Everything Everywhere Daily: History, Science, Geography & More - The Fibonacci Sequence and the Golden Ratio

Starting point is 00:00:00 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're actually tightly intertwined. While they've been known since the ancient world, they're still highly relevant today and can be found almost everywhere. And best of all, despite being important mathematical concepts, they're 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.

Starting point is 00:00:46 What if your perceptions about the past were wrong? ThruLine is a podcast that takes you back in time to uncover the parts of the story that may have gone unnoticed. It effectively turned day into night. And how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. Before I get into the history and the applications of the Fibonacci sequence, of the golden ratio. I should probably explain what they are because they're actually pretty

Starting point is 00:01:20 easy to understand. The Fibonacci sequence is formed by starting with the numbers 0 and 1, and then adding each pair of previous numbers to get the next one. So 0 plus 1 is 1. 1 plus 1 is 2. 2 plus 1 is 3, 3 plus 2 is 5, 5 plus 3 is 8. And you can just keep doing this forever, adding the last two digits. The next would be 13, 21, 34, 55, 89, 144, 233, 377, etc. And that's 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 the number 1.618033-9887 extending out to a infinity in a non-repeating series of numbers. Simple addition and an irrational number hardly seem like they have something in common,

Starting point is 00:02:23 but as we'll see, they actually do. The mathematical relationship that we call the golden ratio was actually known to ancient civilization 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 notice 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

Starting point is 00:03:02 appears in Indian mathematics as early as the sixth 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 Gopala 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 10 centuries when Baghdad's House of Wisdom became a center for scholarly exchange. These Indian documents, such as those describing the work of Virahanka, were translated into Arabic,

Starting point is 00:03:45 where scholars like Al-Khalil Ibn Ahmed and later Abu Kamil applied similar additive principles to problems in algebra, geometry, and combinatorics. Although they didn't use the sequence in the same stylized form that we're used to, and they didn't name it, these mathematicians preserved and expanded upon the underlying reoccurrence relationship, integrating it into broader studies of arithmetic. progressions, number patterns, and practical calculations. The man for whom 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, Phileas Banachi, meaning son of Benachi.

Starting point is 00:04:27 Fibinacci introduced the sequence to Western mathematics in his 1202 book titled Liber Abaki, or the Book of Calculation. The work's primary goal was to popularize the, the Hindu Arabic numeral system in Europe, which I've previously done an episode on, but it also contained a wide variety of mathematical problems. One of these was a now famous puzzle about rabbit populations. Here is how Fibonacci framed his famous rabbit problem. Suppose you start with one pair of newborn rabbits.

Starting point is 00:04:59 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. So how many pairs of rabbits will you have after several months? In month one, you have one pair of newborn rabbits. In month two, you still have one pair as they're not mature yet. In month three, your original pair produces offspring, so you have two pairs. In month four, the original pair produces another set of offspring, and the pair born in month three is now mature, so you have three pairs. Can you start to see the pattern emerging? The sequence, goes, 11-12358, 1321, 345, 8, 8, 944. Each number is just the sum of the two preceding

Starting point is 00:05:47 numbers. And 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, Edward Lucas, who studied its properties in depth. Just before, I said that the Fibonacci sequence was related, and in fact strongly related to the golden ratio. How was 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. So try dividing 13 by 8, and you get 1.625.

Starting point is 00:06:30 Now try dividing 21 by 13, which is approximately 1.615. And keep going. 55 divided by 34 equals 1.618. As 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 a relative. relatively modern one. The ancient Greeks called it various names, but the specific term sectio or a golden section was first used by mathematician Martin Ollm in 1835. The Greek letter

Starting point is 00:07:13 fee used to represent the golden ratio was chosen by American mathematician Mark Barr in the early 1900s, likely in honor of the Greek sculptor Phidius, who used this proportion in his work. It turns out there's more than just the golden ratio and the Fibonacci sequence. There's also something called the golden angle. The golden angle is the smaller of two angles that divides a circle according to the golden ratio. If you take a full circle 360 degrees and divide it such 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 degrees. The great mathematician Blaise Pascal created Pascal's triangle.

Starting point is 00:08:00 which is based on the Fibonacci sequence. Pascal's triangle is a triangle arrangement of numbers where each number is the sum of two numbers directly above it in the previous row. It begins with a single one at the top and then continues with rows like 1-1, then 1-21, then 1-331, and so on. Each row corresponds to the coefficients

Starting point is 00:08:23 in the binomal expansion of A plus B to the power of N, 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 the process of growth and deficient packing. In many plants and biological structures,

Starting point is 00:08:56 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 degrees, over time the pattern of their arrangement produces counts that match the Fibonacci sequence. Many flowers have a number of petals that is a Fibonacci number. For example, lilies have three petals, buttercups have five, chicory has 21, and daisies can have 34, 55, or even 89 petals. This arrangement often optimizes exposure to sunlight for each pedal.

Starting point is 00:09:41 The spiral patterns in sunflower seed heads and pinecone scales follow Fibonacci numbers. If you count the spirals curving in one direction and then 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 also appear in pineapples. The pattern of branches and leaves on many trees 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 the tree can capture.

Starting point is 00:10:28 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's 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. So it really shouldn't come as a surprise that the ancient Greeks found the divine proportion to be so aesthetically appealing.

Starting point is 00:11:08 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

Starting point is 00:11:44 often reflected proportions close to the golden ratio, even if the artist didn't consciously know they were doing 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. Sandra Bonicelli's The Birth of Venus contains figure placement and spacing that approximates golden rectangles. In architecture, the facade of the Notre Dame Cathedral in Paris and the proportions of the great mosque of Kirawan show relationships close to the ratio as well. Modern architects such as Lacrobusier incorporated it into building.

Starting point is 00:12:20 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 Dali 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 Bella Bartok, are structured so that climatic moments fall at golden ratio points in time. The 2001 song Lateralis by the band Tool was based on the Fibonacci sequence, and it was named

Starting point is 00:12:58 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's 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. The executive producer of Everything Everywhere Daily is Charles Daniel. The associate producers are Austin Otkin and Cameron Kiefer.

Starting point is 00:13:41 My big thanks go to everyone who supports the show over on Patreon. Your support helps make this podcast possible. And I also want to remind everyone about the community groups on Facebook and Discord. That's where everything happens that's outside the podcast. And links to those are available in the show notes. As always, if you leave a review on any major podcast app or in the above community groups, you too can have it read in the show.

Everything Everywhere Daily: History, Science, Geography & More - The Fibonacci Sequence and the Golden Ratio

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