Category
Golden Ratio Basics
What Is the Golden Ratio? A Plain-Language Introduction
The golden ratio, φ ≈ 1.618, is one number defined by one simple property. Here's what it actually is, where the value comes from, and where the popular claims about it start to overreach.
How the Fibonacci Ratio Converges to the Golden Ratio
Divide consecutive Fibonacci numbers and the ratio homes in on φ startlingly fast — this is the one popular golden-ratio claim that's not just approximately true, it's exactly provable. Here's the full run of the numbers.
Continued Fractions and Why Phi Is the Most Irrational Number
Phi's continued fraction is an endless chain of nothing but 1s — the simplest one possible, and the precise, provable reason it resists rational approximation better than any other irrational number.
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.