Fibonacci Numbers and the Golden Ratio, math that surprises you lesson

Start with $0$ and $1$ and keep writing the sum of the last two numbers. The Fibonacci numbers $0, 1, 1, 2, 3, 5, 8, 13, 21$ grow without end, yet dividing each term by the one before it, starting with $1/1$ , gives $1$ , $2$ , $1.5$ , about $1.667$ , $1.6$ , $1.625$ and about $1.615$ , values that crowd closer and closer to a number near $1.618$ . Why that number, and do the ratios ever reach it? This page shows that the addition rule itself forces the golden ratio, proves that the ratios approach it from alternate sides without ever landing on it, and compares the familiar spiral drawn in Fibonacci squares with the exact golden spiral.

Key terms

Fibonacci sequence

Definition The Fibonacci sequence $F_0, F_1, F_2, \ldots$ is defined by

\begin{gathered} F_0 = 0, \qquad F_1 = 1, \\[0.4em] F_n = F_{n-1} + F_{n-2}, \end{gathered}

where the addition rule holds for every $n \ge 2$ . Its terms are $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots$ . Every term after $F_0$ is positive, and no term is smaller than the one before it. The surprise is that the ratio $F_{n+1}/F_n$ of each term to the one before it, for $n \ge 1$ , comes as close as we like to the golden ratio $\varphi = (1 + \sqrt{5})/2 \approx 1.618$ as $n$ grows. The reason is that the addition rule sends each ratio $r$ to the next ratio $1 + 1/r$ , that $\varphi$ is the one positive number this rule leaves unchanged, and that the rule moves every ratio of the sequence closer to $\varphi$ . Every ratio is a fraction of whole numbers, while $\varphi$ is irrational, so the ratios approach $\varphi$ without ever equaling it. Some books start the list at $1, 1$ and leave out the $0$ , which changes none of the ratios. The sequence is named after Leonardo of Pisa, known as Fibonacci, whose book Liber Abaci of 1202 contains a problem about breeding rabbits that leads to these numbers.

In plain words Each new number is the sum of the two before it. The numbers grow without end, but the ratio of each number to the one before it settles toward one fixed value, about $1.618$ .

Example Climb a staircase taking one or two steps at a time, and let $W_n$ be the number of different routes up a staircase of $n$ steps. A staircase of one step has one route, and a staircase of two steps has two, two single steps or one double step. For $n \ge 3$ , every route up $n$ steps ends either with a single step from step $n - 1$ or with a double step from step $n - 2$ . No route ends both ways, so

W_n = W_{n-1} + W_{n-2}.

This gives $W_3 = 3$ , the routes $1 + 1 + 1$ , $1 + 2$ and $2 + 1$ , then $W_4 = 5$ , the routes $1 + 1 + 1 + 1$ , $1 + 1 + 2$ , $1 + 2 + 1$ , $2 + 1 + 1$ and $2 + 2$ , and then $W_5 = 8$ . The route counts start with $W_1 = 1 = F_2$ and $W_2 = 2 = F_3$ and follow the same addition rule, so $W_n = F_{n+1}$ for every $n \ge 1$ .

Golden ratio

Definition The golden ratio is the number

\begin{aligned} \varphi &= \frac{1 + \sqrt{5}}{2} \\[0.4em] &\approx 1.6180339887, \end{aligned}

the positive solution of $x^2 = x + 1$ . Dividing that equation by $x$ , which is not zero, shows that $\varphi$ is also the positive solution of $x = 1 + 1/x$ . The other solution of both equations is $(1 - \sqrt{5})/2 \approx -0.618$ , which equals $-1/\varphi$ . A length cut into a longer part $a$ and a shorter part $b$ is cut in the golden ratio when the whole is to the longer part as the longer part is to the shorter, $(a + b)/a = a/b$ , and then $a/b = \varphi$ . The golden ratio is irrational, because $\sqrt{5}$ is irrational. It satisfies $\varphi^2 = \varphi + 1 \approx 2.618$ and $1/\varphi = \varphi - 1 \approx 0.618$ .

