Autumn%202012

FIBONACCI EVERYWHERE!!

I

n mathematics and the arts, two quantities are in the golden ratio (φ) if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The figure on the right illustrates the geometric relationship. Expressed algebraically:

where the Greek letter phi represents the golden ratio. Its value is:

In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence: 0,1,1,2,3,5,8,13,21,34,55,89,144...

The Fibonacci numbers are Nature’s numbering system.

They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple, to compound eye structures in insects. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

In nature one can see numerous examples of the Golden Ratio in the number of petals in plants

Plants do not know about this sequence - they just grow in the most efficient ways.

Many plants show the Fibonacci numbers in the arrangement of the leaves around the stem. Some pine cones and fir cones also show the numbers, as do daisies and sunflowers. Sunflowers can contain the number 89, or even 144.

Many other plants, such as succulents, also show the numbers.

Some coniferous trees show these numbers in the bumps on their trunks. And palm trees show the numbers in the rings on their trunks.

In the seeming randomness of the natural world, we can find many instances of mathematical order involving the Fibonacci numbers themselves and the closely related “Golden” elements.

Humans exhibit Fibonacci characteristics, too. The Golden Ratio is seen in the proportions in the sections of a finger. It is also worthwhile to mention that we have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.

The ratio between the forearm and the hand is the Golden Ratio!

The Greek sculptor Phidias sculpted many things including the bands of sculpture that run above the columns of the Parthenon.

BATMAN GRAPH n snails are moving along flat ground along straight lines at constant speed. The straight lines that the In this world famous painting, we see each not it…even the famous Apple snails move along are such that no two of them are parallel, major section divided in accordance to the And That’sLogo seems to obey this. golden ratio. and no three are concurrent. Naturally, there are (n(n-1))/2 points of intersection, at any of which if two snails are present simultaneously, then there is a possibility of a crossing. Suppose we are told that k crossings have happened (but are told nothing about which particular ones, or in what order). What is the minimum value of k such that we can with certainty deduce that all the crossings have taken, or will take place?

QUESTION OF THE EDITION

TROLL

VOLUME III, AUTUMN ISSUE

Souradeep Purkayastha

What are numbers anyway?

W

e all have been using numbers ever since we were little, everywhere; We first learnt that 2+3 gives 5, that (surprise) 7*9 was the same thing as 9*7, and, a little later down the education line, we learnt of rational numbers, real numbers, complex numbers, and operations on them. We can use numbers well, but many a time we have shortcomings.

The thing is the human mind (more correctly, most humans’) has predominantly always associated numbers with some tangible real object or property, like length or weight or number of discrete quantities; indeed, that most probably is how numbers were thought of in the first place. Our main tool in this regard is geometry, which enables us to reduce complex numbers to points and products to volumes or areas. Like for example, the property whether you keep a rectangle of sides a and b units along a length or along a breadth, its area is the same , enables one to see intuitively why a*b is b*a.

But we can’t visualise something like an irrational or imaginary power very well.

One can say if asked what a cube root is, “well, it’s the length of a side of a cube having a volume equal to the number you mention.” but we can’t say similarly what say a πth root is. For we don’t know what a cube with π sides looks like. So, must we stop using mathematical concepts if we are unable to think of them clearly? Not at all. However clear thinking does bring about a sense of ‘completeness’ and that pretty much needed in an exact science as mathematics is.

There is a way to get out of such a mental state of dissatisfaction, and this way, or rather set of procedures, was conceived of in its present form only relatively recently in mathematical history (say a few hundred years). This ‘sub-branch’ of

mathematics is colloquially known most commonly as abstract mathematics.

What abstract mathematics does is think of numbers as a set of objects obeying some fixed initial ‘rules’ that are called axioms. What a number is is not a prudent question here, the only concern is the properties we postulate for the numbers and how we can build on those postulates more statements that we call theorems. For example, let us say we want to define what a natural number is. We try some ideas. Defining a natural number as ‘something that counts’ is no doubt what appeals best to the mind: two apples, five sheep, etc-what could be more natural?

But if we next want to define a negative integer, it doesn’t make sense to speak of ‘-5 sheep’ or ‘an average of 5.4 apples in the apple trees in this grove’ for we can’t picture them well enough in our minds.

Even more difficulties come when we come across numbers like the ratio of a circle’s circumference to it diameter, which we denote by π. How exactly does a number with no end to the randomness of its digits get expressed in our minds?

