Fibonacci series

Geometric series

Irrational

values

Rational

Approx.

Irr.-Rat. residue

Commensurate

divisor

Residue/ divisor

Metric

${\tau }^{m-1}$

$c_s$

$m$

$F_m$

$F_{m+1}$

$\tau$

$a+\text{3}b/2$

$/F_{m+1}$

$\Delta =\tau -3/2$

$1/\left(1+\Delta \right)$

0

$\tau$

${\tau }^0$

=

1

1

0

0

1

0

1

$\tau$

=

$\tau$

=

1.618034...

1.5

0.11803

1

0.118034

0.894427

2

1

+

1

$\tau$

=

${\tau }^2$

=

2.618034…

2.5

0.11803

1

0.118034

0.894427

3

1

+

2

$\tau$

=

${\tau }^3$

=

4.236068…

4

0.23607

2

0.118034

0.894427

4

2

+

3

$\tau$

=

${\tau }^4$

=

6.854102

6.5

0.3541

3

0.118034

0.894427

5

3

+

5

$\tau$

=

${\tau }^5$

=

11.09017

10.5

0.59017

5

0.118034

0.894427

6

5

+

8

$\tau$

=

${\tau }^6$

=

17.944272

17

0.94427

8

0.118034

0.894427

7

8

+

13

$\tau$

=

${\tau }^7$

=

29.034443

27.5

1.53444

13

0.118034

0.894427

8

13

+

21

$\tau$

=

${\tau }^{\text{8}}$

=

46.978715

44.5

2.47872

21

0.1180341

0.894427

9

21

+

34

$\tau$

=

${\tau }^{\text{9}}$

=

76.013159

72

4.01316

34

0.1180341

0.894427

10

34

+

55

$\tau$

=

${\tau }^{\text{1}0}$

=

122.99188

116.5

6.49188

55

0.1180341

0.894427

11

55

+

89

$\tau$

=

${\tau }^{\text{11}}$

=

199.00504

188.5

10.505

89

0.1180341

0.894427

12

89

+

144

$\tau$

=

${\tau }^{\text{12}}$

=

321.99691

305

16.9969

144

0.1180341

0.894427