In mathematics, a meander or closed meander is a self-avoiding closed curve which intersects a line a number of times. Intuitively, a meander can be viewed as a road crossing a river through a number of bridges.
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Given a fixed oriented line L in the Euclidean plane R2, a meander of order n is a non-self-intersecting closed curve in R2 which transversally intersects the line at 2n points for some positive integer n. Two meanders are said to be equivalent if they are homeomorphic in the plane.
The meander of order 1 intersects the line twice:
The meanders of order 2 intersect the line four times:
The number of distinct meanders of order n is the meandric number Mn. The first fifteen meandric numbers are given below (sequence A005315 in OEIS).
Given a fixed oriented line L in the Euclidean plane R2, an open meander of order n is a non-self-intersecting oriented curve in R2 which transversally intersects the line at n points for some positive integer n. Two open meanders are said to be equivalent if they are homeomorphic in the plane.
The open meander of order 1 intersects the line once:
The open meander of order 2 intersects the line twice:
The number of distinct open meanders of order n is the open meandric number mn. The first fifteen open meandric numbers are given below (sequence A005316 in OEIS).
Given a fixed oriented ray R in the Euclidean plane R2, a semi-meander of order n is a non-self-intersecting closed curve in R2 which transversally intersects the ray at n points for some positive integer n. Two semi-meanders are said to be equivalent if they are homeomorphic in the plane.
The semi-meander of order 1 intersects the ray once:
The semi-meander of order 2 intersects the ray twice:
The number of distinct semi-meanders of order n is the semi-meandric number Mn (usually denoted with an overline instead of an underline). The first fifteen semi-meandric numbers are given below (sequence A000682 in OEIS).
There is an injective function from meandric to open meandric numbers:
Each meandric number can be bounded by semi-meandric numbers:
For n > 1, meandric numbers are even: