GOLDENSPIRAL
The mathematics of beauty — Fibonacci, the golden ratio, and the patterns that shape nature, art, and design.
From computing a Fibonacci or Lucas term and splitting a length in golden proportion to generating a typography scale, a sunflower-style phyllotaxis pattern, or a rule-of-phi crop guide, GoldenSpiral pairs free, formula-driven calculators with practical guides and a richly illustrated museum. The math behind the pattern, made usable.
Golden Ratio & Fibonacci Calculators
Free tools for the Fibonacci sequence, the golden ratio, and proportion in nature and design — Fibonacci calculator, golden ratio calculator, modular scale generator, golden spiral calculator, Lucas number calculator, and golden ratio grid.
Fibonacci Calculator
Compute any term of the Fibonacci sequence with Binet's closed-form formula, plus the full sequence and its running sum
Golden Ratio Calculator
Split any length into golden proportion (a ÷ b = φ), and see the matching golden-rectangle dimensions
Modular Scale Generator
Generate a typography or layout size scale from a base size and a ratio — φ by default, or any custom ratio
Golden Spiral & Phyllotaxis Calculator
Generate a sunflower-style seed pattern using the golden angle (137.5°), with exact coordinates for any seed count
Lucas Number Calculator
Compute the Lucas sequence and verify the Fibonacci–Lucas identity L(n) = F(n−1) + F(n+1)
Golden Ratio Grid
Generate rule-of-phi gridlines and a golden-rectangle crop suggestion for any frame or canvas size
The Golden Ratio Museum
Wander an illustrated museum of the patterns and proportions that connect mathematics to the world — the nautilus shells and sunflower heads behind phyllotaxis, the Parthenon and Renaissance works popularly (and not so popularly) linked to φ, the mathematicians and manuscripts behind the Fibonacci sequence, and the typography and design systems that use proportion on purpose today, each in its own wing, so you can get to know the story behind every pattern: how it was discovered, what’s actually verified about it, and where it shows up today.
Tour the museum →From the Journal
Original guides on the golden ratio in nature and design, the Fibonacci sequence and its math, and the myths and honest history behind the world’s best-known proportion.
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.
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.
The Fibonacci Sequence: How It's Built and Why It Shows Up Everywhere
0, 1, 1, 2, 3, 5, 8, 13… The rule behind the Fibonacci sequence is almost embarrassingly simple. What's genuinely interesting is why that one simple rule keeps resurfacing across unrelated fields.
Binet's Formula: Computing Fibonacci Numbers Without Recursion
Every Fibonacci number is a whole number built entirely from addition. Binet's formula computes any one of them directly from a single irrational constant — no earlier terms required.
Modular Scales: Using the Golden Ratio to Size Type and Layout
Picking heading sizes by feel rarely produces a coherent page. A modular scale generates every size from one base number and one ratio — here's how the technique actually works.
Keep Learning
Plain-language reference entries on Fibonacci and golden-ratio mathematics, phyllotaxis and pattern in nature, and the design and typography terms that build on the same proportions — everything you need to go a level deeper.
Visit the Learn hub →