[Vibo4Kids]The Fascinating Fibonaccis

The Fascinating Fibonaccis

Author: Dr. Shonali Chinniah

Illustrator: Hari Kumar Nair

Numbers.

We use them everyday.

To count, measure, call friends on the phone and even to find out what something costs.

But did you know you can also use numbers to create patterns - geometrical shapes, rangoli designs, and more?

Did you know number patterns can be seen within patterns in nature?

But first, what is a 'Number Pattern'?

A Number Pattern is a sequence of numbers where each number is connected to the previous one in ONE specific way.

Take this very simple number pattern: 0, 1, 2, 3, 4...

How is each number in this sequence connected to the one before it?

Well, every number in this sequence is the previous number with 1 ADDED to it.

Here's another number pattern: 14,12,10, 8, 6...

Each number in this sequence is the previous number with 2 SUBTRACTED from it.

Now for a slightly more tricky pattern: 0, 1, 3, 6, 10, 15...

How does this sequence work?

Let's see. 0 + 1 = 1 1 + 2 = 3 3 + 3 = 6 6 + 4 = 10 10 + 5 = 15 Do you see the pattern here?

What will the next number in this sequence be?

Yes, 21, because 15 + 6 = 21.

Now, let’s take the 'number pattern' we just discussed: 1, 3, 6, 10, 15... , and see if we can create a 'SHAPE pattern' from it.

We can!

We now have a 'shape pattern' of triangles that get bigger and bigger as we increase the number of dots according to our number pattern!

A number pattern has become a shape pattern!

If you found that interesting, it's time you were introduced to a beautiful number sequence called the Fibonacci (or Hemachandra) Sequence of numbers.

The Fibonacci Sequence of numbers goes like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...

Can you find the pattern that connects these numbers?

Yes!

Every number in the Fibonacci Sequence is the sum of the two numbers before it!

Like this. 0+1 = 1 1+1 = 2 2+1 = 3 3+2 = 5 5+3 = 8 8+5 = 13 13+8 = 21 21+13 = 34 Got it?

Good.

Now for the REALLY interesting part - linking this number pattern to patterns in nature.

The number of petals flowers have are often linked to Fibonacci numbers!

Can you think of flowers with 1, 3 and 5 petals?

(These are all Fibonacci numbers.)

Here are some examples to help you along. 1 petal - 1.

Anthurium; 2.

Calla lilies 3 petals - 3.

Bougainvillea; 4.

Clovers 5 petals - 5.

Temple tree; 6.

Hibiscus; 7.

Jasmine

Flowers with 2 petals are not very common.

The Crown of Thorns, which you see here, is one example.

Flowers with 4 petals (4 is NOT a Fibonacci number) are also rare.

Count the petals of flowers that you come across and see for yourself!

The most interesting flower of all, where the Fibonacci sequence is concerned, is the daisy.

Different daisy species have 13, 21, or 34 petals - which are all Fibonacci Numbers!

There are even more complex and stunning patterns in nature that appear to be based on the Fibonacci numbers.

If you are willing to do a little math, you can see it for yourself.

Shall we try it out?

Now, what would we get if we squared* each of the numbers in the Fibonacci sequence?

Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, etc.

If we 'squared' each of these numbers, we would get: 1 x 1 = 1 Squared or 1 2 = 1 2 x 2 = 2 Squared or 2 2 = 4 3 x 3 = 3 Squared or 3 2 = 9 5 x 5 = 5 Squared or 5 2 = 25 8 x 8 = 8 Squared or 8 2 = 64 13 x 13 = 13 Squared or 13 2 = 169 So the Fibonacci Sequence Squared: 1 – 4 - 9 – 25 – 64 – 169 - etc.

*When you multiply a number by itself, the number is 'squared'.

Now, just like we converted a number pattern into a shape pattern with the triangles before, let's try to convert the Fibonacci Sequence Squared into a shape pattern.

Let's try to DRAW 1 2 , 2 2 , 3 2 and so on. 1 2 is easy enough – it is just one square. 2 2 is drawn like this - 2 squares across and 2 squares down.

We know that 2 2 = 4, and there are 4 squares in the figure (we call this figure a 'grid').

Similarly, 3 2 is drawn as 3 squares across and 3 squares down.

Again, we know that 3 2 = 9, and there are 9 squares in the grid. 5 2 is drawn as 5 squares across and 5 squares down, making a grid with 25 squares, 8 2 as 8 squares across and 8 squares down, making a grid with 64 squares, 13 2 squared is drawn as a grid with 169 squares, and so on.

Now, let's push all the grids we've drawn so far towards each other, and arrange them like in the picture.

Done?

Now draw a smooth curved line from one corner of the smallest grid to its opposite end, as shown in the figure.

Now take the same curved line through each of the other grids, from smallest to biggest, from corner to opposite corner, ending with the 13 squared grid.

What we get is a lovely spiral pattern.

What is the link between this spiral pattern created by squared Fibonacci numbers, and nature?

Well, the exact same Fibonacci Spiral can be found in nature!

Where?

Let's see, shall we?

Here's the Fibonacci Spiral with one more grid - 21 2 - ad...

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