Golden Ratio

Imagine you are a plant. You grow from a single tip, and every so often you need to place something new there: a leaf, a petal, a seed. You cannot plan the whole pattern ahead. You can only follow one simple rule: after placing each new piece, turn by a fixed angle and place the next one.

Which angle should you choose?

If you turn by half a circle, your seeds form two straight lines pointing in opposite directions. If you turn by a third, you get three lines. A turn of $3/10$ gives ten lines. Any simple fraction of a turn produces "spokes", with large empty wedges between them. That wastes space, and a sunflower that wasted space would carry fewer seeds.

The angle that avoids spokes best turns out to be linked to one particular number, the golden ratio, written $\varphi$ (phi):

This number has gathered a lot of mystique. It has been credited with the beauty of the Parthenon, the composition of the Mona Lisa, the proportions of the human body and the shape of seashells. Most of those claims do not hold up when checked. ^[1] The real reason $\varphi$ matters is less mystical and more interesting: in a precise mathematical sense, it is the irrational number that is hardest to approximate with fractions, and that property explains why it shows up in growing plants.

One Number, Two Definitions

Cut a line into two pieces so that the whole line relates to the longer piece in the same way the longer piece relates to the shorter one. There is exactly one way to do this, and the ratio you get is $\varphi$.

Euclid described this construction around 300 BCE under the name "division in extreme and mean ratio". The adjective "golden" is modern; it became common only in the nineteenth century.

If the longer piece has length $\varphi$ and the shorter has length $1$, the definition says

Multiplying both sides by $\varphi$ gives the equation that defines the number:

This short equation contains most of the number's odd properties:

• Squaring $\varphi$ is the same as adding 1: $\varphi^2 \approx 2.618$.
• Dividing both sides by $\varphi$ gives $\varphi = 1 + 1/\varphi$, so $1/\varphi = \varphi - 1 \approx 0.618$. Taking the reciprocal is the same as subtracting 1, so $\varphi$ and $1/\varphi$ share the same digits after the decimal point.
• The quadratic $x^2 - x - 1 = 0$ has two roots, $\varphi$ and $\psi = (1-\sqrt5)/2 \approx -0.618$. The second root is $-1/\varphi$. Both describe the same geometry, which is why some authors call $0.618\ldots$ "the golden ratio" too.

The Fibonacci Connection

The Fibonacci numbers start with $1, 1$, and each new term is the sum of the two before it:

Divide each term by the one before it: $2/1 = 2$, $3/2 = 1.5$, $5/3 \approx 1.667$, $8/5 = 1.6$, $13/8 = 1.625$, $21/13 \approx 1.615$. The ratios swing above and below a limit, and that limit is $\varphi$.

The reason is short. If the ratio $F_{n+1}/F_n$ settles to some value $x$, then dividing the recurrence by $F_n$ gives

which is again $x^2 = x + 1$. The connection can be made exact with Binet's formula,

where $\psi = -1/\varphi$. Because $|\psi| < 1$, the term $\psi^n$ shrinks rapidly, and $F_n$ is simply the whole number nearest to $\varphi^n/\sqrt5$. The Fibonacci sequence is the whole-number shadow of the golden ratio. This is a mathematical fact, not an empirical claim.

Why φ Is the "Most Irrational" Number

A rational number is a fraction, $p/q$. An irrational number, such as $\sqrt2$ or $\pi$, cannot be written as a fraction. But irrational numbers are not all equally far from fractions. Some can be approximated extremely well by small fractions; others resist.

A scenario with π

Write $\pi$ as a whole number plus a remainder: $\pi = 3 + 0.14159\ldots$. The remainder is less than 1, so write it as one over something: $0.14159\ldots = 1/7.0625\ldots$. Repeat the process with $7.0625\ldots$, and keep going. The result is a continued fraction:

Stopping early gives fractions that approximate $\pi$. Stopping after the 7 gives $22/7$. Stopping just before the 292 gives $355/113$, which is correct to six decimal places. The large number 292 is the reason: a large term means the next correction is tiny, so the fraction just before it was already unusually accurate.

So a large term in a continued fraction signals that the number sits very close to a simple fraction. A number that stays far from every fraction should have only small terms. The smallest possible term is 1. The number whose terms are all 1 is

Because the pattern repeats forever, the part after the first "1 +" is again $x$. So $x = 1 + 1/x$, and $x = \varphi$. The fractions you get by cutting this continued fraction short are exactly the ratios of consecutive Fibonacci numbers, and they approach $\varphi$ as slowly as any continued fraction can.

