It’s just so beautiful

16 04 2010

“Mathematics possesses not only truth, but also supreme beauty”

-Bertrand Russell (English Logician and Philosopher 1872-1970)

Bertrand Russell

What did he mean by this? Do you believe him? Have you ever seen anything in maths that you think is beautiful?

When considering what to put in this post, I had to sift through literally dozens of beautiful and mysterious things that I’ve seen in maths including:

  • The idea that all natural numbers (except one) can be made by just multiplying together prime numbers. What is a prime number?
  • That insects called cicadas spend most of their lives underground as grubs but then pupate and come out of their burrows after 13 or 17 years (prime numbers) after which they live for only a few weeks. Why is this interesting? It is thought that they do this because it would make it hard for predators to have evolved that could specialise in eating only the cicadas. If instead they appeared in intervals which were not prime numbers, say every 15 years, predators appearing every 1, 3 and 5 years would definitely meet (and eat) them. Amazing!
  • Pythagoras’ Theorem about how you can work out the length of sides in right angled triangles. The history of this one is amazing, seeing how he proved it with only geometry rather than with algebra.
  • The idea of irrational numbers. Those are the ones that have an infinite number of non repeating numbers after the decimal point. I find this fascinating because infinity is such a hard idea to get your head around.

All these things are interesting, but, for me there has always been one thing that totally amazed me and it’s to do with a really simple bit of maths that seems to appear in all sorts of different places in nature: Fibonacci Numbers.

What are Fibonacci Numbers?

Fibonacci (Leonardo of Pisa)

Look at this sequence of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377…

Can you spot a pattern? Can you work out how you get from one number to the next?

Did you work it out? To get from any number in the sequence to the next one you just take the number and add on the number before it. For example to get the number after 5 you do 3 + 5 = 8, to get the number after 55 you do 34 + 55 = 89.

This number sequence is called the Fibonacci Sequence and all the numbers in it are called Fibonacci Numbers. Who was Fibonacci? He was actually called Leonardo of Pisa! He lived 1170 to 1250 and didn’t actually discover this number sequence at all! The sequence had been discovered in India and Fibonacci quoted it in his book Liber Abaci which he wrote to introduce the symbols 0 to 9 into Europe. Before then we’d all been using Roman numerals like I, IV, XI etc.

Where do they show up in nature?

Everywhere! Look at a pine cone from underneath. You will see spirals heading off in both clockwise and anti-clockwise directions. Count the spirals in each direction, I bet they’ll be two numbers next to each other (consecutive numbers) in the Fibonacci Sequence! Now do the same with pineapples and you’ll notice the same thing!

34 spirals in this direction. Picture courtesy of hosted-sites/R.Knott/Fibonacci/ fibnat.html

55 spirals in this direction. Picture courtesy of hosted-sites/R.Knott/Fibonacci/ fibnat.html

Now look at the seeds in the head of this sunflower. You can count the number of spirals in each direction and, hey presto! 34 and 55 are consecutive numbers in the Fibonacci Sequence! It turns out that these special spiral pattern is the most efficient way of packing seeds into the head of the sunflower (you get most in using this pattern). Whether you believe in a creator of the universe, in evolution, or both, you can’t deny how beautiful it is than nature has come up with such an efficient solution to the seed packing problem and how the maths hidden inside the solution is so simple!

Fibonacci Numbers appear in other areas relating to how plants grow. Go outside and count how many petals you find on lots of different flowers. A very large majority of them, will have a number of petals that is a Fibonacci Number!

