Space filling curves

Curve

http://en.wikipedia.org/wiki/Curve

N.B. Different disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context.

It was an exciting event in the history of mathematics when in 1890, the Italian mathematician Peano constructed a continuous curve which passed through every point in the unit square.

Examples of such `space-filling' curves were later constructed by Hilbert (in 1891),

Moore (in 1900) , Lebesgue (in 1904), Sierpinski (in 1912) and Schoenberg (in 1938) . These have come to be known as Peano curves

http://en.wikipedia.org/wiki/Space-filling_curve

Hilbert curve

http://en.wikipedia.org/wiki/Hilbert_curve

An interesting question: Where would you be if you started at the lower right and traveled a third of the way along his recursively defined curve?

Hint: http://www.jasondavies.com/hilbert-curve/

Some combinatorial applications of spacefilling curves

http://www2.isye.gatech.edu/~jjb/mow/mow.html

Hilbert Space Filling Curve Abstract Geometric Art

http://www.donrelyea.com/hilbert_algorithmic_art_menu.htm

Multidimensional Space-Filling Curves

http://people.csail.mit.edu/jaffer/Geometry/MDSFC

Sierpiński curve

http://en.wikipedia.org/wiki/Sierpi%C5%84ski_curve

For example, it has been used as a basis for the rapid construction of an approximate solution to the Traveling Salesman Problem (which asks for the shortest sequence of a given set of points): The heuristic is simply to visit the points in the same sequence as they appear on the Sierpiński curve

Lebesgue 3D Curves

http://demonstrations.wolfram.com/Lebesgue3DCurves/

Schoenberg Plane-Filling Curve

http://demonstrations.wolfram.com/SchoenbergPlaneFillingCurve/

An Elementary Proof that Schoenberg's Space-Filling Curve Is Differentiable

http://www.jstor.org/pss/2690494

Plane Filling Curves: Hilbert's & Moore's

http://www.cut-the-knot.org/Curriculum/Geometry/PlaneFillingCurves.shtml