve> ,..."
If this statement means anything, ALL spacefilling "curves" "densely self-intersect".
The sane phraseology is: All spacefilling functions revisit a dense set (whose closure
is the whole image) three times. --rwg
Then please update the wiki page! -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com