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:
- Mn ≤ Mn ≤ M2n
For n > 1, meandric numbers are even:
- Mn ≡ 0 (mod 2)