Bernold Fiedler

Μeanders: past, present, and future

Μeanders: past, present, and future

Think of a curve which does not intersect itself. In the plane, we call this a Jordan curve M. In how many ways can M cross a horizontal line, say, N=5 times? Draw this now, and you may be surprised ...

When the Russian mathematician V.I. Arnold called M a meander, he thought of the horizontal line as a "road" with bridges crossing the "river" M. (It is not unscientific, at all, to think in such "primitive" terms: our mind prefers them to the dust of egghead jargon.) Meanders pop up in Babylonian cuneiform, in mystic labyrinths of the Templars, in the pendulum, in algebra, in billiards, in ordinary and partial differential equations, in black holes -- but never in math classes. Well, old Gauss looked at them, and so will we. For example, enumerate the N crossings along the meandering "river". Then read your labels along the horizontal "road". Suddenly you have a permutation, and math starts.

New results are joint work with Carlos Rocha (IST Lisboa) and others. See more here and here