The formal statement

This intuition has a precise form, proved by Adolf Hurwitz in 1891. ^[2] For every irrational number $\alpha$, there are infinitely many fractions $p/q$ with

Here $q$ is the denominator of the fraction, and the right-hand side says how close the fraction must come, relative to its size. The constant $\sqrt5$ cannot be made any larger. If you replace it with anything bigger, the statement fails, and $\varphi$ is the number that makes it fail. In this sense $\varphi$ (together with numbers closely related to it) is the irrational number that is worst approximated by fractions.

"Most irrational" is a nickname, not a formal term. The underlying theorem is established mathematics.

From Irrationality to Sunflowers

Now return to the plant turning by a fixed fraction of a circle between seeds.

• If the fraction is rational, $p/q$, every $q$-th seed lands on the same ray, and the pattern forms $q$ spokes.
• If the fraction is irrational but very close to a simple fraction, the pattern looks almost like spokes. A turn of $1/\pi$ of a circle, for example, produces 22 slightly curved arms, because $\pi$ is so close to $22/7$.
• If the fraction is as far as possible from every simple fraction, no set of spokes ever dominates. Seeds keep landing in the largest remaining gaps.

This progression is easy to watch in animated simulations, such as the one in Numberphile's video on the topic, where slowly changing the turning fraction makes spokes appear and dissolve as the fraction passes near $1/3$, $2/5$ and other simple values. ^[15]

That last case is a turn of $1/\varphi^2 \approx 0.382$ of a circle (equivalently $0.618$ in the other direction). In degrees, this is the golden angle:

In 1979 Helmut Vogel proposed a simple model of a sunflower head in which seed number $n$ sits at angle $n\,\theta$ and radius proportional to $\sqrt n$, so that each seed occupies roughly the same area. ^[3] With $\theta$ equal to the golden angle, the model packs seeds evenly with no gaps and no spokes, and it reproduces the familiar crossing spirals. Changing the angle by a fraction of a degree makes gaps and spokes reappear.

Those crossing spirals are called parastichies. When you count the spirals running clockwise and those running anticlockwise, you tend to find two neighbouring Fibonacci numbers, such as 34 and 55, or 55 and 89. The reason is the continued fraction above: the Fibonacci ratios are the best fractional approximations to $\varphi$, and each best approximation shows up as a visible family of spirals at some distance from the centre.

Evidence from real sunflowers

The idea that sunflowers show Fibonacci spirals is often repeated but seldom tested with transparent data. A 2016 citizen-science study led by Jonathan Swinton and Erinma Ochu collected 657 sunflower heads grown by members of the public. ^[7] In the most reliable subset, 565 of 768 spiral counts were Fibonacci numbers, and a further 67 showed related Fibonacci-type structure. Roughly one in five counts did not fit a simple Fibonacci pattern, and some heads showed more complex arrangements than had been reported before.

Established fact: Fibonacci spiral counts are the dominant pattern in sunflowers. Also established: they are not universal, and a theory of phyllotaxis has to explain the exceptions as well.

Why Plants Find the Golden Angle

Plants cannot measure angles, and they do not know about continued fractions. So how would a plant end up turning by 137.5°?

The biological mechanism

In the 1860s the botanist Wilhelm Hofmeister observed that a new leaf or flower bud (a primordium) tends to form at the growing tip in the largest available gap, as far as possible from the existing primordia. Modern work identified the chemistry behind this rule. The plant hormone auxin is pumped through the tissue of the growing tip; existing primordia drain auxin from their surroundings, and a new primordium forms where auxin builds up, away from its neighbours. ^[5] This produces a local rule: each new piece appears where there is the most room.

Graeme Mitchison showed in 1977 that such a space-filling rule, applied at a growing tip, naturally generates Fibonacci spiral patterns. ^[6]

The physics experiment

The strongest demonstration came from physics rather than biology. In 1992 Stéphane Douady and Yves Couder built a small laboratory "plant". ^[4] They dropped tiny droplets of ferrofluid, a liquid that responds to magnets, one at a time at the centre of a dish of oil in a magnetic field. The droplets repelled each other, and the field pushed them outward, just as older primordia are carried away from the growing tip by growth.

Each new droplet settled where the repulsion from its neighbours was weakest, which is simply the local "most room" rule. Nothing in the setup mentioned $\varphi$. Yet when the drops were released slowly relative to how quickly they moved outward, the angle between successive drops converged to about 137.5°, and the droplets formed Fibonacci spirals. When drops were released quickly, the pattern switched to alternating at 180°, like leaves on some grasses. Numerical simulations of the same rule gave the same results.

