Sizu Riatto
Grow by Numbers · Chapter 7

The Golden Hint

φ is about 1.618. Multiply by it for a golden rectangle, watch Fibonacci's neighbours close in on it, and see why 137.5° packs a sunflower head and 137° does not.

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Opens on the book's Microgreen Shelf. Change any number to your own and press Calculate.

Labels on a sheet

The best fractions for a number

A dot sunflower

Enter a length, then press Calculate.

The formula

  • φ = (1 + √5) ÷ 2 ≈ 1.618034; its partner both ways: length × φ and length ÷ φ
  • Fibonacci: each number is the sum of the two before; their ratios 8/5, 13/8, 21/13 … close in on φ
  • Labels on a sheet = (sheet width ÷ label width, rounded down) × (sheet height ÷ label height, rounded down); try both ways round
  • Best fractions: the continued fraction of the number, cut off after each step
  • A dot sunflower (Vogel): seed n at angle n × the turn, distance √n from the centre
The book's worked example, Chapter 7. The shelf's label is 50 mm wide and 50 × φ = 80.9 mm tall (8/5 gives 80 mm, 0.9 mm short). 12 fit on a US letter sheet standing up, only 10 lying down, and 12 on A4. The golden shape is an aesthetic choice, not a growing rule. π's best fractions: 3, 22/7, 333/106, 355/113.
Check it. Divide the long side by the short side: a golden rectangle gives 1.618. Cut a square off it and the piece left has the same shape.

Honest limits. Microgreens are cut at the seed-leaf stage, two opposite leaves, with no spiral to see; the famous spirals are in mature heads, and about 1 in 5 counted sunflower spirals are not Fibonacci numbers (Turing's Sunflowers). A-series paper is √2, not φ. The Microgreen Shelf was worked out on paper from published figures, not grown by us: weigh and measure your own. Prices are illustrative US prices from October 2026, shown without a currency sign; put in your own. Everything stays in your browser: no sign-up, nothing stored.

These calculators go with Grow by Numbers (Make It Add Up, Book 1), where every number on this page is worked by hand.