Cool Fibonacci: A Mathematical Sequence That Occurs in Nature
What if I told you there was a mathematical sequence that occurs in nature–a pattern that connects plants, living creatures, the stars, and even our own bodies?
You’d think this would be hard, Einstein-y type math, but it isn’t. It’s a mathematical pattern so simple, even a second grader can understand it!
It’s called the Fibonacci sequence of numbers, and it’s a mathematical mystery that you can easily explore and share with your child.
This hidden math sequence shows up in the most unexpected places in nature. But maybe even more incredible is how it links and connects all the different parts of our world.
In this article, you’ll learn about the most famous mathematical sequence that occurs in nature–the Fibonacci sequence–and why many people call it “nature’s secret code”.
You’ll also find simple, fun activities scattered throughout this post. Do them with your kids to help them discover the beauty of math in nature.
- Flower patterns in nature
- Who was Fibonacci?
- Fibonacci’s life
- A mathematical sequence that occurs in nature
- Nature’s math spiral
- How to draw a Fibonacci spiral
- The beauty of math in pine cones
- Sunflower Fibonacci
- Helping kids discover the beauty of math in nature
Flower Patterns In Nature
Imagine you are strolling through your garden, counting the number of petals on your flowers.
You count 3 petals, 8 petals, 5 petals. You start to notice the same numbers over and over. Not with all the flowers, but those numbers definitely pop up a lot.
“Is this a coincidence?” you wonder. “Could it be a pattern?”
On the daisy-type flowers, the numbers are bigger: 13 petals, or 21.
At first it all seems a bit random, but as you play around with the flower petal numbers you realize something:
The numbers add up to each other! In fact, together they make up an addition pattern that looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
Each two consecutive numbers add up to the next number in the series.
It’s the mathematical sequence and number pattern named after Leonardo Fibonacci (fi-bo-NACH-ee), an Italian mathematician who lived in the Middle Ages.
You may hear this pattern referred to as the “Fibonacci sequence”, the “Fibonacci series”, or simply “Fibonacci numbers”, but they are all the same thing.
Fun Fact: The man we now call Fibonacci was not known as Fibonacci during his lifetime; he acquired that nickname much later.
Leonardo was a fascinating guy. Let’s learn more about him.
who was fibonacci?
During his lifetime (1170-1250), he was known not as Fibonacci but as the Italian mathematician Leonardo Pisano (Pisano means “from Pisa”).
So why do we call him Fibonacci today?
To answer that question, we need to fast forward about 600 years.
In the 19th century, a group of historians, doing their historical record keeping, discovered not one but TWO famous men from the same town, both named Leonardo Pisano.
How would people tell them apart?
Names back then often said what city or what family a person was from. So the historians looked for another name.
Our Leonardo was the son of Guglielmo Bonacci, and so was sometimes called “son of Bonacci” (which in Italian was “filius Bonacci”).
To keep people from getting confused, the historians went with his family name. Since Leonardo filius Bonacci was a lot to say, they shortened it to the nickname Fibonacci.
Fibonacci always had an eye out for the beauty of math. He loved learning about numbers.
In Italy at the time (and in all of the western world), everyone used Roman numerals for doing calculations.
But Leonardo’s father traveled a lot, and that meant Leonardo studied with people from many countries–not just Italy.
Leonardo and his father lived for awhile in Bugia in North Africa. He studied with an Arab master of calculation, and there he was first introduced to Hindu-Arabic numerals.
It was a big change. This is what the number 18 looks like in Roman numerals and Hindu-Arabic numerals:
Do those Hindu-Arabic numbers look familiar?
Leonardo was blown away by the new number system, and wanted everyone to know about it.
He wrote a book called Liber Abaci, where he talked about Hindu-Arabic numerals, and also the pattern we now call the Fibonacci sequence.
These mathematical ideas weren’t new; they’d been found as early as the 9th century in ancient Sanskrit texts.
Leonardo never claimed to have invented them himself. He simply shared what he’d learned, and in time convinced the people using Roman numerals to try a new way of doing math.
Fibonacci is the reason we do math with the numerals 0-9 today, and not letters like XVIII. Thank you, Fibonacci!
