Meander (mathematics)

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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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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:

Missing image
Meander_M22_jaredwf.png

Meandric numbers

The number of distinct meanders of order n is the meandric number Mn. The first fifteen meandric numbers are given below Template: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:

Missing image
Open_Meander_M2_jaredwf.png

Open meandric numbers

The number of distinct open meanders of order n is the open meandric number mn. The first fifteen open meandric numbers are given below Template: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:

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 Mn (usually denoted with an overline instead of an underline). The first fifteen semi-meandric numbers are given below Template: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:

Mn = m2n−1

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

MnMnM2n

For n > 1, meandric numbers are even:

Mn ≡ 0 (mod 2)

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