This result was important because it showed that the golden angle is a dynamical attractor: a pattern that a simple growth process tends to settle into, whether or not the process is alive. Theoretical work by Leonid Levitov around the same time reached a similar conclusion from an energy-minimization argument, and predicted the same Fibonacci structure in the arrangement of magnetic vortices in layered superconductors. ^[14,16]

What this explains, and what it does not

Accepted theory: a local inhibition rule in a growing system tends to produce the golden angle and Fibonacci spirals. Physical experiments, simulations and plant biology all support this.

Still open: why some plants settle on other patterns, how reliably the rule is realized in different species, and whether evolution also favours the golden angle for other reasons. One proposal, for example, argues that the golden angle also minimizes the cost of restructuring the plant's vascular tissue during growth. ^[8] That is a hypothesis, not a settled result.

It is also worth noticing the direction of the explanation. The plant does not "use" the golden ratio. The golden ratio is what a certain kind of growth produces, because it is the angle that never lines up with anything before it.

Other Real Appearances

The golden ratio also appears in several places where the connection is well founded:

• Regular pentagons and icosahedra. The ratio of a regular pentagon's diagonal to its side is exactly $\varphi$. Any geometry with five-fold symmetry, including the icosahedron and dodecahedron, carries $\varphi$ in its coordinates.
• Quasicrystals. In 1984 Dan Shechtman and colleagues reported an alloy with icosahedral symmetry, a form of order that crystallographers had considered impossible. ^[11] Such quasicrystals, and the related Penrose tilings, are built on ratios involving $\varphi$. This work later received the 2011 Nobel Prize in Chemistry.
• Stability of orbits. In mechanical systems with two competing rhythms, motions whose frequency ratio is hard to approximate by fractions are the most resistant to disruption. Numerical work by John Greene found that, in a standard model of this problem, the motion with the golden-mean frequency ratio is the last to break down as the disturbance grows. ^[13] This is the same "most irrational" property at work.
• A quantum magnet. In 2010 an experiment on a chain of cobalt atoms near a quantum phase transition found two excitation energies whose ratio was close to $\varphi$, matching a prediction from a theory with $E_8$ symmetry. ^[12] Here the golden ratio was predicted in advance by theory, then measured, which is far stronger evidence than finding the number after the fact.

Myths and Misconceptions

The golden ratio has attracted more unsupported claims than almost any other number. George Markowsky's 1992 paper Misconceptions about the Golden Ratio is the standard critical survey. ^[1] Clement Falbo later added careful measurements of seashells. ^[9] The recurring problem is that it is easy to "find" $\varphi$ in almost anything if you are free to choose which points to measure, how accurately, and which results to report.

The Parthenon

The claim that the Parthenon was designed around the golden ratio is extremely common. Markowsky pointed out that the building's proportions do not match $\varphi$ well, that popular diagrams draw golden rectangles that do not line up with the actual edges, and that there is no evidence from the period that the builders intended such a ratio. ^[1] The Parthenon was built around 440 BCE, before Euclid wrote down the construction. The claim is best classified as unsupported.

The Mona Lisa and Leonardo

Leonardo da Vinci did illustrate Luca Pacioli's 1509 book De divina proportione, which is about the golden ratio. That historical link is real. The claim that Leonardo built the Mona Lisa or other paintings around golden rectangles is not supported by documentary evidence. ^[1] Rectangles overlaid on reproductions can be placed almost anywhere, so such diagrams demonstrate the flexibility of the method rather than the painter's intent.

The human body

Many popular sources claim that the ratio of a person's height to the height of their navel is $\varphi$. Real bodies vary a great deal, and the result depends on how you measure. Markowsky noted that among the many ratios one can form from the body, some will inevitably come close to 1.6 by chance. ^[1] There is no established biological reason for the human body to follow $\varphi$.

Beauty and the "most pleasing rectangle"

In the nineteenth century, Gustav Fechner asked people to choose the most pleasing rectangle and reported a preference near the golden ratio. Later research has been mixed. A 1995 review by Christopher Green concluded that the effect, where it appears at all, depends heavily on how the question is posed, and is far from the strong universal preference often claimed. ^[10] The status of this idea is contested, not established.

The nautilus shell is not a golden spiral

The chambered nautilus is probably the most famous "golden spiral" in popular culture, and it is wrong. A golden spiral grows by a factor of $\varphi$ every quarter turn, which means by $\varphi^4 \approx 6.85$ each full turn. Falbo measured shells of Nautilus pompilius in the collection of the California Academy of Sciences and found ratios between about 1.24 and 1.43, averaging about 1.33 by his measure, well away from 1.618. ^[9] Roughly speaking, the nautilus shell grows by a factor of about 3 per turn, not 6.85.

The shell is a logarithmic spiral, a spiral that keeps the same shape as it grows. That part is real. But logarithmic spirals come in every growth rate, and the nautilus does not use the golden one.

