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Fibonacci Sequence

Fibonacci Sequence

There are some things that all living things on Earth have in common. It's not a hard-to-find secret or a connection to something from another world; it's just a simple fact. At some point, we all come into existence, and at some point in the future, we all leave existence.

We each have our own experiences between the beginning and the end. Some of our problems are shared with other people, while others are only our own. However, they all affect us in some way. All of these things just happen by chance, or do they all have a pattern and a reason?

A lot of philosophers, teachers, mathematicians, and people who think about life in general think that nothing is random. This is a man who was born in Italy more than 800 years ago. Leonardo Pisano Bogollo was his name, but today we call him Fibonacci. He liked numbers from a young age, and this interest led to a discovery that could change the way people think about life and the world around them.

Fibonacci's Numbers Come to Life

Some people think that you can find a pattern in anything if you look hard enough. This may be true, but Fibonacci was able to use his knowledge of numbers to find a specific pattern and show that it could, in theory, go on forever. In his first book, Liber Abacci, he wrote about the Fibonacci Sequence and gave everyone a math-based word problem to think about..

"A man put two rabbits in a place with a wall all around them. How many pairs of rabbits can be made from that pair in a year if each pair makes a new pair every month, which starts making rabbits in the second month?

Your mind is about to be blown. Now might be a good time to make a pot of coffee and figure out where the aspirin is.

Okay, here we go! Fibonacci figured that the rabbits would grow up and mate in about a month, so after a month, there was still only one pair. Since they also have a month-long pregnancy, the second month ended with only one pair. At the end of that third month, however, another pair showed up. Female rabbits should be able to have a litter once a month after their first one. The first pair had another litter in the fourth month after their first one.

When month five came around, pair one had another child, and pair two did the same. At this point, there were five pairs in all. In short, by the end of the year, 377 couples were living inside the wall.

Taking a closer look

Fibonacci did make a few pretty big leaps before he was able to predict how well this mess would work out. His plan was based on a set of highly controlled circumstances. As an example,

  • One male and one female rabbit were born with each new litter.
  • All were physically able to have children.
  • Each female was able to start having babies at one month old and have a new litter every month after that.
  • None of the rabbits got sick or were kicked out of the compound by the other rabbits during the whole year.

Some rabbits can take up to four months or longer to grow up. A single litter can have up to 12 babies, and not all of them are sure to live. When you really think about it, though, natural selection and uncontrolled numbers could easily cancel each other out. Fibonacci's well-thought-out rules could easily cover all the possibilities and keep the scales in balance.

Putting it into parts

All of these ideas might sound a little vague when they are written down. Fibonacci's math is perfect, of course. When you put the numbers where the words are, it's a little easier to understand. Starting with 0 and 1, the Fibonacci sequence goes on from there. To put it simply, it goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.

Each new number in the series is made by adding the two numbers that came before it.








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You could keep going as long as you wanted by adding the last two numbers to find the next one. Even though the explanation sounds long and hard to understand, it's really not that hard. From here on out, though, things get quite a bit more complicated.

The idea starts to take shape.

We're about to take this to an entirely new level. Think of the numbers in the Fibonacci Sequence as squares, like on a piece of graph paper. Find a place near the middle of the page to start, and color in one square to show the first number in the sequence (1). Do the same thing for the second one next to it. Then, since two is the third number, shade in a square that is 2 by 2 with a different color. From there, find a different color to fill a 3x3 square with, and so on. You will have a lively version of this example:

Do you see how they fit together perfectly, like puzzle pieces? The next number in the series would be 13, and the width of the shape here is 13 squares. Even though the numbers get bigger, the pattern stays the same. Each new square will be bigger than the one before it, but if you follow the numbers in the Fibonacci Sequence, they will always fit together perfectly.

Now, if you're writing down what I say, get a pen. Start in the top corner of your first single square and draw the first step of a spiral all the way to the corner on the other side. This will cut the square in half. Continue drawing your spiral across the diagonal center of each larger square as you move from one single block to the next. The result will be something like this:

Because of their link to the Golden Ratio, Fibonacci's spiral and the rectangle made by his sequence of numbers are called the Golden Rectangle. As a number, this magic number is 1.61803, and the Greek letter Phi () stands for it. It might seem like we're jumping around a bit, but everything we're talking about is important.

In geometry, you get the Golden Ratio when you cut a line into two parts, but not just any two parts. The line has to be split just right, so that the length of the whole line divided by the length of the longest part of the split is exactly the same as the length of the longest part of the split divided by the length of the shortest part. You guessed it: the sum of the two numbers must be 1.61803.

