Meandric systems
three.js visualizations of order-4 meandric systems
A meandric system of order n is an ordered
pair of noncrossing matchings of the marked points
{1, 2, …, 2n}. Place the marked points on the equator
of a sphere; draw each arc as the half-circle of the chord, going up into
the top hemisphere or down into the bottom hemisphere depending on which
of the two matchings the arc belongs to. The two matchings together
cut the boundary of the sphere into a disjoint union of closed loops,
the connected components of the system. Each component
gets a glowing ball that races along it.
The 14 noncrossing matchings of {1,…,8} give
142 = 196 systems of order 4. The pages below
are different views of those 196.
Multiplication table →
All 196 order-4 systems on a 14 × 14 grid. Rows index the top matching, columns the bottom.
Mobile →
Same 196 systems, rearranged as 14 vertical strands hanging from the 14 vertices of the 3-D Stasheff associahedron K5.
Single system →
One system at a time, with controls for n and the two matchings. URL-parameterizable for sharing specific examples.