In plain words The golden ratio is the one proportion that survives adding the parts together. If a long piece and a short piece are in the ratio $\varphi$ , then the two pieces together and the long piece are in the ratio $\varphi$ as well.

Example To check $\varphi^2 = \varphi + 1$ directly, square the formula,

\begin{aligned} \varphi^2 &= \frac{1 + 2\sqrt{5} + 5}{4} \\[0.4em] &= \frac{3 + \sqrt{5}}{2} = \varphi + 1. \end{aligned}

A golden rectangle has sides $1$ and $\varphi$ . Cutting off a $1$ by $1$ square leaves a rectangle with sides $\varphi - 1$ and $1$ , whose long side divided by its short side is $1/(\varphi - 1) = \varphi$ , so the leftover rectangle has the same shape as the original. The Fibonacci rectangles work the other way around. Attaching a square of side $F_{n+1}$ to the long side of an $F_n$ by $F_{n+1}$ rectangle makes an $F_{n+1}$ by $F_{n+2}$ rectangle. So the rectangles $1$ by $2$ , $2$ by $3$ , $3$ by $5$ and $5$ by $8$ have long side over short side equal to the Fibonacci ratios, and their shapes approach the golden rectangle.

Ratio convergence

Definition Write $r_n = F_{n+1}/F_n$ for $n \ge 1$ . The ratios converge to $\varphi$ , which means that the gap $\lvert r_n - \varphi \rvert$ becomes and stays smaller than any positive number we choose, once $n$ is large enough. Since $F_{n+2} = F_{n+1} + F_n$ , the ratios follow the update rule

\begin{aligned} r_{n+1} &= \frac{F_{n+1} + F_n}{F_{n+1}} \\[0.4em] &= 1 + \frac{1}{r_n}, \end{aligned}

starting from $r_1 = 1$ . If the ratios have a positive limit $L$ , then $L = 1 + 1/L$ , so $L = \varphi$ . That alone does not show that a limit exists, since a sequence can stay away from a value that its update rule leaves unchanged. The rule $r \mapsto 2r$ , for example, leaves $0$ unchanged, but starting from $1$ it gives $2, 4, 8, \ldots$ . The proof uses $\varphi = 1 + 1/\varphi$ to compare each gap with the next one,

\begin{aligned} r_{n+1} - \varphi &= \frac{1}{r_n} - \frac{1}{\varphi} \\[0.4em] &= \frac{\varphi - r_n}{r_n \varphi}. \end{aligned}

Every ratio is at least $1$ , because $F_{n+1} \ge F_n$ . So each step flips the sign of the gap and multiplies its size by $1/(r_n \varphi)$ , which is at most $1/\varphi \approx 0.618$ . The first gap has size $\varphi - 1 = 1/\varphi$ , so

\lvert r_n - \varphi \rvert \le \varphi^{-n},

which goes to $0$ . The ratios with odd $n$ lie below $\varphi$ , those with even $n$ lie above it, and none equals it.

In plain words The ratio never lands on the golden ratio. Each new ratio overshoots or undershoots by less than the one before, so the ratios zigzag in toward $\varphi$ .

Example The update rule produces each ratio from the previous one without the Fibonacci numbers themselves. From $r_4 = 5/3$ it gives $r_5 = 1 + 3/5 = 8/5$ , and from $8/5$ it gives $1 + 5/8 = 13/8$ . The table lists the first eight ratios, rounded to four decimal places.

$n$	$r_n$	Decimal	Gap $r_n - \varphi$
1	$1/1$	$1.0000$	$-0.6180$
2	$2/1$	$2.0000$	$0.3820$
3	$3/2$	$1.5000$	$-0.1180$
4	$5/3$	$1.6667$	$0.0486$
5	$8/5$	$1.6000$	$-0.0180$
6	$13/8$	$1.6250$	$0.0070$
7	$21/13$	$1.6154$	$-0.0026$
8	$34/21$	$1.6190$	$0.0010$

The signs alternate as predicted. The size of each gap divided by the size of the one before settles near $0.382$ , which is $1/\varphi^2$ , because the factor $1/(r_n \varphi)$ approaches $1/\varphi^2$ as $r_n$ approaches $\varphi$ .

