Cross validation — where it appears
A disc is a cylinder
Vogel's seed head makes the rise fall as one over radius squared, so a disc is not one lattice but a family of them. Feed that into the cylinder's ladder and it predicts where a sunflower's spiral counts change — with nothing fitted, and against a counter that never sees either model.
Transitions a factor of φ² apart
The ladder's rungs are a factor of 1/φ² apart in rise. A disc's rise falls as one over radius squared and a cone's as one over distance, so the same rungs land a factor of φ apart on a seed head and a factor of φ² apart on a cone — measured, on both, by a counter that has never heard of either.
Named alongside it
The objects these essays reach for when they reach for this one.
CylinderLadderRiseTransitionsCounting blindConeDivergence angleFibonacciGeometric ladderφ, the golden ratioParastichyParastichy pair