Meander (mathematics)

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.

Meander

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.

Examples

The meander of order 1 intersects the line twice:

The meanders of order 2 intersect the line four times:

Meander M21 jaredwf.png Meander M22 jaredwf.png

Meandric numbers

The number of distinct meanders of order n is the meandric number M_n. The first fifteen meandric numbers are given below (sequence A005315 in the OEIS).

M₁ = 1

M₂ = 2

M₃ = 8

M₄ = 42

M₅ = 262

M₆ = 1828

M₇ = 13820

M₈ = 110954

M₉ = 933458

M₁₀ = 8152860

M₁₁ = 73424650

M₁₂ = 678390116

M₁₃ = 6405031050

M₁₄ = 61606881612

M₁₅ = 602188541928

Open meander

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.

Examples

The open meander of order 1 intersects the line once:

The open meander of order 2 intersects the line twice:

Open meandric numbers

The number of distinct open meanders of order n is the open meandric number m_n. The first fifteen open meandric numbers are given below (sequence A005316 in the OEIS).

m₁ = 1

m₂ = 1

m₃ = 2

m₄ = 3

m₅ = 8

m₆ = 14

m₇ = 42

m₈ = 81

m₉ = 262

m₁₀ = 538

m₁₁ = 1828

m₁₂ = 3926

m₁₃ = 13820

m₁₄ = 30694

m₁₅ = 110954

Semi-meander

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.

Examples

The semi-meander of order 1 intersects the ray once:

Semi-meander M1 jaredwf.png

The semi-meander of order 2 intersects the ray twice:

Semi-meandric numbers

The number of distinct semi-meanders of order n is the semi-meandric number M_n (usually denoted with an overline instead of an underline). The first fifteen semi-meandric numbers are given below (sequence A000682 in the OEIS).

M₁ = 1

M₂ = 1

M₃ = 2

M₄ = 4

M₅ = 10

M₆ = 24

M₇ = 66

M₈ = 174

M₉ = 504

M₁₀ = 1406

M₁₁ = 4210

M₁₂ = 12198

M₁₃ = 37378

M₁₄ = 111278

M₁₅ = 346846

Properties of meandric numbers

There is an injective function from meandric to open meandric numbers:

M_n = m_2n−1

Each meandric number can be bounded by semi-meandric numbers:

M_n ≤ M_n ≤ M_2n

For n > 1, meandric numbers are even:

M_n ≡ 0 (mod 2)