Category
The Golden Ratio in Nature
The Golden Ratio in Nature: Nautilus Shells, Sunflowers, and Pinecones
One of these is a solid, well-documented case of golden-ratio math in a living organism. The other is a popular myth that keeps getting repeated. Here's how to tell them apart.
Why Phyllotaxis Uses the Golden Angle (137.5°), Not a “Nicer” Number
A sunflower head rotates each new seed by roughly 137.5 degrees from the last. That's an oddly specific number for nature to land on — until you see what happens with the tidier alternatives.
Generating a Phyllotaxis Pattern by Hand: What the Numbers Mean
One square root, one modulo, two trig calls per point — the entire sunflower-spiral algorithm, worked through point by point with the calculator's own real output, plus what happens if you round the angle even slightly wrong.
Drawing a Golden Spiral Properly: The Real Curve vs. the Fibonacci-Square Approximation
The nested-squares picture everyone calls a golden spiral is a close approximation, not the real thing — a true logarithmic golden spiral grows by exactly phi every quarter turn, from the very first step, with no convergence involved.