The way leaves grow on plants may seem random when you first look at it but actually, it’s very ordered and guess what, more than 90% of plants grow leaves in a pattern that follows the Fibonacci Sequence! When plants grow leaves they make sure that the leaves are not positioned right above each other. If they were, then the bottom leaf wouldn’t get much light which wouldn’t be very helpful for growth! Plants grow by spreading their leaves out so they all get as much sunlight as possible. As you move up a plant stem you’ll find that eventually leaves are positioned above each other like leaves 0 and 8 in the picture below. What you’ll find is that if you count how many leaves you pass as you move up the plant until you meet another one right above the one you started from you get a Fibonacci Number! The magic doesn’t stop there! If you count how many times you have to spiral around the plant to pass all the leaves on the way up until you get to the leaf directly above the one you started with you’ll find it too is another Fibonacci Number! For example, if you look at the picture you can see that you have to spiral round 5 times to pass all the leaves between leaf 0 and 8. 5 is of course a Fibonacci Number! There’s even more too! If you do the spiral going the other way you’ll find that the number of whole turns is actually yet another Fibonacci Number! Look at the picture below and think about the spirals going the other way. There would be 3 full turns between leaves 0 and 8. 3 is a Fibonacci Number! So what have we found? In plants, if you count how many leaves there are between ones that are directly above each other, if you count how many spirals you have to go round to pass all the leaves in between and all the spirals in the other direction too that they are all Fibonacci Numbers! But that’s still not the end of the magic… When we looked at the plant in the picture we found there were 8 leaves between leaves that were above each other, 5 spirals in one direction and 3 in the other. 3, 5, 8 are three numbers that are next to each other in the Fibonacci Sequence! You’ll notice the same thing in more than 90% of plants! The numbers might be different, they might be 5, 8, 13 but they are always three numbers next to each other in the Fibonacci Sequence. That is beautiful!

Fibonacci Numbers in leaf growth patterns on plants. Picture courtesy of http://britton.disted.camosun

A different area of nature where we see some Fibonacci magic is to do with how shells grow. Look at the pattern of squares that is growing in the picture below. It’s made by starting off with a square with sides of length 1 unit (lets say 1cm) and then you draw another square to the left of it with sides of length 1cm, then another above it with sides of length 2cm, then another to the right with sides of length 3cm, then another below it with sides of length 5cm and so on. You carry on creating this pattern by moving around clockwise each time you add a square and making the new square have sides of length that are the next number in the Fibonacci Sequence.

Fibonacci Spiral pattern. Picture courtesy of http://www.maths Fibonacci/ fibnat .html

Fibonacci Spiral pattern. Picture courtesy of http://www. maths.surrey hosted-sites/R.Knott /Fibonacci/ fibnat.html

The next thing you do is that you draw a spiral that starts in the first square and goes round touching two corners of each of the next squares you made. Look at the picture above to see what this spiral looks like. Notice how it looks the same as the spiral patterns you get in sea shells? Check out the picture below taken from a great website showing lots of beautiful pictures of Fibonacci patterns in the real world.

Fibonacci Spiral in a nautilus shell. Picture courtesy of english/fibon/index.html

Think about music. There are 13 notes in an octave, 8 in a scale and the 1st, 3rd and 5th notes in a scale make up the basic chords! 1, 3, 5, 8, 13. Recognise those numbers?

What about DNA? In every cell in our body it is the DNA double helix which stores all the genetic information that makes us who we are. If you measure how wide the double helix is you’ll find it’s 21 angstroms (21 hundred-millionths of a centimetre). If you measure how long one full turn of the double helix is you’ll find it’s 34 angstroms. Consecutive numbers in the Fibonacci Sequence in the very stuff that makes us human!

21 angstroms wide by 34 angstroms long- The DNA double helix that makes us who we are. Picture courtesy of files/u11/dna_helix.jpg

Maths has weaved its way into every part of the universe including the smallest building blocks that make us, the way plants and shells grow and even into our music! Things like the pattern of seeds in the head of a sunflower seem complex at first sight but then we look at the maths and find they are remarkably simple. Maths gives us the power to describe these things in a language that everyone can understand. Enjoy it and learn more! Here we’ve just had the smallest glimpse at the beauty and mysterious character 0f maths. Lets go on together and enjoy this journey of finding out more about this great subject!

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