ACTIVITY: Check out the book Blockhead: The Life of Fibonacci from the library.
It’s a great book for elementary kids of all ages, and even little ones will enjoy searching for Fibonacci numbers and spirals hidden in the illustrations.
A MATHEMATICAL SEQUENCE THAT OCCURS IN NATURE
How do you make the Fibonacci pattern? You start with a 0 and a 1.
Add them together to get the next number. Then keep adding the final two numbers to find the next number in the pattern:
Each pair of consecutive Fibonacci numbers gets added together to make the next number in the sequence.
Or as another way to put it, he sum of the two preceding numbers becomes the following number in the pattern.
When you’re helping your child discover the Fibonacci sequence, don’t just come right out and tell it to them. Show them the first few numbers and give them some guided questions and hints, a little at a time:
- What do you notice?
- Do you see a pattern?
- Hint: There’s some adding involved.
- What’s 1+1? What’s 1+2? What’s 2+3? What do you notice?
Most kids in grades 2 and up will figure it out themselves as you give guided hints. Once they “get it”, ask them to tell you more about what they notice.
They’ll be excited to find Fibonacci numbers everywhere!
ACTIVITY: Take your child on a scavenger hunt outside. Look for Fibonacci numbers in flower petals, lobes on leaves, and so on.
nature’s Math spiral
Do a Google search of “the beauty of math in nature”, and one of the first things that will pop up is pictures of nautilus shells.
For a long time, I thought this was just because it was pretty–a lovely spiral shape, a geometrical masterpiece.
And it was all that, but what I didn’t realize until much later was that particular spiral shape was made of a very simple math pattern. The Fibonacci sequence, to be exact.
Fibonacci spiral patterns are a bit different from what we typically think of as a spiral. Spirals at are human creations tend to go round and round in equal distances from each other.
This is called an Archimedean spiral.
But nature’s curve is different. In what is sometimes called the perfect spiral, the Fibonacci math spiral starts closer together toward the center, then gets farther apart as it grows.
One of the most surprising things about the beauty of math is the connections you’ll find in nature.
The tail of a chameleon looks like the tail of a seahorse. A fern opening mirrors the curve of an elephant’s trunk. A snail’s shell, a hurricane, and even our spiral galaxy are connected by math!
How to Draw a Fibonacci Spiral
To make a Fibonacci spiral, you start with squares.
Start with two 1×1 squares right next to each other. Then draw a 2×2 square below them.
Next comes the 3×3 square. You’re creating a spiral shape, starting with the 1s, then the 2, then 3 and so on.
It helps to draw the spiral with your finger to know which side to draw the 3×3 square.
Following the spiral shape, add a 5×5 square, an 8×8 square, and so on until you run out of squares on your paper.
You’ll notice as you draw that the most recent square always lines is always the same width as the last to squares you drew.
All the squares together make a Fibonacci rectangle–a specific shape and proportion that many people call a perfect rectangle.
It doesn’t matter how many squares you add; the proportion of the rectangle always stays exactly the same.
When you’ve filled your paper, draw a curved line to connect the opposite corners of each square, to make a perfect Fibonacci nature spiral.
Drawing a Fibonacci spiral has its tricky bits, but overall it’s easier than it looks. With just a little help, kids as young as 7 years old can make one!
ACTIVITY: Show kids how to draw squares of Fibonacci numbers: 2×2, 3×3, 5×5 and so on. Write the number inside each square.
(This is a great excuse for a lesson on square numbers!)
Have them cut out their squares, and maybe color them. Challenge them to put the squares together into a spiral shape.
This is a fun first step to drawing a Fibonacci spiral, and it really helps younger kids so they can focus on making squares and physically moving them around.
Older kids can go straight to drawing the spiral shape on graph paper.
Kids typically need a little help with 1) drawing squares instead of rectangles, 2) knowing where to draw the next square, and 3) drawing the spiral line at the end.
Your child may ask you to draw the finished spiral–the hardest part for kids. I recommend drawing the spiral in pencil, and letting your child trace over it themselves.
