Mathaematics.

For my exam time absence~.

(Photos, videos, concepts, or interesting problems are well appreciated)

Formal definition.

Let \(\phi :[0,1]\to T\) (where \( T\) is a topological space) be a continous function (i.e. \(\phi ^{-1}\) of any open set \( O\) of \( T\) is an open set of the unit interval) from the unit interval to a topological space. Such map is said to be an space-filling curve if \(\phi\) is onto (i.e. if \phi passes through every element of \( T\) ).

Note

Generally, space filling curves are represented on spaces homeomorphic to \( E^{n}\) euclidian spaces. So to say, the plane, or any geometric solid, since these are the easiest ways to visualize a space filling curve. The most popular example, perhaps, is the two-dimensional Hilbert curve.

Images

A Hilbert curve from the unit interval to the unit cube

Peano Curve from the unit interval to the unit cube.

Flow Snake sapce filling curve.

Last three school weeks. Final projects, exams, and kind-of complicated topics.

When I end my Group Theory, advanced linear algebra courses and the overwhelming amounts of multivariable calc and mechanics I’ll be back. In the meantime, I’ll ony be posting pretty images.