← Everything Everywhere Daily: History, Science, Geography & More
Everything Everywhere Daily: History, Science, Geography & More
The Fibonacci Sequence and the Golden Ratio
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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. The what they are and how they were discovered on this episode of Everything Everywhere Daily. This episode is sponsored by Fiji Water. You've probably heard of Fiji Water and have seen it in stores. Well, Fiji Water really is from the islands of Fiji. Drop by drop, Fiji Water is filtered through volcanic rock 1600 miles away from the nearest continent in all its pollution, protected and preserved naturally from external elements. 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The Quint's website literally showed me how much an equivalent sweatshirt of the same color and same material would have cost from other fashion designers and my savings were between 78 to 94%. I've been telling you for months now about how Quint's brings you quality items at a fraction of the price and it's not just a marketing slogan. You can go to their website and see the savings for yourself by working directly with top artisans and cutting out the middleman. Quint gives you luxury pieces without the markup and they pass the savings on to you. Keep it classic and cool with long lasting staples from quince. Go to quince.com daily for free shipping on your order and 365 day returns. That's Q U I-N-E.com daily to get free shipping and 365 day returns quince.com 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're actually pretty 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 1.6180339887, 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 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 noticed that when you divide a line segment into two parts, the 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 prosodhi, 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 10th 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, where scholars like El 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 recurrence 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 Filius Bonacci, meaning son of Bonacci. Fibonacci introduced the sequence to Western mathematics in his 1202 book titled Liber Abaci, or the Book of Calculation. The work's primary goal was to popularize the Hindu Arabic numeral system in Europe, with 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. 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 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 8, 9, 1, 4, 4. Each number is just the sum of the two preceding 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 Edouard 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. 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 relatively modern one. The ancient Greeks called it various names, but the specific term sectio aurea, or golden section, was first used by mathematician Martin Ulm 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 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, 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, 2, 1, then 1, 1, 3, 3, 1, and so on. Each row corresponds to the coefficients in the binomial 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 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 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 petal, 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 following Fibonacci numbers at multiple scales. Similar spiral arrangements also appear in pineapples. The pattern in 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. 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 nonliving 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. 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 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. Sandro Botticelli'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 Kirowan show relationships close to the ratio as well. 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 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 such as the compositions of Bela 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 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. 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 on the show.