Simplest Fibonacci series | Ratios between consecutive terms | Fibonacci with integers and reals | Ratios | Geometric series |
fn + 1 = fn + fn − 1 | fn/fn − 1 | gn | ||
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 |