Limitations

To keep the picture honest, it helps to separate the kinds of claims involved:

A general warning applies to all claims of this kind. A number near 1.6 will appear somewhere in almost any sufficiently complex object. A finding is only meaningful if it was predicted before looking, if the measurement points were fixed in advance, and if the agreement is better than chance would give.

The Pattern Without the Myth

The golden ratio is a good example of a true story being overshadowed by a more glamorous false one.

The false story says that $\varphi$ is a secret code of beauty, placed deliberately in temples, paintings and bodies. The evidence for that story is weak.

The true story is more surprising. A number defined by a simple equation, $\varphi^2 = \varphi + 1$, turns out to be exceptionally hard to approximate by fractions. Exact fractional turns form spokes, while turns close to simple fractions leave persistent near-alignments. In many growing systems, the local rule "put the next piece where there is most room" settles near the golden angle because it strongly resists both. A dish of magnetic droplets does the same thing. No design is needed. The pattern follows from growth and geometry.

The mathematics is exact, and experiments support the link between local growth rules and golden-angle spirals. Similar patterns can emerge wherever a system adds new units one at a time while avoiding old ones. Biology still has to explain the full range of arrangements plants produce, including the many that depart from the Fibonacci pattern.

For a related case where a simple rule produces structure that looks designed, see the God of the gaps, and for how a small set of equations can generate rich physical behaviour, see Maxwell's equations. For another number that has attracted symbolic meaning, see numerology.

References

[1] Markowsky, G. (1992). "Misconceptions about the Golden Ratio." The College Mathematics Journal, 23(1), 2–19. https://doi.org/10.2307/2686193

[2] Hurwitz, A. (1891). "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche." Mathematische Annalen, 39, 279–284. https://doi.org/10.1007/BF01206656

[3] Vogel, H. (1979). "A better way to construct the sunflower head." Mathematical Biosciences, 44(3–4), 179–189. https://doi.org/10.1016/0025-5564(79)90080-4

[4] Douady, S., & Couder, Y. (1992). "Phyllotaxis as a physical self-organized growth process." Physical Review Letters, 68, 2098–2101. https://doi.org/10.1103/PhysRevLett.68.2098

[5] Reinhardt, D., Pesce, E.-R., Stieger, P., Mandel, T., Baltensperger, K., Bennett, M., Traas, J., Friml, J., & Kuhlemeier, C. (2003). "Regulation of phyllotaxis by polar auxin transport." Nature, 426, 255–260. https://doi.org/10.1038/nature02081

[6] Mitchison, G. J. (1977). "Phyllotaxis and the Fibonacci Series." Science, 196(4287), 270–275. https://doi.org/10.1126/science.196.4287.270

[7] Swinton, J., Ochu, E., & The MSI Turing's Sunflower Consortium (2016). "Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment." Royal Society Open Science, 3(5), 160091. https://doi.org/10.1098/rsos.160091

[8] Okabe, T. (2015). "Biophysical optimality of the golden angle in phyllotaxis." Scientific Reports, 5, 15358. https://doi.org/10.1038/srep15358

[9] Falbo, C. (2005). "The Golden Ratio: A Contrary Viewpoint." The College Mathematics Journal, 36(2), 123–134. https://doi.org/10.1080/07468342.2005.11922119

[10] Green, C. D. (1995). "All That Glitters: A Review of Psychological Research on the Aesthetics of the Golden Section." Perception, 24(8), 937–968. https://doi.org/10.1068/p240937

[11] Shechtman, D., Blech, I., Gratias, D., & Cahn, J. W. (1984). "Metallic Phase with Long-Range Orientational Order and No Translational Symmetry." Physical Review Letters, 53(20), 1951–1953. https://doi.org/10.1103/PhysRevLett.53.1951

[12] Coldea, R., Tennant, D. A., Wheeler, E. M., et al. (2010). "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry." Science, 327(5962), 177–180. https://doi.org/10.1126/science.1180085

[13] Greene, J. M. (1979). "A method for determining a stochastic transition." Journal of Mathematical Physics, 20(6), 1183–1201. https://doi.org/10.1063/1.524170

[14] Levitov, L. S. (1991). "Energetic Approach to Phyllotaxis." Europhysics Letters, 14(6), 533–539. https://doi.org/10.1209/0295-5075/14/6/006

[15] Numberphile (2018). "The Golden Ratio (why it is so irrational)." Video, YouTube. Further watching. https://www.youtube.com/watch?v=sj8Sg8qnjOg

[16] Levitov, L. S. (1991). "Phyllotaxis of flux lattices in layered superconductors." Physical Review Letters, 66(2), 224–227. https://doi.org/10.1103/PhysRevLett.66.224