Fibonacci fraction

A Fibonacci fraction is a rational number of the form $\\frac{F_{n}}{F_{m}}$ where $F_{i}$ is the $i$ th number of the Fibonacci sequence and $n$ and $m$ are integers in the relation $n<m$ . In the Fibonacci fractional series, each $m=n+2$ :

\\frac{1}{2},\\frac{1}{3},\\frac{2}{5},\\frac{3}{8},\\frac{5}{13},\\frac{8}{21},%\\frac{13}{34},\\frac{21}{55},\\frac{34}{89},\\ldots

The most important application of Fibonacci fractions is in botany: plants arrange the leaves on their stems (phyllotaxy) in many different ways, but “only those conforming to a Fibonacci fraction allow for efficient packing of leaf primordia on the meristem surface.” There is also an application in optics.

References

1 P. A. David “Leaf Position in Ailanthus Altissima in Relation to the Fibonacci Series” American Journal of Botany 26 2 (1939): 67
2 R. W Pearcy & W Yang “The functional morphology of light capture and carbon gain in the Redwood forest understorey plant Adenocaulon bicolor Hook” Functional Ecology 12 4 (1998): 551
3 H. C. Rosu, J. P. Trevino, H. Cabrera & J. S. Murguia, “Self-image effects for diffraction and dispersion” Electromagnetic Phenomena 6 2 (2006): 204 - 211