Computational Verification of Vogel's Model in Phyllotaxis Patterns

The Fibonacci sequence and its generalizations, known as k-nacci sequences, exhibit spatial distribution properties that have inspired engineering applications through biomimetics. This study presents a computational exploration of k-nacci arrangements (k = 2–10) applied to the optimal placement of solar panels within a circular area, generalising Vogel's phyllotaxis model. Using a Python simulation validated with NumPy, we compare four quantitative metrics — minimum inter-element distance, mean nearest-neighbour distance, distribution uniformity, and shadow index — across k-nacci, rectangular grid, hexagonal, and random layouts (N = 200 panels, circular area of radius 240 px). Results show that the Fibonacci arrangement (k = 2) achieves 98.9% uniformity and zero shadow index, closely matching hexagonal packing while avoiding its periodic alignment gaps. Higher-order k-nacci sequences show progressive degradation. These findings confirm that the Fibonacci case is the optimal member of the k-nacci family for circular panel distribution. Python simulation code and interactive HTML tool included.