The class was silent. The teacher was asking a question. He asked the children, if I have three bananas, how to distribute these bananas equally to three peoples. The answer from children was one. Then teacher again asked, if I have 1000 bananas, how to distribute equally for 1000 peoples. The answer is again one from the children. But one student stood up asked a question to the teacher, Sir if I have no bananas, how to distribute equally to 1000 peoples. The class burst into laughter thinking it is a silly question. But the teacher was very impressed with the question. That student was thinking on a concept of infinity which troubled the mathematicians for over centuries. The student was Sriniavasa Ramanujan, later he went on to became one of the great mathematician.
Ramanujan was born in a small city called Erode in Tamil Nadu on December 22, 1887. He did his early education in India, he was very fascinated with mathematics at the age of 10, at one stage in his college education he studied only mathematics, forgetting all other subject and failed in first year college! He even worked with G. H. Hardy, another great mathematician at that time. Seeing Ramanujan interest on mathematics Hardy once remarked that, Ramanujan was a Natural Genius like Eular and Gauss. His contribution to mathematics is number theory, infinity series, mathematical analysis and fractions. Later Ramanujan was elected as a Fellow of Royal Society in 1918; he was the second Indian to be honored. After two years unfortunately, Ramanujan fell victim to tuberculosis, malnutrition and liver infection. He died of illness in the year 1920 at the age of 32.
Within a short span of his life Ramanujan earned repute and became one of the greatest mathematician of our country. In recognition of his work on mathematics and this year will see the 125^{th} Birth anniversary of Ramanujan, Government of India declared this year as “National Year of Mathematics” and Dec 22, 2012 will be celebrated as “National Mathematics Day”.
More on Ramanujan and his work.
http://en.wikipedia.org/wiki/Srinivasa_Ramanujan
Well, now we will come to Numbers, Once Hardy went to meet Ramanujan at his residence in taxi, numbered 1729. Hardy remarked that, the number 1729 is uninteresting. Suddenly Ramanujam said, 1729 is a very interesting number, it is the smallest natural number which can be represented by sum of cube of two numbers in two different ways!!
Every number is interesting, only when we know in which angle we need to visualize those numbers. But now I am here to introduce, you to some of other interesting numbers which actually expalins nature for no reason. That is, these numbers just appear in the nature, and they explain certain observed features. Believe it or not certain number gave scientist a hint to make new discoveries. Let us see some of those…
 Galileo’s law of odd numbers.
It was believed that Galileo did one of the famous experiment of dropping two object of different mass from a leaning tower of Pisa. These object falls to the ground at same time. Later Newton said, Earth gravity pulls every object at the same rate. That is Apple, stone, Tennis ball, everything falls at the same rate to Earth in a free fall.
Now take tennis ball, when you drop a tennis ball at some hight ‘h’, the tennis ball will undergo free fall and finally hits to the ground. Now if you want to calculate the distance traversed by teenis ball under a free fall, in a equal interval of time, it turns out that they are in the ration of
1 : 3 : 5 : 7 : 9 : 11 : … ..
That is in a free fall for 1 second tennis ball traverse 1unit of distance, next second it traverese 3units, next second it traverse 5units of distance and goes on…
This is constant for all the objects which is at free fall condition. Isn’t it amazing, a simple odd numbers explaining distance traversed by an object in a free fall! This feature was first identified by Galileo, so it is called Galileo’s law of odd numbers.
2. Bode’s Law
The year was 1700, a young Gernman astronomer Johann Elert Bode, proposed a simple rule to calculate the distance of planets from sun. This rule is popularly known as Bode’s Law.
His rule says like this.
 Write down these sequence of numbers 0, 3, 6, 12, 24 … … here number after the second number is simple twice the preciding number.
 Add 4 to each numbers in the serious.
 Divide each number by 10
 The result you get is the distances of planets from sun in Astronomical Unit.
Sequence 
Planets 
Actual Distance in (AU) 
(0+4)/10=0.4 
Mercury 
0.39 
(3+4)/10=0.7 
Venus 
0.72 
(6+4)/10=1.0 
Earth 
1.00 
(12+4)/10=1.6 
Mars 
1.52 
(24+4)/10=2.8 
? 

