In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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:

Meandric numbers

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

M₁ = 1

M₂ = 2

M₃ = 8

M₄ = 42

42 (number)

42 is the natural number immediately following 41 and directly preceding 43. The number has received considerable attention in popular culture as a result of its central appearance in The Hitchhiker's Guide to the Galaxy as the "Answer to the Ultimate Question of Life, the Universe, and...

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 .

m₁ = 1

m₂ = 1

m₃ = 2

m₄ = 3

m₅ = 8

m₆ = 14

14 (number)

14 is the natural number following 13 and preceding 15.In speech, the numbers 14 and 40 are often confused. When carefully enunciated, they differ in which syllable is stressed: 14 vs 40...

m₇ = 42

42 (number)

42 is the natural number immediately following 41 and directly preceding 43. The number has received considerable attention in popular culture as a result of its central appearance in The Hitchhiker's Guide to the Galaxy as the "Answer to the Ultimate Question of Life, the Universe, and...

m₈ = 81

81 (number)

81 is the natural number following 80 and preceding 82.-In mathematics:Eighty-one is the square of 9 and the fourth power of 3. Like all powers of three, 81 is a perfect totient number. It is a heptagonal number and a centered octagonal number. It is also a tribonacci number, and an open meandric...

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

M₁ = 1

M₂ = 1

M₃ = 2

M₄ = 4

M₅ = 10

10 (number)

10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...

M₆ = 24

24 (number)

24 is the natural number following 23 and preceding 25.The SI prefix for 1024 is yotta , and for 10−24 yocto...

M₇ = 66

66 (number)

66 is the natural number following 65 and preceding 67.Usages of this number include:-Mathematics:*66 is a sphenic number, a triangular number, a hexagonal number, and a semi-meandric number...

M₈ = 174

174 (number)

174 is the natural number following 173 and preceding 175.-In mathematics:* 174 is an even number* 174 is an abundant number with the abundance of 12* 174 is a composite number* 174 is a nontotient number* 174 is an odious number...

M₉ = 504

M₁₀ = 1406

M₁₁ = 4210

M₁₂ = 12198

M₁₃ = 37378

M₁₄ = 111278

M₁₅ = 346846

Properties of meandric numbers

There is an injective function

Injective function

In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...

 from meandric to open meandric numbers:

M_n = m_2n−1

Each meandric number can be bounded

Upper bound

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...

 by semi-meandric numbers:

M_n ≤ M_n ≤ M_2n

For n > 1, meandric numbers are even

Even and odd numbers

In mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...

:

M_n ≡ 0 (mod 2)

External links

• "Approaches to the Enumerative Theory of Meanders" by Michael La Croix