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^{} $$. In the Fibonacci fractional series, each $m=n+2$:
$$\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},\mathrm{\dots}$$ |
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
Title | Fibonacci fraction |
---|---|
Canonical name | FibonacciFraction |
Date of creation | 2013-03-22 18:03:18 |
Last modified on | 2013-03-22 18:03:18 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 4 |
Author | PrimeFan (13766) |
Entry type | Definition |
Classification | msc 11B39 |