We hence are forced to look for alternate ways of thinking of numbers, so that we can have a better grasp of them without falling into physical limitations. In such an axiomatic build up of mathematics one can start off by defining first the natural numbers and their properties, and then slowly introducing one by one all the number systems we are familiar with today. As for example, one can start by saying a natural number is a member of the set {0,1,2,3....} (yes, 0 can also be considered a natural number) following a few definite laws, that are carefully selected to ensure that our built number system has no loopholes or paradoxes in it, and works exactly the way the same number system when thought in terms of physical objects. These laws can be of many forms, but among the most popular forms is the Peano Axioms. With these one can build rigorously a solid number system that can be extended, again with more axioms, to cover all of the mathematics that is relatable to the world around us, and to much more.

An immensely beneficial side of this approach is, besides making numbers as we know them, we can extend this

process of rigorous treatment to many other categories of objects in mathematics, such as vectors, matrices or operators, and build on them the same way.

It needs to be said that the natural sciences, especially physics, and applied sciences like engineering and economics, don’t necessarily need a rigorous treatment of the mathematics needed in them, for there what matters more is the mathematical postulates one comes up with satisfy experimental observations. As noted mathematician Terence Tao opines, a mathematician can use any of the models of mathematics (geometrical or axiomatic or any other) when convenient, to aid to one’s intuition and understanding, but also can discard some of those models when they begin to get in the way. As such knowing a fair bit of how abstract mathematics goes about certainly helps scientists, and may even help the common man in day to day affairs, even if it happens to appear at first glance ‘weird’ and ‘complicated’.

TOL MOL KE BOL Years it took to reach a market audience of 50million #1- Radio-38 years #2- TV-13 years #3- Internet-4 years #4- iPod-3 years #5- Facebook-2 years The largest shadow, cast by a mountain on the canary island, is 200 miles long.

The word Google is derived from GOOGOL. Googol is the mathematical term for a 1 followed by 100 zeros. 10^100 At a jet plane’s speed of 1,000 km (620mi) per hour, the length of the plane becomes one atom shorter than its original length.

The average lead pencil will draw a line 35 miles long or write approximately 50,000 English words.

BATMAN EQUATION

1

Harish-Chandra: The Transformation “In mathematics we agree that clear thinking is very important, but fuzzy thinking is just as important as clear thinking.”

T

he turning point in Harish-Chandra’s life was when he realised what his true bearings were. Among the most influential abstract mathematicians of the 20th century, it might appear to be strange that he had started off as young enthusiastic physics student. He soon came to realise that what he was truly interested in was the mathematical aspects of physical theories, and not physical intuition in itself, and he eventually changed his professional designation to that of a mathematician.

Born in Kanpur in the year 1923, he matriculated at the University of Allahabad, and was influenced by the great physicist P.A.M. Dirac to study theoretical physics. After a brief stint at the Indian Institute of Science as a research assistant to Homi Bhabha, he moved to Cambridge, where, with Bhabha’s recommendation, he was able to become a research student of Dirac, who eventually supervised his PhD thesis. In 1947 he became Dirac’s assistant at the Institute for Advanced Study at Princeton. But he soon began to feel that his work lacked mathematical rigor, and this feeling was further solidified by the personality of Dirac; in his own words: “Soon after coming to Princeton I became aware that my work on the Lorentz group was based on somewhat shaky arguments… I once complained to Dirac about the fact that my proofs were not rigorous and he replied, ‘I am not interested in proofs but only in what nature does.’ This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics.”

There is in fact this interesting anecdote about a conversation Harish-Chandra had with Freeman Dyson, one of the central figures behind the development of Quantum Electrodynamics. Harish-Chandra had said to him, “I am leaving physics for mathematics, I find physics messy and unrigorous, elusive.” Dyson had replied, “I am leaving NUMB3R FACTS mathematics for physics for exactly the same reasons.”

The total money spent by the And so Harish-Chandra began to delve into pure mathematics. At Princeton he took courses from nine IPL franchisees on the play- esteemed mathematicians Emil Artin, Claude Chevalley, and Hermann Weyl. In 1949-50, he spent ers turns out to be $73335000. a year at Harvard to study algebraic geometry with Oscar Zariski. Harish-Chandra then became a Whereas Cristiano Ronaldo was professor at Columbia in New York (1950-63) before returning to the Institute for Advanced Study, transferred from Manchester where he remained for the rest of his life as the IBM-von Neumann Professor of Mathematics.

