In 1890, for the first meeting of the German Math Society, Hilbert had constructed
a continuous curve whose image is a filled square. Although the double
points of that curve were known to be dense, Hilbert constructed a sequence of polygonal
approximations without double points. (A double point of a parametrized curve c(t)
is a point such that t1≠t2 but c(t1) = c(t2).)
In 1890 the limit curve was the spectacular thing. Today the finite approximations
without double points are taught to computer graphics students, because these polygons
run through the square in a very useful way.