The Fibonacci Sequence and the Golden Ratio

Podcast Transcript

Two of the most important concepts that can be found in the world of mathematics and nature are the Fibonacci Sequence and the Golden Ratio.

These two concepts seem separate, but they are actually tightly intertwined.

While they have been known since the ancient world, they are still highly relevant today and can be found almost everywhere.

Best of all, despite being important mathematical concepts, they are also among the easiest to understand.

Learn more about the Fibonacci Sequence and the Golden Ratio, what they are, and how they were discovered on this episode of Everything Everywhere Daily.

Before I get into the history and the applications of the Fibonacci Sequence and the Golden Ratio, I should probably explain what they are, because they are easy to understand.

The Fibonacci sequence is formed by starting with 0 and 1, then adding each pair of previous numbers to get the next one.

So 0+1 is 1

1+1 is 2

2+1 is 3

3+2 is 5

5+ 3 is 8

You can keep doing this forever, just adding the last two numbers together.

13, 21, 34, 55, 89, 144, 233, 377, etc.

That is all there is to it. Any child who knows basic addition can calculate the Fibonacci sequence.

The Golden Ratio is an irrational number that is close to 1.6180339887….extending out to infinity in a non-repeating series of numbers.

Simple addition and an irrational number hardly seem like they have something in common, but as we’ll see, they do.

The mathematical relationship we now call the Golden Ratio was actually known to ancient civilizations long before Fibonacci was born. The ancient Greeks, particularly around the 5th century BC, were deeply fascinated by what they called the “divine proportion.”

They noticed that when you divide a line segment into two parts such that the ratio of the whole line to the longer part equals the ratio of the longer part to the shorter part, you get a special number – approximately 1.618.

The pattern of numbers we now call the Fibonacci sequence appears in Indian mathematics as early as the 6th century. Indian scholars were studying prosody, the arrangement of syllables in Sanskrit poetry, and discovered that the number of possible rhythmic patterns for a given length followed this sequence. The mathematician Virahanka described the pattern, and later scholars such as Gop?la and Hemachandra expanded on it.

Early Islamic mathematicians then encountered the pattern through translations of Indian mathematical works during the Abbasid Caliphate, particularly in the 8th to 10th centuries, when Baghdad’s House of Wisdom became a center for scholarly exchange.

Indian documents, such as those describing the work of Virahanka, were translated into Arabic, where scholars like al-Khalil ibn Ahmad and later Abu Kamil applied similar additive principles to problems in algebra, geometry, and combinatorics.

Although they did not use the sequence in the same stylized form and didn’t name it, these mathematicians preserved and expanded upon the underlying recurrence relationship, integrating it into broader studies of arithmetic progressions, number patterns, and practical calculations.

The man who the sequence is named after is Leonardo of Pisa, who is more commonly known as Fibonacci.

Fibonacci is a shortened form of the Italian phrase filius Bonacci, meaning “son of Bonacci.”

Fibonacci introduced the sequence to Western mathematics in his 1202 book Liber Abaci or “The Book of Calculation”. The work’s primary goal was to popularize the Hindu–Arabic numeral system in Europe, which I’ve covered in a previous episode, but it also contained a wide variety of mathematical problems.

One of these was a now-famous puzzle about rabbit populations.

Here’s how Fibonacci framed his famous rabbit problem: Suppose you start with one pair of newborn rabbits. Each month, every mature pair produces a new pair of rabbits. Rabbits mature after one month, so they can reproduce starting in their second month of life. How many pairs of rabbits will you have after several months?

In month 1, you have 1 pair (newborns). In month 2, you still have 1 pair (they’re not mature yet). In month 3, your original pair produces offspring, so you have 2 pairs. In month 4, the original pair produces another set of offspring, and the pair born in month 3 is now mature, so you have 3 pairs. Can you see the pattern emerging?

The sequence goes: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… Each number is the sum of the two preceding numbers. This became known as the Fibonacci sequence.

For centuries, the sequence was little more than a curiosity in number theory. It was first called “the Fibonacci sequence” in the 19th century by the French mathematician Édouard Lucas, who studied its properties in depth.

Just before I said that the Fibonacci sequence was related, strongly related, to the Golden Ratio.

How is that so?

It was a relationship that Fibonacci himself didn’t even realize.

If you take any Fibonacci number and divide it by the previous Fibonacci number, you get a ratio. Let’s try this: 13 divided by 8 equals 1.625. Now try 21 divided by 13, which equals approximately 1.615. Keep going: 55 divided by 34 equals about 1.618.

See what’s happening?

As the Fibonacci numbers get larger, these ratios get closer and closer to the Golden Ratio!

Or to put it another way, as the Fibonacci sequence grows to infinity, the ratio converges on the Golden Ratio.

The term “Golden Ratio” itself is relatively modern. The ancient Greeks called it various names, but the specific term “sectio aurea” or golden section was first used by mathematician Martin Ohm in 1835.

