Fibonacci Sequence Mathematics in the Modern World

Editor's Notes

• #2: Now that we’ve talked about the Fibonacci Sequence, let’s learn about the man behind it: Leonardo Fibonacci – one of the greatest European mathematicians of the Middle Ages. 🔹 He was born in 1170 and died in 1240. Even though he lived hundreds of years ago, his work still influences math today. 🔹 Fibonacci is best known for two major contributions: 1️⃣ Introducing the Arabic number system to Europe – Before that, people in Europe were using Roman numerals, which were hard to calculate with. – Thanks to Fibonacci, Europe started using the numbers 0–9, just like we do today. 2️⃣ Discovering the Fibonacci Sequence – In his book Liber Abaci, he introduced the sequence through a problem about rabbit population growth. – That simple pattern of adding the two previous numbers became one of the most famous sequences in mathematics.
• #3: Did you know that the Fibonacci Sequence was actually discovered through a problem about rabbits? 🔹 In his book Liber Abaci, Leonardo Fibonacci asked this question: 'How many pairs of rabbits will be produced in a year if you start with one pair, and each pair produces another pair every month, starting from their second month of life?' When he worked it out, he noticed a pattern: Month 1: 1 pair Month 2: 1 pair Month 3: 2 pairs Month 4: 3 pairs Month 5: 5 pairs Month 6: 8 pairs ...and so on. 🔹 Each number in the sequence is the sum of the two previous numbers—just like the Fibonacci Sequence! So, what started as a simple question about rabbit reproduction led to one of the most important number patterns in history.
• #4: "Let’s take a closer look at the famous rabbit problem that led to the discovery of the Fibonacci Sequence. 🔹 The problem goes like this: Suppose a newly-born pair of rabbits (one male and one female) are placed in a field. Rabbits can mate when they are one month old, so at the end of the second month, each pair produces another pair of rabbits (one male and one female). We assume that: Rabbits never die. Every mature pair produces one new pair every month. Each new pair starts reproducing after one month. 🐰 Let’s count the rabbit pairs month by month: MonthNumber of PairsExplanation11Start with 1 pair (newborn)21Still just one pair, now mature32First pair produces a new pair43First pair produces another, second pair still too young55First and second pairs reproduce683 mature pairs produce 3 new ones And the pattern continues... Each number is the sum of the two previous numbers! 🔹 That’s the Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21... 🟢 Wrap-up: "So, this simple rabbit problem shows how population growth can follow a mathematical pattern. It’s a fun and classic example of how real-life situations can lead to amazing discoveries in mathematics and nature."
• #5: Let’s now define the Fibonacci Sequence more clearly. 🔹 The Fibonacci Sequence is a list of numbers that goes like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, ... And it continues forever—it’s an infinite sequence. 🔹 The first two numbers are both 1. After that, each new number is the sum of the two numbers before it. 📌 Simple Rule: 👉 Add the last two numbers to get the next one. Let’s try: 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8 5 + 8 = 13 And so on… 🟢 Wrap-up: "This is what makes the Fibonacci Sequence so special—a very simple rule creates a pattern that shows up in nature, art, architecture, and science. It’s one of the most famous and fascinating patterns in mathematics!"
• #8: "One of the most beautiful examples of the Fibonacci Sequence in nature can be found in the sunflower. 🔹 If you look closely at the center of a sunflower, you’ll notice that the seeds are arranged in spirals—both clockwise and counterclockwise. 🔹 What’s amazing is that the number of spirals in each direction is usually a Fibonacci number—like 34 and 55, or 55 and 89. These numbers aren’t random—they follow the same add-the-last-two rule of the Fibonacci Sequence! 🌿 Why does the sunflower grow this way? 🌻 The spiral arrangement helps the plant: Pack the most seeds into a small space without overlapping Ensure equal spacing so each seed has room to grow Maximize sunlight and nutrient exposure 🔹 This is an example of nature using math for efficiency and beauty. 🟢 Wrap-up: "So, when you see a sunflower, you're not just looking at a flower—you're seeing Fibonacci in action. It’s a perfect example of how mathematics creates patterns that help living things grow in the best way possible."