United to Real Madrid (2009) for Harish-Chandra’s years at Columbia were his most productive, mathematically, as he concentrated on the discrete series representations of semi-simple Lie groups. He is also known for his a whooping $130 million. work with Armand Borel founding the theory of arithmetic groups. Before changing from physics The average height of a Europe- to mathematics, he had already written a paper on infinite-dimensional representations of semian soccer player is 5’11″. Along- simple and reductive groups, which were not of much use in physics but had a great deal of signifside Germany, Sweden have icance in mathematics. Despite having some gaps in his mathematical background, he made up the tallest squad at UEFA EURO for those with his creativity and work-ethic to invent the tools he needed as ‘scaffolding’ for his 2012, with an average height of research, at times unaware that they existed in formal literature. He was among those responsible 1.85m while in Isaksson (1.99m) for transforming infinite-dimensional group representation theory from a topic found on the pethey possess the tallest player. riphery of mathematics and physics into a major field at the heart of leading-edge mathematics.

Eventually, mathematicians moved away from the problem of solving equations to the Spain averaged 681.6 passes study of the abstract groups. One problem they faced was to represent abstractly formulatper 90 minutes at Euro 2012, ed groups by concrete means. With the rise of quantum mechanics infinite groups and maover 100 more than they did at trices had to be considered. To understand very complex interacting systems of highly enthe 2010 World Cup (588) and ergetic elementary particles it is necessary to study the group of symmetries that leave the the most by a team in the his- system unchanged. The groups that are most important to this problem are known as simtory of the European Champion- ple groups. Harish-Chandra constructed the most fundamental infinite dimensional repreship since 1980 (Germany 555 in sentations of all of the simple groups and used them to represent generalized functions in these groups by means of Fourier series. Besides high-energy physics, applications of repre2012). sentation theory of simple groups are found in modern number theory and Galois Theory.

Harish-Chandra once attempted to express his mathematical aesthetics in a painting metaphor, “In mathematics there is an empty canvas before you which can be filled without reference to external reality.” This statement aptly sums up the fundamental aims and nature of pure mathematics. Harish-Chandra, whose health had always been fragile, had his first heart attack in 1969, and from then on his health was a major concern. He suffered a fatal heart attack on October 16, 1983 as he walked in the woods near the Institute. He was considered for the Fields Medal in 1958, but due to internal dissensions among certain mathematicians towards what they termed was ‘Bourbakist’, he was set aside. Nonetheless he was the recipient of the Cole Prize, a Fellow of The Royal Society, and many other honours and accolades.

F

un with Number 5 Rules: This is normally used for multiplication and in this case both the units in digit i.e. the last numbers are 5. Also, the beginning numbers are same. You will understand this as you see some examples.

25 x 25 = 6 | 25 (This one’s easy, right?)

Explanation: As you see both the ending numbers are 5 and also they are beginning with 2. So we have satisfied both the rules. Now how do we get 625. The first step is that you write down 25 on the right hand side. Why we do so? Do not think about that. Just write 25 on

the right hand side. Then we take the number on the left hand side (of 25), in this case 2. And we multiply it with the next higher digit number i.e. 3. Therefore we get the answer 2 x 3 = 6, which we write down on the right hand side.

And the left hand side number should be same. This is very similar, so lets look at some example.

23 x 27 = 6 | 21 (I am sure that,not everyone finds this simple).

VEDIC TRICK OF THE EDITION So we finally get the answer as 625.

Slight Modification to the Number 5 method Rule: The ending number should add up to 10. Infact in the above case too the ending number 5 + 5 added upto 10.

Explanation: Ending numbers 3 + 7 add upto 10. So the first rule is satisfied. And both the number on left hand side are same i.e. 2. First thing you do is, you multiply 3 x 7 = 21. And then like earlier you multiply 2 with the next higher digit number i.e. 3. And we get 2 x 3 = 6 on the left hand side. Final answer is 621.

Weird Theorems Hairy ball theorem The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres.

For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0. In other words, whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball.

The theorem was first stated by Henri Poincaré in the late 19th century.

This is famously stated as “you can’t comb a hairy ball flat without creating a cowlick”, or sometimes “you can’t comb the hair on a coconut”. It was first proved in 1912 by Brouwer.

Stable marriage problem In mathematics and computer science, the stable marriage problem (SMP) is the problem of finding a stable matching between two sets of elements given a set of preferences for each element.