(48+4)/10=5.2 
Jupiter 
5.20 
(96+4)/10=10.0 
Saturn 
9.54 
(192+4)/10=19.6 
Uranus 
19.18 
(384+4)/10=38.8 
Neptune 
30.06 
Now you notice, Bode’s sequence (numbers) almost gives the exact distance of planets from sun up to Uranus. Easy way of remembering right! These sequence of numbers, for no reason they are telling something about the universe. Notice other thing in the table, between Mars and Jupiter, there is no planets, but still Bode’s sequence says, there should be a planet at distance 2.8AU from sun! It’s interesting, even though this law has nothing to do with actual science; some scientist believed and went on searching for a “missing planet” between Mars and Jupiter.
Later in 1801, an astronomer called Piazzi’s noticed a dim object and named ‘Ceres’. The orbit of this object was determined to be lie between the orbits of Mars and Jupiter. Later many such small objects were discovered, and they even identified a belt which contains small objects, orbiting sun at a distance of 2.7AU. This belt is called Asteroid Belt. Coincidentally Bode’s law predicted a planet at a distance 2.7AU, but scientist thought this missing planet has become a broken or exploded!
At 2.7AU we have Asteroid Belt!
Bode’s law is not a law at all, it just a sequence of numbers. But this interesting number gave scientist a clue to discover Asteroid Belt. Isn’t it amazing! Numbers play a role in Nature!!
3. Golden Ratio
Before going to the Golden Ration, I will have to introduce series called Fibonacci series. It was first introduced by Leonardo Pisa also known as Fibonacci. He was a famous mathematician of Middle Age.
The series is as follows..
(First two numbers in series is 0 and 1, remaining numbers are sum of the previous numbers)
Even before Fibonacci introduce these series, the series appears in Indian Mathematics, in connection with Sanskrit Prosody.
Fibonacci number and golden number:
Consider Fibonacci numbers
Now take the ratio of successive number in the series i.e.,
1/1=1, 2/1=2, 3/2=5, 5/3=1.66, 8/5=1.60, 13/8=1.625 . . . . . . . (the ratio’s are called phi)
Now if you plot a graph of phi v/s no of steps…
You see the ratio settles down at 1.6180. This number is called Golden Number or the ratio is called Golden Ratio.
Golden Ratio is represented by Phi
Ratio of the sum of the quantities to the larger quantity is equal to the ratio of larger quantity to the smaller one (see fig). The resulting number is golden number.
Fibonacci Rectangles
This is a representation of Fibonacci series in a picture…
Consider Fibonacci series
Draw two squares of 1 unit dimension, and then draw another square on top of these two square measuring two units. Again draw another square of 3 units touching previous two squares and again draw a square of 5units and it goes on… (See fig)
Now draw spiral in square (quarter of circle in each square) as shown in figure.
Vo this includes lots of math, now I will stop all these maths will look what it signifies.
The question is where this series or Golden Ration appears in Nature.
 Tree Branches
Actually speaking our nature is mathematically beautiful. Have you ever observed a tree carefully, Tree always grows and branch out like Fibonacci series..
Next time when you go out to park, observe it carefully.
 Leaf Arrangement and Petals on Flowers
When you carefully observe a leaf arrangement in planets, it turns out that it shows Fibonacci numbers in their arrangement. For example look at this image..
When you observe carefully, no of leaf for each turn is a Fibonacci number!! There is a reason for this arrangement. If you look down on a plant, the top leaf does not hide the bottom leaf, this kind of arrangement enables leafs to get sufficient amount of sun light! Nature and Math’s!
Petals on Flower:
The number of Petals on Flower is Fibonacci number!
Observe these images
Lily 3petals
Rose 5 petals
Delphinium 8petals
Marigolds 13petals
Etc..
 Spiral Structure!
By using Fibonacci series we plotted the spiral structure, you find this spiral structure in almost everywhere in nature.
See the figures…
and even in Galaxies…
Number and nature connection does not end here.. when you go deep, there are more things which unfolds to you. I leave it you to explore further. Nature knows more mathematics than Humans, its the beauty of Nature.
Next time when went out for trip, observe nature!!
Happy Year of Mathematics!
—
Viswa Keerthy S, 23/08/2012.
More about Fibonacci Numbers and Nature
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibnat.html#seeds
http://en.wikipedia.org/wiki/Fibonacci_number
Enjoyed all of it 🙂
Thank you.