It’s so satisfying seeing a spiral shape appear out of the math!
The Beauty of Math in Pine Cones
What, you’ve never heard of a Fibonacci sequence pinecone?
You’re not alone. Pinecones are one of those common things our eye slides right past–but if we dig a little, there are some wonderful mysteries hidden in this humble nature object!
At first glance, the arrangement of pinecone petals seems haphazard and random, with no Fibonacci numbers in sight.
But if you put that pine cone in water for a couple of hours, the petals (known as “bracts”) close up tight! Clear rows and patterns begin to appear.
Run your finger along any one of these rows, and you’ll notice it makes a spiral shape around the pinecone.
The easiest way to spot a Fibonacci pinecone spiral is by turning the pine cone upside-down to look at the bottom.
By putting your finger in the center point, you can follow a single spiral all the way to its end. There are spirals going the opposite direction as well.
If you count the number of spirals on a pine cone, it is ALWAYS a Fibonacci number.
And if you count the spirals in the opposite direction, it will always be an adjacent Fibonacci number–a “neighbor number” to the right or left in the Fibonacci pattern.
It’s not just pine cones that have a Fibonacci number of spirals, though.
As you look for the beauty of math in nature, you’ll start to see spirals everywhere–various plants, the center of flowers, and succulents.
Count them–the number of spirals will be a Fibonacci number!
Fun Fact: Most things with “pine” in the name, like pinecones and pineapple, have Fibonacci spirals in them.
ACTIVITY: Find some pine cones out in nature, and put them in a big bowl of water. After a couple of hours, take them out of their water bath. What happened to the pine cones? Can you see the spiral patterns better now?
Sunflowers are an amazing Fibonacci flower.
First, there’s the petals. Sunflowers have a lot of petals, but if you were able to count them, you’d almost always get a Fibonacci number.
Sunflowers are one of the few places in nature where you’ll find the larger Fibonacci numbers. There’s something about counting exactly 21, 34, or even 55 flower petals that is SO satisfying!
(Keep in mind that bugs and weather can destroy flower petals. Still, the number will usually be quite close to a Fibonacci number, even if it’s not exact.)
Once you’ve counted petals, look closely at the center of a sunflower. Notice the arrangement of seeds: what do you see?
Yep, another spiral! And just like with pine cones, there are spirals in two directions–some pointed right, others pointed left.
It’s tough to count that many spirals! But if you did, you’d find that the number of sunflower spirals is ALWAYS a Fibonacci number.
Just like pine cones, the number of spirals going in one direction, and the number of spirals going the other direction, will always be right next to each other in the number sequence.
ACTIVITY: Watch this Vi Hart YouTube video about Fibonacci numbers. At the 4:44 minute mark, she counts the spirals in the center of a daisy.
Helping Kids Discover The Mysterious Mathematical Sequence That Occurs in Nature
When you’re looking for ways to share the beauty of math with your child, it’s important to keep their math level in mind.
Kids in KG and first grade are super enthusiastic about counting flower petals and looking for Fibonacci numbers in nature.
KG-1st graders will enjoy the book Swirl By Swirl: Spirals In Nature by Joyce Sidman, and drawing pictures of Fibonacci flowers.
Kids in second, third, fourth, and fifth grades are at just the right age to explore Fibonacci.
They quickly pick up on growing patterns like the Fibonacci sequence, and their fine motor skills are developed enough to draw a Fibonacci spiral.
I highly recommend the printable workbook Nature Math: Fibonacci for Kids Ages 7-11.
Part nature exploration and part real-world math, your child will explore many more wonders of Fibonacci numbers, including the human body, fruits and vegetables, and more.
In this article, I’ve focused on Fibonacci in nature up to about grade 5. But there is more to explore, especially once kids have mastered division and decimals.
For example, the golden ratio (also known as the golden mean) can be a fascinating exploration for middle schoolers, and and they can challenge their logical brain by solving Fibonacci’s famous rabbit problem.
The more you’ve experienced the beauty of math in nature yourself, the easier it is to share that experience with your kids.
Take this as your invitation to get outside, open your eyes to wonder, and enjoy the little miracles all around you!