Fibonacci Sequence and the Golden Ratio in Nature
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It may be impossible for some students to believe, but math can be more than just sets of numbers and equations. Turning abstract concepts into everyday observations is one of the best math strategies for struggling students. Two great examples of this involve the Fibonacci Sequence and the Golden Ratio.
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An Overview of the Fibonacci Sequence
In the 13th century, an Italian mathematician, Leonardo Pisano, identified one of the most interesting mathematical patterns. This pattern appears in nature and connects to other interesting mathematical properties and ratios. He named it after his nickname, Fibonacci, and then introduced the Fibonacci Sequence to the world.
What Is the Fibonacci Sequence?
The Fibonacci Sequence is a series of numbers that progress by adding each number to the number before it. While it may seem like a random string of numbers, the more you dive in to understand it, the more you’ll realize this is a universal pattern. You’ll see it while visiting art museums, watching Netflix, and strolling through a local park.
What Is the Fibonacci Spiral?
The Fibonacci Spiral relates directly to the Fibonacci Sequence. Using the Fibonacci Sequence, you can convert each number in the chain into a square. Starting from 1, each square will get larger in proportion to the sequence. When you draw and connect a quarter circle arc through each square, it creates a never-ending spiral.
Mathematical Explanation
The Fibonacci Sequence begins with 0 and 1. If you add 0 and 1 together, you get 1. To continue the sequence, add 1 and 1 together to get 2. Then, you add 1 and 2 together to get 3. So the Fibonacci Sequence looks like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. You can continue to add the two sequential numbers forever.
To convert this to the Fibonacci Spiral, make two squares, one box by one box (1x1). Next to those, make a square that is 2x2, then 3x3, then 5x5. Continue making squares with the numbers from the Fibonacci Sequence. Drawing an arc from corner to corner of each square will create a pattern that shows up in many real-world instances.
The Fibonacci Sequence in Nature
Nature uses the Fibonacci Sequence in various ways. This sequence helps organisms like plants and animals maximize their growth. Plants may arrange leaves with golden angles derived from the Fibonacci Sequence so they can all receive sunlight without blocking each other. The petals of flowers also follow this pattern to minimize space and maximize seeds.
An Overview of the Golden Ratio
In 300 B.C., you’ll find mention of the Golden Ratio in the book Elements by the ancient Greek mathematician Euclid. Because this ratio has piqued the interest of millions of people over thousands of years, it’s a good idea to learn what it’s all about.
What Is the Golden Ratio?
The Golden Ratio is 1.618033988749895. Many use the Greek letter Phi for this number in mathematical equations and refer to it in conversation as the divine proportion, golden proportion, or golden number. Like the Fibonacci Sequence, you’ll see the Golden Ratio everywhere once you know to look for it.
What Is the Golden Spiral?
You can draw the Golden Ratio to create the Golden Spiral. This spiral continues infinitely, gets continuously wider, and keeps the same shape. It is aesthetically pleasing and has caught the attention of artists, architects, and many others over the years.
Mathematical Explanation
Imagine the perfect rectangle. It can’t look too much like a square or too long like a line. The Golden Rectangle is ideally shaped because the longer part (A) divided by the smaller part (B) equals A plus B divided by A. This equation of A+B/A is the Golden Ratio, meaning the Golden Rectangle has a length of 1.618 and a width of 1.
If you cut a perfect square out of the Golden Rectangle, you’re left with another Golden Rectangle. Cut a perfect square out of that rectangle, and you’ll have another Golden Rectangle. This process goes on forever. Drawing a quarter-circle arc through each square will connect to make the Golden Spiral.
The Golden Ratio in Nature
The Golden Ratio’s contribution to nature helps organisms grow larger by optimizing space and proportions. When you see balanced natural formations, this may be the Golden Ratio at work. Butterfly wings, sea shells, and DNA helices use the Golden Ratio to maximize structure and function.
How Is the Golden Ratio Related to the Fibonacci Sequence?
The Golden Ratio and Fibonacci Sequence might seem very similar, but they’re not identical. If you take two sequential numbers from the Fibonacci Sequence and divide the larger number by the smaller one, you will get closer and closer to the Golden Ratio as the numbers get larger.
Because both the Golden Ratio and the Fibonacci Sequence are unending, there will never be a pair of numbers in the Fibonacci Sequence that you can divide to get the Golden Ratio. However, this means the Golden Ratio will always define the mathematical expansion of the Fibonacci Sequence.
Examples of the Fibonacci Sequence and the Golden Ratio in Nature
The natural world has no shortage of examples of the Fibonacci Sequence and the Golden Ratio. Next time you leave the house, you’ll see divine proportions, continued fractions, and the Golden Mean in the natural world.
Examples of the Fibonacci Sequence in Nature
One place you’ll find Fibonacci numbers in nature is at the park. Next time you bend down to smell the flowers, admire their petals, too. Many flowers use the Fibonacci Sequence to optimize their growth potential. Buttercups have five petals, corn marigolds have 13, and daisies will have 34 or 55, all in line with the numerical pattern. Similarly, pinecones grow spirals that reflect the Fibonacci pattern.
You might also spot bees flying from flower to flower. The family tree of a honeybee uses the pattern too. Male bees develop from an unfertilized egg laid by the queen bee, meaning they only have one parent. The males then fertilize eggs from the queen to create female bees. Because of this, the female bee has two parents but three grandparents. As this family tree continues branching out, the number of great-grandparents is five and great-great-grandparents is eight, perfectly simulating the Fibonacci sequence in nature.
Examples of the Golden Ratio in Nature
The Golden Ratio and Golden Spiral are also easy to find. Head to the beach and pick up a nautilus shell. You can trace the spiral and see how the chambers expand in perfect balance to expand spatially. If you’re unlucky, a hurricane might ruin your day at the beach. However, as you’re running for cover, you can think about how the spiral pattern of a hurricane also emulates the Golden Spiral.
Human proportions also show off the Golden Ratio. The ratio of a forearm to a hand and the proportions of each finger match the math. You’ll also see it when you look in the mirror. The positioning of facial features like your nose, mouth, and chin is another example of where you’ll find the Greek letter Phi in nature.
Practical Applications of the Fibonacci Sequence and the Golden Ratio
Because the Fibonacci Sequence and the Golden Ratio are so prevalent in nature, it shouldn't be surprising that you’ll also see it in manmade items. Looking at the works of great painters or photographers, you may find that they use the Golden Spiral to balance and compose their pictures.
Remember this next time you look at Leonardo da Vinci’s painting The Last Supper. You’ll also hear the harmonic patterns from the Fibonacci Sequence used by Bach and Mozart in their music. If you travel to Paris and visit the Notre Dame Cathedral, you’ll see a structure based on the proportions of the Golden Ratio.
If you’re more into finance than art, you might be interested to know that stock traders often use Fibonacci ratios to analyze stock trends and determine support and resistance levels. Or, if you're a gardener, you might want to use a Golden Spiral design for your garden layout to optimize the water distribution.
Make Math Fun With Nature’s Patterns
Exploring how math translates into natural sequences and structures can transform it from a dry classroom subject to a world of endless possibilities that can help your child with math. At Alexander Tutoring, we help students see how these abstract mathematical concepts translate into beautiful snail shells, famous paintings, and music, making math a fun endeavor with fascinating outcomes.
Our tutors provide personalized classes and study plans for students. We emphasize conceptual understanding of mathematical principles over boring repetition and memorization. We help each student gain the confidence and passion to improve their skill sets and use their newfound knowledge in all areas of their lives.
Learn more about how we make math fun and engaging for our students.