Logarithmic spiral

Definition Fix a center point $O$ and a starting ray from $O$ . Describe a point by its distance $r$ from $O$ and the angle $\theta$ turned counterclockwise from the starting ray, measured in radians, so that a full turn is $2\pi$ , and let $\theta$ keep growing past $2\pi$ , or go below $0$ , as a curve winds around $O$ . A logarithmic spiral is a curve

r = a e^{b\theta},

where $\theta$ ranges over all real numbers, $a > 0$ and $b \ne 0$ are fixed numbers and $e \approx 2.718$ is the base of natural logarithms. Turning through any fixed angle $\Delta$ multiplies the distance from $O$ by the same factor $e^{b\Delta}$ . As a result, enlarging the spiral about $O$ by the factor $e^{b\Delta}$ gives the same spiral turned through the angle $\Delta$ , and the spiral crosses every ray from $O$ at the same angle. For $b > 0$ it winds outward without end as $\theta$ grows, and as $\theta$ decreases it winds around $O$ infinitely many times, coming as close to $O$ as we like without reaching it.

In plain words Each time a logarithmic spiral turns through the same angle, it grows by the same factor. Zooming in or out about its center only rotates it.

Example The spiral $r = 2^{\theta/(2\pi)}$ is the logarithmic spiral with $a = 1$ and $b = \ln 2/(2\pi) \approx 0.110$ , since $2 = e^{\ln 2}$ . Its distance from $O$ doubles with every full turn, with $r = 1, 2, 4, 8$ at $\theta = 0, 2\pi, 4\pi, 6\pi$ , and every quarter turn multiplies it by $2^{1/4} \approx 1.189$ . Turning backward, $r = 1/2$ at $\theta = -2\pi$ and $r = 1/4$ at $\theta = -4\pi$ , so the spiral circles $O$ forever without reaching it. The spiral $r = \theta$ for $\theta \ge 0$ is different. It adds the same distance $2\pi$ with every full turn instead of multiplying by the same factor, so it is not a logarithmic spiral.

Golden spiral

Definition The golden spiral is the logarithmic spiral whose distance from its center grows by the factor $\varphi$ with every quarter turn,

r = a\,\varphi^{2\theta/\pi}, \qquad a > 0.

In the form $r = a e^{b\theta}$ it has $b = 2\ln\varphi/\pi \approx 0.306$ , one full turn multiplies the distance by $\varphi^4 \approx 6.854$ , and the curve crosses every ray from its center at an angle of about $73^\circ$ . The Fibonacci spiral, drawn as quarter circles in squares of sides $1, 1, 2, 3, 5, 8, \ldots$ , is a different curve. Each of its pieces is a quarter of a circle centered at a corner of its own square, so the drawing changes center from square to square and its radius jumps from one Fibonacci number to the next. The golden spiral turns around a single center, and its distance from that center changes continuously. Far out, the squares grow by factors close to $\varphi$ per quarter turn, the same growth as the golden spiral, which is why the two pictures look alike. Spiral patterns in shells, plants and art are worth measuring before they are named. Some follow Fibonacci counts or a growth factor close to $\varphi$ , and others do not.

In plain words The square and arc drawing is a good sketch of the golden spiral made from pieces of circles. The golden spiral itself grows by exactly the factor $\varphi$ in every quarter turn, starting from any point on it.

Example Compare the growth over a quarter turn. On the golden spiral it is always $\varphi \approx 1.618$ . In the square drawing the first two arcs both have radius $1$ , a growth of $1$ . Later the arc of radius $3$ is followed by an arc of radius $5$ , a growth of $5/3 \approx 1.667$ , and then come radii $8$ and $13$ , with growth $8/5 = 1.6$ and $13/8 = 1.625$ . Inside a single arc there is no growth at all, since every point of the arc in the square of side $5$ is exactly $5$ from that arc’s center, while the distance from the golden spiral to its center increases all along the curve. The growth factors of the drawing are the Fibonacci ratios, so they approach $\varphi$ , but each arc is still a piece of a circle and the drawing never becomes the golden spiral.

