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GEEK GAZETTE AUTUMN 2017 COVER STORY : THE SHAPE OF THINGS TO COME

BIG STORY Hacker Emblem Page 26

INTERVIEW Understanding Planetary Sciences Page 12

EDITORIAL A Resonance Through the Ages Page 6

Association for Computing Machinery Indian Institute of Technology Roorkee

ISSUE 18


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CONTENTS

16 26 28

The Quest for Elegance

Hacker Emblem

YouTube Recommends

Obscure Mathematical Theorems

05

A Resonance Through the Ages

06

Hard Coding Cognition

08

Is There Any Great Mathematical Truth?

10

Understanding Planetary Sciences (Interview)

12

Why Do We Do Math?

14

Cypherpunks and the Surveillance Dystopia 18 The Shape of Things to Come

20

Figuring Out Competitive Coding (Interview)

24

Nature of Truth in Literature

32

All New Particles, No New Physics

34

From Donuts to Coffee Cups

36


W

hen we first discussed the possibility of having mathematics as the central theme of our magazine, the idea seemed to be almost absurd. Mathematics was too vast a field to be covered in a few articles. In addition, we had apprehensions whether our readers were already introduced to the ideas and the intuitions of the subject. Our personal judgements were similar too, irrespective of whether we loved or hated the subject, we felt that we knew maths. Fortunately for us, we were proved wrong. As soon as we started working on the issue, the initial task was to get beyond the layers of specificities of different fields of mathematics and to understand it as a comprehensive whole. Mathematics, at its core, is an idea and in this idea lies its beauty. This issue of Geek Gazette is all about that.

GEEK SPEAK

Our article, The Quest for Elegance, is a discussion on the existence of elegant proofs and the very meaning of mathematical beauty. Our cover story titled The Shape of Things to Come cogitates on the possibilities of existence of higher dimensions and our understanding of the same. Numbers and theorems are the syllabaries of mathematics and the articles on π and Obscure Mathematical Theorems are an attempt to understand them better. Why do we do Math? explores the motivations behind true pursual of mathematics, customarily believed to be a challenging and abstract subject. A Resonance Through the Ages embodies our convictions of mathematics being an art form by drawing out the four ages of mathematical revolution similar to the ones that we had in music. Articles on the Hacker Emblem and the Cypherpunks advocate the need of having a hacker culture and understanding the implications of lack of privacy. The article All New Particles, No New Physics is an enthusiast’s interpretation of the role played by the LHC post the discovery of Higgs Boson. An ironic yet important concept of fiction—truth is discussed in the article Nature of Truth in Literature. For the professor interview, we talked to Dr Nachiketa Rai who exuberantly introduced us to the field of planetary sciences while for the student interview, it was the ingenious trio of Vaibhav, Saharsh and Adarsh with whom we talked, at length, about Competitive Programming. The idea of this magazine has moved through a lot of stages in its journey till now, right from its conceptualization to printing. Now that it is sitting in your hands as a tangible product, it is you, our readers, whom we have entrusted to conclude this grand tour. We really hope that you have fun reading this issue. Team Geek


Obscure Mathematical Theorems

A

xioms, theorems, and logic. Mathematics has always presented itself as a field grounded on the lines of objectivity, pillared by logic such that it transcends individual civilizations and specific languages. But there still lie loopholes and obscurities which disturb its en rapport with perfection. These are often the paradoxes and esoteric theorems, which from time immemorial, have confounded the minds of many. Map colouring often relates itself with a feeling of nostalgia, taking us back to our Elementary Geography classes. Francis Guthrie, while colouring the map of countries of England, might have got aroused by the same feeling, but he saw a little far beyond when he formulated the ‘Four Colour Theorem’ in the year 1852. The theorem states that given any separation of a plane into contiguous regions, thus, producing a figure called map, no more than four colours are required to colour the regions of the map, so that, no two adjacent regions have the same colour. For around 125 years, during which the theorem remained unsolved, it witnessed an array of false proofs, counterexamples, and parallel conjectures, all pertaining to the complexity of the proof that the theorem holds. The research to find an aesthetically pleasing solution still goes on. While some of the paradoxes are actually fallacies where the error is hard to spot, others, like the Banach-Tarski Autumn 2017

