Ratio & Proportion · 2D
Golden Spiral
Fibonacci squares and quarter arcs
Squares whose sides follow the Fibonacci numbers, each with a quarter circle. The spiral widens by φ every quarter turn.
- Growth / turn
- φ⁴ ≈ 6.85
- Fₙ₊₁ ÷ Fₙ
- → φ
Watch it drawn with compass and straightedge, hear the chord of its lines, and turn the Dimension dial to raise its form in space.
Facts
- Fibonacci: 1, 1, 2, 3, 5, 8, 13, … each number is the sum of the two before.
- The ratio of neighbours approaches φ ≈ 1.618 ever more closely.
- Remove a square from a golden rectangle and a smaller golden rectangle remains.
- Nautilus shells are logarithmic spirals too, but usually not with the golden growth rate.
History and meaning
Leonardo of Pisa (Fibonacci) introduced the sequence to Europe in 1202 through a puzzle about breeding rabbits. It had been known in India centuries earlier in the study of poetic meter.
Across the dimensions
- Slice of Golden ShellThe spiral gains a body that widens by φ every quarter turn: a shell whose middle slice is the flat spiral.
Learn it step by step
- Construct it The Golden CutDivide AB at G so that AB : AG = AG : GB.
- Construct it Golden RectangleExtend the square into a golden rectangle.
- Journey The Golden ThreadStep 2: Squares that grow
- Journey The Plane RisesStep 7: A shell
