Sacred geometry · Grow it yourself
Golden Angle Garden
The golden angle is 360° ÷ φ² = 137.5°, the turn a sunflower makes between one seed and the next. At exactly that angle the seeds pack evenly with no gaps; move it even half a degree and spokes and empty wedges appear. Drag the angle, nudge it, or snap it to a simple fraction of a turn, and watch the garden regrow. Below it, step through the Fibonacci numbers and watch their ratio close in on 1.618.
137.508°
- Share of a turn
- As a fraction
- Nearest Fibonacci ratio
- Golden angle137.5077640500° = 360° ÷ φ²
F(n)
F(n) ÷ F(n − 1)
The angle
What is the golden angle?
The golden angle is the smaller of the two pieces you get when you cut a full turn in the golden ratio: 360° ÷ φ² = 137.5077...°. MathWorld gives it as “137.507... degrees.” The bigger piece is 360° ÷ φ = 222.4922...°, and the two add up to 360°. The cut is golden because the whole turn is to the big piece as the big piece is to the small one: 360 ÷ 222.49 = 222.49 ÷ 137.51 = 1.618, the golden ratio φ.
Two other ways to write the same number. Because φ² = φ + 1, the share of a turn is 1 ÷ φ² = 2 − φ = 0.3819660113... So the golden angle is also 360° × (2 − φ). In radians it is 2π ÷ φ² = 2.39996...
How the garden is drawn. Seed number n is placed n × the angle round the circle, at a distance from the middle that grows with the square root of n, so every seed gets the same share of space. That is all. Every spiral, spoke and gap you see comes from the angle alone.
Why it packs best
Fractions make spokes, φ makes none
Pick a simple fraction of a turn and the seeds line up. At 3/8 of a turn (135°), seed 8 has turned exactly 3 full circles and lands straight behind seed 0, seed 16 behind seed 8, and so on: 8 straight spokes with empty wedges between them. Any fraction p/q does the same with q spokes. Press 3/8 turn above and see it.
An irrational angle never repeats, but some come close. 137.0° is not 3/8, but it sits very near 8/21 of a turn, so its seeds bend into 21 curved arms with gaps between. 138.0° sits near 5/13 and shows 13 arms. The closer an angle sits to a simple fraction, the sooner its gaps open up.
The golden ratio is the number furthest from every fraction. MathWorld calls φ “the "worst" real number for rational approximation because its continued fraction representation [1,1,1,...] ... has the smallest possible term (1) in each of its infinitely many denominators.” Hurwitz’s theorem makes that exact: every irrational number a has infinitely many fractions p/q with |a − p/q| < 1/(√5 q²), and the √5 can only be improved by leaving out the numbers tied to the golden ratio. So seeds turned by the golden angle never settle into spokes, and the head fills evenly.
Plants and physics agree. MathWorld reports that van Iterson “showed in 1907 that points separated by [the golden angle] on a tightly bound spiral tends to produce interlocked spirals winding in opposite directions, and that the number of spirals in these two families tend to be consecutive Fibonacci numbers.” In 1992 Douady and Couder got the same spirals from drops of magnetic fluid, and put it down to the system’s “trend to avoid rational (periodic) organization, thus leading to a convergence towards the golden mean.” How often real sunflowers, pine cones and pineapples follow it is on the Fibonacci in nature page.
The calculator
Why F(n) ÷ F(n − 1) closes in on 1.618
Divide any Fibonacci number by the one before it and the answer gets closer to φ = 1.6180339887... every step, landing above and below in turn. 2 ÷ 1 = 2, 3 ÷ 2 = 1.5, 5 ÷ 3 = 1.667, 8 ÷ 5 = 1.6, 13 ÷ 8 = 1.625. MathWorld: the ratios “approaches the golden ratio ... as first proved by Scottish mathematician Robert Simson in 1753”.
The gap is exact. Binet’s formula writes every Fibonacci number with φ and ψ = 1 − φ = −0.618... From it follows F(n) − φ × F(n − 1) = ψⁿ⁻¹, so the ratio misses φ by exactly 0.618...ⁿ⁻¹ ÷ F(n − 1), above φ when n is odd and below when n is even. The calculator shows that gap for any n up to 2,000, with every digit of F(n) worked out exactly. At n = 25 the ratio is 1.618033988957..., against φ = 1.618033988749...
The same numbers make the garden. The fractions 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34... are Fibonacci numbers two apart, and they close in on 0.381966, the golden angle’s share of a turn. The Nearest Fibonacci ratio line shows which one your angle sits closest to. More on the sequence itself is on the Fibonacci sequence page.
Questions
The golden angle
What is the golden angle?
About 137.5°: the smaller part of a full turn cut in the golden ratio, 360° ÷ φ² = 137.5077...°. It is the turn between one seed and the next in a sunflower head drawn the Fibonacci way.
Why is 137.5 the golden angle?
Because 360° ÷ 137.5° ≈ 2.618 = φ², and the other part of the turn, 222.5°, is 360° ÷ φ. Whole turn to big part equals big part to small part, 1.618 each: a golden cut of the circle.
How do you calculate the golden angle?
Divide 360° by φ² (2.6180339887...), or multiply 360° by 2 − φ (0.3819660113...). Both give 137.5077640500°. In radians it is 2π ÷ φ² = 2.3999632297.
Why do sunflowers use the golden angle?
Each new seed turned 137.5° from the last never lines up with the earlier ones, because φ is the number hardest to approximate with fractions. So the seeds fill the head evenly with no spokes or gaps, and the spirals you see come out in neighbouring Fibonacci numbers.
What happens if the angle is a simple fraction of a turn?
The seeds pile up on straight spokes. At p/q of a turn, every qth seed points the same way, so you get q spokes with empty wedges between: 2 at 1/2 turn, 3 at 1/3, 5 at 2/5, 8 at 3/8.
Is 3.14 a golden ratio?
No. 3.14159... is π, the ratio of a circle’s distance round to its width. The golden ratio is φ = 1.6180339887..., and the golden angle built from it is 137.5° or 2.39996 radians.
Sources
Where this comes from
- Eric W. Weisstein, “Golden Angle,” MathWorld, for 137.507° and van Iterson 1907: mathworld.wolfram.com.
- Eric W. Weisstein, “Golden Ratio,” MathWorld, for φ and the “worst” approximable number: mathworld.wolfram.com.
- Eric W. Weisstein, “Hurwitz’s Irrational Number Theorem,” MathWorld: mathworld.wolfram.com.
- Eric W. Weisstein, “Fibonacci Number,” MathWorld, for Simson 1753: mathworld.wolfram.com.
- Eric W. Weisstein, “Binet’s Formula,” MathWorld: mathworld.wolfram.com.
- Eric W. Weisstein, “Phyllotaxis,” MathWorld: mathworld.wolfram.com.
- S. Douady and Y. Couder, “Phyllotaxis as a physical self-organized growth process,” Physical Review Letters 68(13) (1992): 2098-2101: doi.org.
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