Farey sequence

In mathematics, the Farey sequence of order

n

{\displaystyle n}

is the sequence of completely reduced fractions which have denominators less than or equal to

n

{\displaystyle n}

, arranged in order of increasing size. Some authors restrict the fractions to the interval between 0 and 1.

A Farey sequence is sometimes called a Farey series, although the terms are not summed.

Examples

The first eight Farey sequences are:

F

1

=

{

0

1

,

1

1

}

,

F

2

=

{

0

1

,

1

2

,

1

1

}

,

F

3

=

{

0

1

,

1

3

,

1

2

,

2

3

,

1

1

}

,

F

4

=

{

0

1

,

1

4

,

1

3

,

1

2

,

2

3

,

3

4

,

1

1

}

,

F

5

=

{

0

1

,

1

5

,

1

4

,

1

3

,

2

5

,

1

2

,

3

5

,

2

3

,

3

4

,

4

5

,

1

1

}

,

F

6

=

{

0

1

,

1

6

,

1

5

,

1

4

,

1

3

,

2

5

,

1

2

,

3

5

,

2

3

,

3

4

,

4

5

,

5

6

,

1

1

}

,

F

7

=

{

0

1

,

1

7

,

1

6

,

1

5

,

1

4

,

2

7

,

1

3

,

2

5

,

3

7

,

1

2

,

4

7

,

3

5

,

2

3

,

5

7

,

3

4

,

4

5

,

5

6

,

6

7

,

1

1

}

,

F

8

=

{

0

1

,

1

8

,

1

7

,

1

6

,

1

5

,

1

4

,

2

7

,

1

3

,

3

8

,

2

5

,

3

7

,

1

2

,

4

7

,

3

5

,

5

8

,

2

3

,

5

7

,

3

4

,

4

5

,

5

6

,

6

7

,

7

8

,

1

1

}

.

{\displaystyle {\begin{aligned}F_{1}&={\Big \{}{\frac {0}{1}},{\frac {1}{1}}{\Big \}},\\[5pt]F_{2}&={\Big \{}{\frac {0}{1}},{\frac {1}{2}},{\frac {1}{1}}{\Big \}},\\[5pt]F_{3}&={\Big \{}{\frac {0}{1}},{\frac {1}{3}},{\frac {1}{2}},{\frac {2}{3}},{\frac {1}{1}}{\Big \}},\\[5pt]F_{4}&={\Big \{}{\frac {0}{1}},{\frac {1}{4}},{\frac {1}{3}},{\frac {1}{2}},{\frac {2}{3}},{\frac {3}{4}},{\frac {1}{1}}{\Big \}},\\[5pt]F_{5}&={\Big \{}{\frac {0}{1}},{\frac {1}{5}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{5}},{\frac {1}{2}},{\frac {3}{5}},{\frac {2}{3}},{\frac {3}{4}},{\frac {4}{5}},{\frac {1}{1}}{\Big \}},\\[5pt]F_{6}&={\Big \{}{\frac {0}{1}},{\frac {1}{6}},{\frac {1}{5}},{\frac {1}{4}},{\frac {1}{3}},{\frac {2}{5}},{\frac {1}{2}},{\frac {3}{5}},{\frac {2}{3}},{\frac {3}{4}},{\frac {4}{5}},{\frac {5}{6}},{\frac {1}{1}}{\Big \}},\\[5pt]F_{7}&={\Big \{}{\frac {0}{1}},{\frac {1}{7}},{\frac {1}{6}},{\frac {1}{5}},{\frac {1}{4}},{\frac {2}{7}},{\frac {1}{3}},{\frac {2}{5}},{\frac {3}{7}},{\frac {1}{2}},{\frac {4}{7}},{\frac {3}{5}},{\frac {2}{3}},{\frac {5}{7}},{\frac {3}{4}},{\frac {4}{5}},{\frac {5}{6}},{\frac {6}{7}},{\frac {1}{1}}{\Big \}},\\[5pt]F_{8}&={\Big \{}{\frac {0}{1}},{\frac {1}{8}},{\frac {1}{7}},{\frac {1}{6}},{\frac {1}{5}},{\frac {1}{4}},{\frac {2}{7}},{\frac {1}{3}},{\frac {3}{8}},{\frac {2}{5}},{\frac {3}{7}},{\frac {1}{2}},{\frac {4}{7}},{\frac {3}{5}},{\frac {5}{8}},{\frac {2}{3}},{\frac {5}{7}},{\frac {3}{4}},{\frac {4}{5}},{\frac {5}{6}},{\frac {6}{7}},{\frac {7}{8}},{\frac {1}{1}}{\Big \}}.\end{aligned}}}

Farey sunburst

Plotting the numerators versus the denominators of a Farey sequence gives a shape like the one to the right, shown for

F

6

{\displaystyle F_{6}}

.

Reflecting this shape around the diagonal and main axes generates the Farey sunburst, shown below. The Farey sunburst of order n connects the visible integer grid points from the origin in the square of side

2

n

{\displaystyle 2n}

, centered at the origin. By Pick's theorem, the area of the sunburst is

4

|

F

n

|

−

4

{\displaystyle 4|F_{n}|-4}

, where

|

F

n

|

{\displaystyle |F_{n}|}

is the number of fractions in Fn.

History

The history of 'Farey series' is very curious — Hardy & Wright (1979)

... once again the man whose name was given to a mathematical relation was not the original discoverer so far as the records go.