• #9: "Another great example of the Fibonacci Sequence in nature is found in something very common—pine cones! 🔹 If you look at the scales or bumps on a pine cone, you’ll notice they form spirals that curve upward in two directions—one clockwise and one counterclockwise. 🔹 When you count the number of spirals going each way, they are usually Fibonacci numbers—like 5 and 8, or 8 and 13. 🌿 Why do pine cones grow this way? This spiral pattern helps the cone: Pack its scales efficiently Distribute seeds evenly Allow space for growth as it matures It’s all about natural efficiency and balance—and it follows the same simple rule we learned: 📌 Add the last two numbers to get the next one. 🟢 Wrap-up: "So even in a pine cone, we can see mathematics at work. The Fibonacci sequence helps nature create patterns that are both functional and beautiful. This shows us once again that math isn’t just in books—it’s built into the world around us."
• #10: "Let’s take a look at another fruit where we can see the Fibonacci Sequence in action: the pineapple. 🔹 If you look closely at the surface of a pineapple, you’ll notice that the small diamond-shaped ‘eyes’ or scales are arranged in spiral patterns. 🔹 These spirals run in three different directions: Clockwise Counterclockwise Vertically or diagonally 🔹 When you count the number of spirals in each direction, they are often Fibonacci numbers—for example, 8, 13, and 21. 🌿 Why does this happen? Just like in sunflowers and pine cones, this spiral pattern allows the pineapple to: Grow evenly Distribute its scales efficiently Maximize space and stability as it grows It’s nature’s way of using simple math to solve complex problems like packing, growth, and structure. 🟢 Wrap-up: "So next time you see a pineapple, try counting the spirals! You’ll be amazed to find that something so ordinary is actually following the Fibonacci Sequence—another example of how mathematics shapes the natural world."
• #11: "Now let’s talk about another fascinating number found in nature and design—the Golden Ratio. 🔹 The Golden Ratio is a special number that’s often written as the Greek letter Φ (uppercase Phi) or φ (lowercase Phi). 🔹 Its approximate value is 1.618. That means, if you divide a line into two parts—a longer part (A) and a shorter part (B)—the Golden Ratio happens when: A / B = (A + B) / A ≈ 1.618 In other words, the ratio of the whole to the larger part is the same as the ratio of the larger part to the smaller part.
• #12: "The Golden Ratio can be thought of as a special relationship between two numbers. 🔹 It’s when the ratio of the larger number to the smaller number is the same as the ratio of their sum to the larger number. Let’s say we have two numbers: A (the larger one) and B (the smaller one). The Golden Ratio happens when: 📌 A / B = (A + B) / A And this ratio is always approximately equal to: 👉 1.618 That’s what makes it so special! 🔢 Example in Numbers: Let’s try A = 8 and B = 5: A / B = 8 / 5 = 1.6 (A + B) / A = 13 / 8 = 1.625 Both are very close to 1.618 — so this is very close to the golden ratio! 🟢 Wrap-up: "So, the Golden Ratio is not just about shapes or art—it’s a mathematical relationship between two numbers that creates a feeling of balance and harmony. That’s why we find it in nature, design, and even the human body." It took 16 centuries before Leonardo of Pisa came up with a sequence of numbers that was pretty special. Every number was the sum of the previous two, and the sequence was infinite. This sequence of numbers was named after his nickname: the Fibonacci sequence. Astronomer Johannes Kepler finally discovered the link between the golden ratio and the Fibonacci sequence. He found out how the division between a Fibonacci number by the previous number in the sequence would approach the golden ratio with increasing accuracy. The well know square based on the golden ratio. The Fibonacci sequence numbers are added. It all fits like a glove.
• #13: The Fibonacci Spiral projected over the paintings of the Mona Lisa by Leonardo da Vinci (public domain, Creative Commons) and the Girl With a Pearl Earring by Johannes Vermeer (public domain, Creative Commons).