Fibonacci: from a simple sequence to the golden ratio

Evelina Petitto

“How many pairs rabbitsof rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?”

This is the problem that mathematician Fibonacci (Leonardo of Pisa) proposed in 1202, in his treatise “Liber Abaci”. The problem assumes the following conditions:

  • The first couple of rabbits have just been born
  • Rabbits reach sexual maturity after one month
  • The gestation period is also one month
  • After reaching sexual maturity, the female rabbit give birth every month
  • The female rabbit always gives birth to one male and one female
  • The rabbits never die

Following these indications, it is easy to see that, for the first two months, we will have only 1 pair of rabbits; in the third month, we will have one new pair, bringing the total number of pairs to 2. In the fourth month, there will be 3 pairs, in the fifth month, 5 pairs and so on, until one year passes and there will be 233 pairs of rabbits.

What we get is the following series of numbers:

1,1,2,3,5,8,13,21,34,55,89,144,233,….

This is an example of a recursive sequence, where to find out what the next term is, one simply adds up the preceding two numbers:

F(1) = 1

F(2) = 1

F(n) = F(n-1) + F(n-2)

This simple recursive sequence has fascinated mathematicians for centuries; it also appears in nature, from sunflowers to galaxies. For example, consider this diagram:

golden-rectangle

Each square has sides whose lengths correspond to the terms of the Fibonacci sequence – here, they are arranged in an “outwardly spiralling” pattern. A Fibonacci spiral is a series of connected quarter-circles drawn inside an array of squares with Fibonacci numbers for dimensions. Because of the nature of the sequence, where the next number is equal to the sum of the two before it, the squares fit perfectly together. Furthermore, any two successive Fibonacci numbers have a ratio very close to the Golden Ratio, which is approximately 1.618034. This number was known to the ancient Greeks, who assigned it the letter ϕ (phi). They believed that the proportion ϕ : 1 was aesthetically perfect, and the majority of their artwork, sculptures and architecture was built using this proportion as a foundation. A rectangle whose sides have this proportion is called a golden rectangle.

As we mentioned before, the Fibonacci sequence and the consequent Golden Ratio appear in nature, in many instances. Here are some examples:

1. Flower petals

The number of petals on a fbuttercaplower consistently follows the Fibonacci sequence. Examples include the lily, which has three petals, buttercups, which have five petals (see picture), the chicory’s 21 petals, the daisy’s 34 petals, and so on. Phi appears in petals on account of the ideal packing arrangement, as dictated by Darwinian selection; each petal is placed at 0.618034 per turn (of a 360° circle) allowing for the best possible exposure to sunlight and other factors.

2. Tree branchestree

The Fibonacci sequence can also be seen in the way tree branches form or split. A main trunk will grow until it produces a branch, which creates two growth points. Then, one of the new stems branches into two, while the other one lies dormant. This pattern of branching is repeated for each of the new stems.

3. Seeds and pinecones

seeds

Also, the heads of flowers are subjected to Fibonaccian processes. Typically, seeds are produced at the centre, and then migrate towards the outside to fill all of the available space. Sunflowers provide a great example of these spiralling patterns: the numbers of spirals in each direction are invariably two consecutive Fibonacci numbers. Similarly, the seed pods on a pinecone are arranged in a spiral pattern. Each cone consists of a pair of spirals, with each one spiralling upwards in opposing directions. The number of steps will almost always match a pair of consecutive Fibonacci numbers.

4. GalaxiesGolden-Ration_Galaxy

Spiral galaxies also follow the familiar Fibonacci pattern. The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees.

5. Facesface

Faces, both human and non-human, are abound with examples of the Golden Ratio. For example, the mouth and nose are each positioned at golden sections of the distance between the eyes and the bottom of the chin.

6. DNAdna

Fibonacci also affects the microscopic domain. DNA molecules measure 34 angstroms long by 21 angstroms wide for each full cycle of their double helices. These numbers, 34 and 21, are numbers in the Fibonacci series, and their ratio 1.6190476 closely approximates Phi, 1.6180339.

References:

http://platonicrealms.com/encyclopedia/Fibonacci-sequence

http://www.livescience.com/37470-fibonacci-sequence.html

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

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Zero, nought, nil, O (Oh). So many names for… nothing!”

Evelina Petitto

“He’d always felt he had a right to exist as a wizard in the same way that you couldn’t do proper maths without the number 0, which wasn’t a number at all but, if it went away, would leave a lot of larger numbers looking bloody stupid.”  (Terry Pratchett, Interesting times)

zero

Zero is a very strange and interesting number with a lot of amazing properties and, for many reasons, the most significant number of all. From its origins to the paradoxes that it creates, we will explore the facts that make 0 so special.