People say that rectangles made with the Golden Ratio are the most beautiful things you'll ever see. We're back to Fibonacci again. It turns out that if you take any two consecutive numbers in his sequence and find their ratio by dividing the larger number by the smaller one, it's very close to the ancient golden standard. Look at this!

21/13 = 1.61538

89/55 = 1.47272

121393/75025 = 1.61803

Take a look! We chose the winner! Even with all of that, the Fibonacci Sequence, the Golden Rectangle, and the spiral inside of it are everywhere. You're almost done learning how and where.

Spiraling into Complete Control

Before you keep reading, you should be sure you're ready to make this jump. This is just some advice from a friend. You can't forget it once you've seen it, and it will change your life!

Fibonacci's Sequence makes the Golden Rectangle, which has been seen all over the world for hundreds of years. The Parthenon in Greece is a great example. There are many examples of the Golden Ratio and the Golden Rectangle both inside and outside, and they can be big or small.

The Mona Lisa by Leonardo da Vinci could also be seen as an example of the near-Golden ratio at work. It is said to be 30 inches long and 21 inches wide. Using the standard plan, if you divide the longer side of the rectangle by the shorter side, you get the number 1.42857. It doesn't match perfectly, but it's close.

When we go to Egypt, the Great Pyramid of Giza is another possible example. This one is the biggest of the three. It is 756 feet wide and 481 feet tall. When you do the math, the ratio comes out to be 1.57172, which is again right on the line.

The Fibonacci Sequence is also used in some paintings by the artist Piet Mondrian. In some of his works, you can see the puzzle-piece squares that make up the Golden Rectangle. If you look at it long enough, you might even start to see the spiral in your mind.

These are just a few examples of how this pattern has been used in art, architecture, and engineering over many years. A lot of people say it's just a coincidence, pointing out that if you look for patterns, you can find them in almost anything. In the same way, some of these things happened long before Fibonacci was born.

Many people say that it's all plain to see. Some of the smartest and most creative people in history are behind the creation of these masterpieces. Even if they don't realize it, it makes sense that they would use the most pleasing visual elements in their work.

You can make sense of what's going on in a lot of different ways, but the problem goes even deeper. Some things can't be ignored or made to make sense.

It makes sense.

If you go outside and look around, you'll also see Fibonacci numbers in nature, and not just in one or two places. One good example is a sunflower. These bright yellow petals surround a large group of seeds. The seeds aren't thrown together in a random way, though. Instead, they spiral out from the center of the seed head.

Pinecones are another example that is often used. You can look at them either clockwise or counterclockwise and see that their spikes are set up in spirals. Pineapples are the same way. If you have a holly bush or a similar bush nearby, you should look at it. At first, it looks like the leaves are growing all over the place or even in circles. When you look at them more closely, you'll see that they're not all in a straight line. Instead, they run in rough spirals from the tip to the trunk.

Small parts of the Golden Rectangle can also be seen in people. The average adult face is about 8 to 9 inches long and 6 to 7 inches wide. When you divide the two extremes by each other, you get 1.5, which is close to the Golden Ratio. From a central point, hair tends to grow outward in spirals. From now on, you'll start to see different versions of the sequence everywhere. Don't say that we didn't warn you! "

Figuring it out

As with most math problems, there are rules for how to figure out the numbers in this series. It comes with its own set of different formulas. Let's start by looking at an example of the general rule behind the Fibonacci Sequence:

xn = xn-1 plus xn-2

Here, "n" gives each number in the Fibonacci Series a number from our usual sequence: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...

"Xn" stands for the Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...

We can figure out a number in the sequence by following the rule. Let's try number seven:

x7 = x7-1 + x7-2

x7 = x6 + x5

x7 = 8 + 5

x7 = 13

It's just another way of saying, "A number in the Fibonacci sequence is equal to the sum of the two numbers before it."

From here on out, things will only get more complicated and harder. There are a lot of equations and explanations about mathematical induction, linear recurrence, plotting the sequence on graphs, etc.

For those who try to jump right into the advanced level, these formulas are Greek, if you'll excuse the pun. If you don't know much about how to solve problems in this situation, it's best to ease into it. Once you know the basics, like the above formula, it's easy to take baby steps from there.


We all share some things. Most people don't agree on whether they happen by chance or because of underlying mathematical sequences and mysteriously ordered chaos. No matter where you stand on the issue, you can't just ignore or forget about certain patterns. Fibonacci saw this as a child, and his work set off a chain of events that can't be stopped.


This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2022 Waqar Anwar

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