Simplest Fibonacci series

Ratios between consecutive terms

Fibonacci with integers and reals

Ratios

Geometric series

f_{n + 1} = f_n + f_{n − 1}

f_n/f_{n − 1}

g_n

τ^-infinity

.

.

τ^−2

τ^−1

0

1

τ^0

1

infinity

τ

τ

τ^1

1

1

1 + τ

τ

τ^2

2

2

1 + 2τ

τ

τ^3

3

1.5

2 + 3τ

τ

τ^4

5

1.6666...

3 + 5τ

τ

τ^5

8

1.6

5 + 8τ

τ

τ^6

13

1.625

8 + 13τ

τ

.

21

1.615

13 + 21τ

τ

.

.

.

.

.

.

.

.

.

.

.

.

.

.

τ = 1.618034....

τ

τ^infinity

Not observed

As observed in

quasicrystals

Redundant 6D

Definitely 3D