 

HISTORY

The first numeric systems such as the ones used in ancient Egypt or Greece didn’t have a zero. Quantities were expressed using combinations of symbols or letters, which made it very difficult to write big numbers and even harder to perform also simple calculations.

The firsts to develop a positional numerical system were the Sumerians in 3000 BC; the value of a symbol was given by the relative position of that symbol in relation to the others. This system was later further developed by the Babylonians in 2000 BC, who created a mark to indicate that a number was missing from one position (units, decimals, hundreds and so on).

Brahmagupta, an Indian mathematician and astronomer, was the first to formulate the concept of zero as a number around 650 AD. He also gave rules for its use, explaining how to use it in additions with positive and negative numbers (“The sum of two positives is positive, of two negatives negative; of a positive and a negative the sum is their difference; if they are equal it is zero. The sum of a negative and zero is negative, that of a positive and zero positive, and that of two zeros zero”), in subtractions (“A negative minus zero is negative, a positive minus zero positive; zero minus zero is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added”) in multiplications (“The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero”) and divisions, although his description differs from the one we have today (“A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is also negative. A negative or a positive divided by zero has that zero as its divisor, or zero divided by a negative or a positive has that negative or positive as its divisor”). What differs from nowadays conception of division by zero is that the result is now considered undefinable… we will see later what this means.

But it still took a few centuries for zero to arrive in Europe.  It first reached the Middle East, where Arabian mathematicians based their numbers upon the Indian system. Finally, thanks to the conquest of Spain by the Moors, the zero reached Europe, where mathematicians and philosophers such as Fibonacci, Descartes, Newton and Leibniz further developed its understanding and conceptualisation.

HOW CAN WE DESCRIBE ZERO?

The first thing that makes us understand that zero is a special number is the fact that it’s not easy to describe it the same way we describe every other number: is zero positive or negative? Is it even or odd? These simple questions, in fact, don’t have a straightforward and intuitive answer.

Let’s start from the first issue: positive or negative?

Positive and negative are the two big categories in which we divide real numbers. But, to be completely correct, we should divide them into three categories: positive, negative and zero. Zero, in fact, is neither positive nor negative and the demonstration for that is very simple. Let’s suppose that we want to demonstrate that zero is positive using this simple equation:

0 x -3 =

= +0 x -3 =

= -0

We end up with a negative zero. For this reason, we say that zero is neutral.

The other question is: is zero even or odd?

Zero is an even number. In fact, it follows the properties of all even numbers such as: it’s divisible by 2, it’s adjacent to odd numbers on both sides, and a set of 0 objects can be divided into two equal sets.

numb

However, people find it hard to classify it, and reaction times are much higher compared to those for other numbers. In an experiment conducted by Dehaene in the 1990s, numbers and number words were flashed on a monitor, and the subjects had to press a button for odd and another for even. A computer registered the reaction times for each answer. The results of this experiment showed that people took on average 10% more time to classify the number 0, regardless from age, nationality, linguistic background. The only discriminating factor was mathematical expertise.

MATHEMATICAL PROBLEMS

In mathematics zero is a tricky number. We know that if we multiply any number by zero we have zero as an answer, but what happen if we try to divide a number by zero? Can we do it?

First of all, let’s think about what a division is. Let’s say that we have the number 10 and we want to divide it by 2. This basically means subtracting 2 from 10 until we have 0:

10 – 2 = 8confused

8 – 2 = 6

6 – 2 = 4

4 – 2 = 2

2 – 2 = 0

We did this 5 times, and our answer for 10/2 is therefore 5. But what happens if we want to divide 10 by 0? We have:

10 – 0 = 10

10 – 0 = 10

10 – 0 = 10

…. And so on. Thinking about it we can believe that the division by zero give us infinity. However, infinity isn’t a number, and we can’t treat it like if it was. So infinity is not the correct answer. As we mentioned previously, division by zero is undefined, or in other words it doesn’t make sense.

Let’s now think about 00. This gives problems because:

X0 = 1

0X = 0            where X = any number

What happen when we have 0 on both sides? There isn’t a definitive answer to this: some people argue that the result should be 0, others that it should be 1. Once again, the answer is that 00 is undefined; we can’t give a value for it because the limit varies depending on the number considered.

If you want to know more about the mathematical problems with number 0 check this video:

Bibliography:

http://yaleglobal.yale.edu/about/zero.jsp

http://en.wikipedia.org/wiki/Brahmagupta

http://mathfour.com/arithmetic/is-zero-positive-or-negative

http://en.wikipedia.org/wiki/Parity_of_zero#Mathematical_contexts