paradox that arise from a seemingly incomplete set of axioms, are correct but may seem absurd to some. Put simply, the Banach-Tarski paradox is a theorem that talks about how it is mathematically possible to decompose a three-dimensional sphere in a certain way such that you’re left with two identical spheres which are the exact copies of the original sphere. Some mathematicians take it further and state that it is possible to do the same with any solid object. This theorem, if practical, would solve all our problems related to lack of resources. But, not only does the theoretical proof make certain questionable assumptions, the theorem also contradicts elementary geometric intuition. The questionable assumption is the existence of sets of undefined volume. This assumption is unique to this theorem. Another example of a seemingly absurd yet simple paradox bugging an entire group of mathematicians is a paradox existing from the early days of set theory. One of the first developments in set theory called as Naïve set theory used any natural language to describe a set or any operation on that set. Developed by Gottlob Frege, the system assumed that any definable collection could be called as a set. And this assumption led to a paradox which became popular as the Russell’s Paradox. A popular and easily understood version of it is the Barber’s Paradox. The Barber’s Paradox assumes a hypothetical city with one barber and a general rule that the barber shaves only those who do not shave themselves. The question here is—Who shaves the barber? A contradiction shows up no matter the answer, since if he doesn’t, then he should, and if he does, then he shouldn’t. This is also known as an antinomy and is considered an extreme form of paradox. Perhaps, there will always be some paradoxes which have no universally accepted solution, self-contradictory theories that most mathematicians drop in disgust because they prove that mathematics cannot be consistent. As for Russell’s antinomy, it cannot be as undesirable for anyone as it has been for Gottlob Frege; for it shook the entire foundation of his work on Naïve set theory even before it was published.

05


A RESonance through the ages

A

ccording to the Greek mythology, when Hermes stole fifty of Apollo’s cattle, his anger knew no bounds. He took Hermes to face judgment in front of his father Zeus, the king of the gods, who commanded Hermes to return the cattle. Hermes, the messenger of the gods, with mischief in his eyes and a grin on his face, took out his latest invention, the Lyre. He played music on the Lyre that not only appeased Zeus’s anger but also mesmerized Apollo, the god of logic and reasoning. He offered Hermes his forgiveness in exchange for the Lyre. The music Apollo heard that day appealed not only his heart but also his rationale. The Lyre offered him the answer to a mystery, the mystery of how higher logic and reasoning could be translated into a language of their own. From that day forth, Apollo came to be known as the god of healing, poetry, rational thought, and music.

The American academic Jim Henley put forward the theory that like art, music, and literature, mathematics has also had a very definite evolution through time. Interestingly, the stages of evolution in mathematics coincide with that of music, while literature and art seem to have progressed almost a generation earlier. This further strengthens the belief that mathematics has had an impact on music and vice versa, on a much larger scale than one could have imagined.

Much like Apollo, academics like Pythagoras, Aristotle, and Plato were fascinated by music theory, so much so that the mathematics in ancient Greece comprised of four sections: Number Theory, Geometry, Astronomy, and Music. They considered music as a way to open their minds to an abstract way of thinking, a trait we more commonly attribute to pure mathematical thought and intuition.

Few people know that the mathematical genius was also the first real music theorist. He found a relation between musical intervals and the ratios of integers, purely on the basis of his experiments with pipes, strings, and hammers. Pythagoras claimed that all the observations confirmed his initial hypothesis—music intervals are directly related to some very specific fractions. He started by establishing the frequency ratio for the octave and the fifth to the fundamental note.

06

The Renaissance was a period in human history which witnessed a revival of ideas in the fields of art, music, literature, and mathematics. The work of the ancient Greek writings on music and mathematics were given their long due recognition. The one theory that was revived and studied extensively in this period was the theory of music intervals by Pythagoras.