Check yourself

Continue the sequence up to $F_{13}$ and find the ratio $F_{13}/F_{12}$ as a decimal. Is it above or below $\varphi$ ? Is it equal to $\varphi$ ?

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From $F_8 = 21$ and $F_9 = 34$ the addition rule gives $F_{10} = 55$ , $F_{11} = 89$ , $F_{12} = 144$ and $F_{13} = 233$ . The ratio is

\frac{233}{144} = 1.6180555\ldots,

while $\varphi = 1.6180339\ldots$ , so the ratio is above $\varphi$ by about $0.0000216$ . That fits the pattern, since $233/144$ is $r_{12}$ and the ratios with even $n$ lie above $\varphi$ . The gap is well inside the bound $\varphi^{-12} \approx 0.0031$ . The ratio is not equal to $\varphi$ , because $233/144$ is a fraction of whole numbers and $\varphi$ is irrational. No ratio of Fibonacci numbers is the golden ratio, only the limit of the ratios is.

Keep the addition rule but start from $2$ and $1$ . This gives the Lucas numbers $2, 1, 3, 4, 7, 11, 18, 29, 47, \ldots$ . Do the ratios of neighbors still approach $\varphi$ ? The proof above used that every ratio is at least $1$ . Where does that fail here, and does it matter?

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The ratios are $1/2$ , $3$ , $4/3$ , $7/4$ , $11/7 \approx 1.5714$ , $18/11 \approx 1.6364$ , $29/18 \approx 1.6111$ and $47/29 \approx 1.6207$ , closing in on $\varphi$ from alternate sides. The update rule $r \mapsto 1 + 1/r$ comes from the addition rule alone, so it holds for these numbers too, for example $1 + 1/3 = 4/3$ . Only the first ratio, $1/2$ , is below $1$ . For it the factor $1/(r\varphi)$ is $2/\varphi \approx 1.236$ , which is more than $1$ , and the gap grows from $\lvert 1/2 - \varphi \rvert \approx 1.118$ to $\lvert 3 - \varphi \rvert \approx 1.382$ . Every later ratio has the form $1 + 1/r$ with $r > 0$ , so it is greater than $1$ , and from then on each step shrinks the gap by a factor of at most $1/\varphi$ . So the ratios still approach $\varphi$ . The same argument works for any two positive starting numbers.

Change the rule to $a_n = a_{n-1} + 2a_{n-2}$ , starting from $a_0 = 0$ and $a_1 = 1$ . Which number do the ratios $a_{n+1}/a_n$ approach now, and does any ratio reach it?

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The terms are $0, 1, 1, 3, 5, 11, 21, 43, 85, 171, \ldots$ and the ratios are $1$ , $3$ , $5/3 \approx 1.6667$ , $11/5 = 2.2$ , $21/11 \approx 1.9091$ , $43/21 \approx 2.0476$ , $85/43 \approx 1.9767$ and $171/85 \approx 2.0118$ . Dividing $a_{n+1} = a_n + 2a_{n-1}$ by $a_n$ gives the update $r \mapsto 1 + 2/r$ . A positive value that the update leaves unchanged satisfies $r = 1 + 2/r$ , so

\begin{aligned} r^2 - r - 2 &= 0, \\[0.4em] (r - 2)(r + 1) &= 0, \end{aligned}

and the positive solution is $2$ . The gaps obey

\Bigl(1 + \frac{2}{r}\Bigr) - 2 = \frac{2 - r}{r},

so each step flips the sign of the gap and divides its size by $r$ . The first two ratios are $1$ and $3$ , and every ratio from the third on is $1 + 2/r$ for some $r$ between $1$ and $3$ , so it is at least $5/3$ . From then on each gap is at most $3/5$ of the one before, and the ratios approach $2$ . No ratio equals $2$ . The rule gives $a_{n+1} - 2a_n = -(a_n - 2a_{n-1})$ , and $a_1 - 2a_0 = 1$ , so $a_{n+1} - 2a_n$ is always $1$ or $-1$ and the ratio $a_{n+1}/a_n$ is $2 + 1/a_n$ or $2 - 1/a_n$ . Changing the rule changes the limit, here from the irrational $\varphi$ to the whole number $2$ , but the ratios still never reach it.