A matching is a mapping from the elements of one set to the elements of the other set. A matching is stable whenever it is not the case that both: a) some given element A of the first matched set prefers some given element B of the second matched set over the element to which A is already matched, and b) B also prefers A over the element to which B is already matched In other words, a matching is stable when there does not exist any alternative pairing (A, B) in which both A and B are individually better off than they would be with the element to which they are currently matched.

Six degrees of separation

Greedy algorithm

Six degrees of separation refers to the idea that everyone is on average approximately six steps away, by way of introduction, from any other person on Earth, so that a chain of “a friend of a friend” statements can be made, on average, to connect any two people in six steps or fewer. It was originally set out by Frigyes Karinthy and popularized by a play written by John Guare.

A greedy algorithm is an algorithm that follows the problem solving heuristic of making the locally optimal choice at each stage with the hope of finding a global optimum. In many problems, a greedy strategy does not in general produce an optimal solution, but nonetheless a greedy heuristic may yield locally optimal solutions that approximate a global optimal solution in

The Bernoullis were a family of traders and scholars from Basel, Switzerland. The founder of the family, Leon Bernoulli, immigrated to Basel from Antwerp in Flanders in the 16th century, fleeing Spanish oppression.

The Bernoulli family has produced many notable artists and scientists, in particular a number of famous mathematicians in the 18th century: •Jacob Bernoulli (1654–1705; also known as James or Jacques) Mathematician after whom Bernoulli Numbers are named.

•Nicolaus Bernoulli (1662–1716) Painter and alderman of Basel.

•Johann Bernoulli (1667–1748; also known as Jean) Swiss mathematician and early adopter of infinitesimal calculus.

•Nicolaus I Bernoulli (1687–1759) Swiss mathematician.

•Nicolaus II Bernoulli (1695–1726) Swiss mathematician; worked on curves, differential equations, and probability.

•Daniel Bernoulli (1700–1782) Developer of Bernoulli’s principle and St. Petersburg paradox.

•Johann II Bernoulli (1710–1790; also known as Jean) Swiss mathematician and physicist.

•Johann III Bernoulli (1744–1807; also known as Jean) Swiss-German astronomer, geographer, and mathematician.

•Jacob II Bernoulli (1759–1789; also known as Jacques) Swiss-Russian physicist and mathematician.

Devices and ideas named for members of the family •Bernoulli differential equation •Bernoulli polynomials • Bernoulli distribution •Bernoulli process •Bernoulli number • Bernoulli trial •Bernoulli’s principle

a reasonable time.

For example, a greedy strategy for the traveling salesman problem (which is of a high computational complexity) is the following heuristic: “At each stage visit an unvisited city nearest to the current city”.

This heuristic need not find a best solution but terminates in a reasonable number of steps; finding an optimal solution typically requires unreasonably many steps. In mathematical optimization, greedy algorithms solve combinatorial problems having the properties of matroids.

Mapping a fixed point For fifty years, mathematicians have grappled with a so-called “fixed point” theorem. Take a map of the world. Now put it down on the ground in Central Park, against a rock on Mount Everest, or on your kitchen table; there will always be a point on the map that sits exactly on the actual physical place it represents. Obvious? Not for mathematicians. A more complex theorem, called a “fixed point theorem,” has eluded them since 1963.

“Some ideas seem evident to the human mind, but in reality involve complicated concepts that are difficult to demonstrate mathematically,” says Nicolas Monod. It turns out that the answer was there all along, simple and elegant. To reach it, the team of mathematicians had take a different approach to the problem. Their discovery will impress their fellow mathematicians, of course; but on the longer term, it also will be of interest to physicists and economists.

ANOKHE NUMBERS

Carmichael Numbers We know Fermat’s Little Theorem: a^(p1)≡1(mod p), where a is any positive integer and p is a prime number co-prime to a. However, there are also composite numbers p that satisfy the relation.

These are called the Carmichael Numbers. E.g. 561 Abundant Numbers In number theory, an abundant number or excessive number is a number n for which the sum of divisors σ(n)>2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)>n. The valueσ(n)2n (or s(n)-n) is known as the abundance.

Ex-12, 18, 20, 24.....

Deficient Numbers In number theory, a deficient number or defective number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is called the number’s deficiency.

Ex-1, 2, 3, 4, 5, 7, 8, 9.... #this is not done.remove perfect numbers by some mersenne numbers or happy nmbers…