Golden spiral or Fibonacci spiral: what is the difference?

The difference

One smooth curve, one chain of arcs

The golden spiral is one smooth curve. MathWorld writes it as r = a·e^bθ with b = (2 ln φ)/π ≈ 0.306349. Put that b in and every quarter turn (θ grows by π/2) multiplies the distance from the centre by e^{ln φ} = φ. Wikipedia says the same in words: a golden spiral “gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.” Four quarter turns make φ⁴ = 6.854.

The Fibonacci spiral is joined pieces of circles. Ron Knott at the University of Surrey describes the build: squares are added with “each new square having a side which is as long as the sum of the latest two square’s sides,” and the spiral is “a quarter of a circle in each square.” A circle bends by the same amount all the way round, so the curve’s bend jumps every time it crosses into a bigger square. The golden spiral’s bend changes smoothly instead.

Why they look almost the same. Neighbouring Fibonacci numbers divide out closer and closer to φ: 21 ÷ 13 = 1.6154, 377 ÷ 233 = 1.6180. So the squares grow at nearly the golden rate. Slide to 14 squares above and the last two differ from φ by less than 0.001%.

Why they never become the same. Look at the second readout. We fitted the golden spiral through the two outermost corners and measured the widest gap between the curves from the spiral’s centre. From 8 squares on it sits at about 1.2% of the biggest square and stops shrinking (our own computation, open in this page’s code). Better squares cannot fix the shape: even in a perfect golden rectangle, MathWorld notes that the true spiral “is not actually tangent at these points, however, but passes through them and intersects the adjacent side.” Knott sums it up: the Fibonacci spiral “is not a true mathematical spiral (since it is made up of fragments which are parts of circles and does not go on getting smaller and smaller) but it is a good approximation”.

The inside end tells them apart too. The Fibonacci spiral stops at the first 1 × 1 square. The golden spiral keeps winding inward forever, a quarter turn and a factor of φ smaller each time, toward a centre it never reaches.

Try it on paper

How to draw each spiral

The Fibonacci spiral, with squared paper and a compass.

1. 1. Two 1sDraw a 1 × 1 square and a second one to its right.
2. 2. Keep addingDraw a 2 × 2 square on top of both, a 3 × 3 to the left, a 5 × 5 below, an 8 × 8 to the right, and keep going round. Each new side is the sum of the last two.
3. 3. Set the compassIn each square, put the compass point on the corner where the spiral turns and set it to the square’s side.
4. 4. Quarter circlesDraw a quarter circle from one corner to the opposite corner. Each arc starts where the last one ended.

The golden spiral cannot be drawn with a compass, because no part of it is a circle. Plot it instead. Pick a centre and mark a point 1 unit away. Turn 90° and mark the next at 1.618, then 2.618, 4.236, 6.854, 11.09: each one φ times the last. Join the points with one smooth curve that crosses every line from the centre at the same angle, about 73° (our arithmetic, from MathWorld’s b). The quarter circles in a golden rectangle’s squares give a close copy: per Wikipedia, a curve that, “though not a true logarithmic spiral, closely approximates a golden spiral.”

On a photo. To see a golden spiral over a picture of your own, use the Golden Ratio Lens; it runs on your device and uploads nothing.

History

Spira mirabilis: “I shall arise the same though changed”

The golden spiral is one member of a bigger family, the logarithmic spirals. MathWorld: the logarithmic spiral “is also known as the growth spiral, equiangular spiral, and spira mirabilis.” Any growth rate works; φ per quarter turn is the one that matches the golden rectangle. MathWorld: “Successive points dividing a golden rectangle into squares lie on a logarithmic spiral ... which is sometimes known as the golden spiral.”

Descartes was first. MathWorld: “The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli.” “Torricelli worked on it independently and found the length of the curve.”

Bernoulli wanted it on his grave. The MacTutor history archive at St Andrews: “Jacob had always found the properties of the logarithmic spiral to be almost magical and he had requested that it be carved on his tombstone with the Latin inscription Eadem Mutata Resurgo meaning ‘I shall arise the same though changed’.” The motto fits the math: zoom into a logarithmic spiral, or out, and you get the same spiral back, only turned. MathWorld adds that the engraver “did not draw it true to form.”

Measured, not assumed

Nautilus shells, galaxies and hurricanes

The nautilus is a logarithmic spiral, but not a golden one. In 1999 Clement Falbo measured the nautilus shells at the California Academy of Sciences in San Francisco. Science News reported that their spirals “could be inscribed within rectangles with sides in the ratio of about 1.33,” and “The measured ratios ranged from 1.24 to 1.43.” In growth terms, “the spiral of the chambered nautilus triples in radius with each full turn whereas a golden-ratio spiral grows by a factor of about 6.85 per full turn.” Tripling per turn is 3^1/4 = 1.316 per quarter turn, against φ = 1.618. Press Nautilus rate above to draw it: it winds far more tightly. More on shells and plants in Fibonacci in nature.

Galaxy arms are not one spiral at all. Wikipedia’s golden spiral article: “It is sometimes erroneously stated that spiral galaxies and nautilus shells get wider in the pattern of a golden spiral”. A logarithmic spiral keeps the same pitch angle all the way out, but in a 2013 study of 50 grand-design spiral galaxies, Savchenko and Reshetnikov found: “Spiral arms of most galaxies cannot be described by a single value of the pitch angle.” We found no measurement of any galaxy arm that matched a golden spiral.

Hurricanes: claimed, never measured. A 2003 Space.com article on Hurricane Isabel, built on an interview with astrophysicist Mario Livio, says: “The curious similarity between the shape of hurricanes like Isabel and galaxies far away in space is explained by something called the Golden Ratio.” It quotes no measurement, and Livio’s own quoted words are about logarithmic spirals in general (“They also appear, interestingly enough, when a falcon dives toward its prey”). The same article quotes astronomer Karen Masters: “Hurricanes are structures in the gravitational field of the Earth, while galaxies are self-gravitating objects in space.” We found no measurement of a hurricane’s bands growing by φ per quarter turn.

What the tradition holds

The meaning people give the spiral

The oldest meaning on record is Bernoulli’s. He chose the logarithmic spiral for his tombstone with the words “Eadem Mutata Resurgo”, “I shall arise the same though changed”: a curve that keeps growing and keeps its shape. That is a mathematician’s personal motto, not a religious teaching.

We looked for an older religious or spiritual tradition that names the golden spiral and did not find one we could source. The modern idea that it is a code for nature, galaxies and storms rests on the claims above, and those claims have not held up when measured. If you know a primary source, the measurements on this page are the test: does the thing grow by 1.618 every quarter turn?