The Greek letter ? (phi), used to represent the Golden Ratio, was chosen by American mathematician Mark Barr in the early 1900s, likely in honor of the Greek sculptor Phidias, who used this proportion in his works.

It turned out there is more than just the Golden Ratio and the Fibonacci Sequence. There is also something called the Golden Angle.

The golden angle is the smaller of the two angles that divide a circle according to the Golden Ratio. If you take a full circle (360°) and divide it so that the ratio of the larger arc to the smaller arc is the same as the ratio of the whole circle to the larger arc, the smaller arc measures about 137.5°.

The great mathematician Blaise Pascal created Pascal’s Triangle, which is based on the Fibonacci Sequence.

Pascal’s Triangle is a triangular arrangement of numbers where each number is the sum of the two numbers directly above it in the previous row. It begins with a single 1 at the top, then continues with rows like 1 1, then 1 2 1, then 1 3 3 1, and so on. Each row corresponds to the coefficients in the binomial expansion of (a + b)?, making it a fundamental tool in combinatorics and probability theory.

What was a mathematical curiosity became something much more when people began to see these numbers in nature. In fact, they appeared everywhere in nature.

The Fibonacci sequence appears so often in nature because it naturally emerges from processes of growth and efficient packing. In many plants and biological structures, growth happens by adding new elements, such as leaves, seeds, or petals, in a way that maximizes access to resources like sunlight or space.

If each new element is placed at a constant angle from the previous one, often close to the golden angle of about 137.5°, over time the pattern of their arrangement produces counts that match Fibonacci numbers.

Many flowers have a number of petals that is a Fibonacci number. For example, lilies have 3 petals, buttercups have 5, chicory has 21, and daisies can have 34, 55, or even 89 petals. This arrangement often optimizes exposure to sunlight for each petal.

The spiral patterns in sunflower seed heads and pine cone scales follow Fibonacci numbers. If you count the spirals curving in one direction and then in the other, you’ll often get two consecutive Fibonacci numbers. This packing maximizes the number of seeds or scales in a given area without wasting space.

Romanesque broccoli displays a striking example, with spirals in its florets following Fibonacci numbers at multiple scales. Similar spiral arrangements appear in pineapples.

The pattern of branches and leaves on many plants follows Fibonacci rules, as new growth often appears at angles that approximate the golden angle. This arrangement minimizes overlap between leaves, maximizing the amount of sunlight capture.

Some shells, such as the nautilus, grow in a logarithmic spiral that relates to the Golden Ratio. Even the spiral of a chameleon’s tail or the horns of certain sheep follow similar growth proportions.

Even in non-living things, there is evidence of this relationship.

Large-scale spirals in nature, such as hurricane cloud bands and spiral galaxies like the Milky Way, often follow logarithmic spirals related to the Golden Ratio. This form allows for a self-similar structure across different scales.

It shouldn’t come as a surprise that the ancient Greeks found the “divine proportion” so aesthetically appealing.

?Psychologists and vision researchers suggest that this appeal may come from how the ratio appears in natural forms, making it familiar to our visual perception. It has a balance between the monotony of perfect symmetry and the chaos of irregular proportions.

In art, the Golden Ratio has been used, sometimes deliberately, sometimes coincidentally, to create compositions with a sense of natural harmony.

The Parthenon in Athens is often cited for its facade proportions.

In medieval and Renaissance manuscript illumination, page layouts and decorative borders often reflected proportions close to the Golden Ratio, even if the artists didn’t consciously calculate it.

Renaissance artists like Leonardo da Vinci explored the ratio in works such as Vitruvian Man and possibly in The Last Supper to position key elements.

Sandro Botticelli’s The Birth of Venus contains figure placement and spacing that approximate golden rectangles.

In architecture, the façade of the Notre-Dame Cathedral in Paris and the proportions of the Great Mosque of Kairouan show relationships close to the ratio.

Modern architects such as Le Corbusier incorporated it into building designs for pleasing spatial relationships, and photographers often frame subjects using divisions based on the Golden Ratio to guide the viewer’s eye.

In the 20th century, Salvador Dalí designed his painting The Sacrament of the Last Supper within a golden rectangle, aligning the central figure and the composition’s geometry to the ratio.

Even musical works, such as the compositions of Béla Bartók, are structured so that climactic moments fall at golden ratio points in time.

The 2001 song Lateralus by the band Tool was based on the Fibonacci Sequence, and it was named the top Heavy Metal song of the 21st century.

Today, the Fibonacci sequence is studied in number theory, combinatorics, computer algorithms, and mathematical modeling, yet it also serves as a cultural symbol of mathematical beauty and natural form.

It is also perhaps the best example of how mathematics isn’t just something that exists in the abstract or in theory. The Fibonacci sequence, the Golden Ratio, and the Golden Angle are all mathematical concepts that we can see embedded in the very world around us.