Geek Gazette


And by using simple mathematics, he obtained much more. Music theorists, even today, are baffled as to why the human brain comprehends these frequencies related by fractions of small integers as “good” music.

x

Jean-Phillipe Rameau, music composer, sound theorist, and one of the finest minds of the 18th century (the time which also saw the genius of Linnaeus and Fahrenheit), decided to enhance on Pythagoras’s theory of music intervals and came up with an important deduction which further bridged the gap between music, a creative science, and mathematics, a logical and deductive one. He discovered that harmonic frequencies are integral multiples of fundamental ones, integral being key here. Quite astonishingly, it is believed that he single-handedly started the Baroque era when he composed a much controversial opera, ‘Samson’ in 1734. And so the term ‘Baroque’ was derived from the Portuguese word ‘Barocco’, meaning ‘misshapen pearl’. In his later work, Rameau argued that since the fundamental objects in mathematics are derived from sequences of positive integers, and since this sequence continues in music, mathematics is itself a part of music! A bold claim indeed, in the face of mathematicians of the 18th century. Quite predictably, the new “free” mathematics faced harsh criticism and generated quite some tension among mathematicians, most of it revolving around the new mathematical language formalised by Fermat and Descartes—Algebra. With the invention of the microphone, a much-improved typewriter and the Morse code, the notion that music should cross national boundaries and be universalized was popularised in the Classic era. This is perhaps, best exemplified by the works of Mozart and Beethoven. Henle argues that Classic Mathematics also had the same effect. Mathematics was no longer locked in the desks of mathematicians; it was enjoying widespread application across Europe and beyond. The music from the Classic era, especially that of Bach and Beethoven, has always had a strong pull on the Mathematically inclined minds. How did Beethoven write many of his beloved compositions after going deaf for the majority of his career? It is believed that Beethoven was a master of sounds at not only an emotional and creative level, but also at a mathematical Autumn 2017

one. The first half of measure 50 of ‘Moonlight Sonata’ consists of three notes in D major, separated by intervals called thirds that skip over the next note in the scale. By stacking the first, third, and fifth notes—D, F#, and A—you would get a harmonic pattern known as a triad. These three frequencies together create ‘consonance’, which sounds naturally pleasant to our ears. The music of the late 19th and 20th centuries can be seen as a form of revolt against social and cultural norms. The horrors of World War I also had an undeniable influence on music, and some composers began exploring darker, harsher sounds. They explored farther and deeper than they ever had before resulting in many new art forms to come up including musical comedies and musical theatre. Unsurprisingly, mathematics followed the same trend in the Romantic era and focused on two main ideas—the impossible and the infinite. The ideas about infinity had existed since as far back as 450 B.C. But in the Romantic era, they were neither shunned (as by the ancient Greeks) nor criticized (as in the Baroque era); they were openly embraced. By simply employing pre-discovered tools more imaginatively, many mathematical dilemmas were solved. Like, if we disregard the first number of the Fibonacci sequence, we get a relation with the key frequencies of the musical scales i.e. 1, 2, 3, 5, 8, 13; the very musical scales invented by Pythagoras, who condemned any study on infinities or the impossible. We face the Modern era and Henle argues it to be impossible to recognize the patterns and trends in an era by a person who lives in it. To him, the past would appear structured and harmonious while the present would seem chaotic. By having followed the progress of music and mathematics through these four eras, the notion that we find constant throughout is ‘change’. A change of perspectives, a change of practices and a change of beliefs. This compels us to ask the question, if music can be studied and understood as a scientific concept, would mathematics too, as Rameau had argued, be considered as a purely artistic invention in the coming ages?

07


HARD coding

cognition A

ll of us, at some point of time, have asked ourselves—why do we think and behave in a particular way? In essence, each individual’s thought process differs from others. This, actually, is the thing that distinguishes us from the rest of the crowd and validates our individuality. But, why is there such a stark difference in the way we think? To answer this question let us consider a simple example, when someone yells the word ‘fire’, our mind automatically associates it with being dangerous and we are filled with a feeling of caution. But, to an infant, who is oblivious to its dangers, it seems like a very intriguing entity which creates in him/her an urge to touch it, feel it and to comprehend its very concept. When the infant actually goes forth and experiences getting burnt, he/she experiences feelings of anguish, which then registers the fact that fire is something that he/she must stay away from. This shows how human thoughts are shaped, through experiences and environment, be it directly or indirectly. The above example, though seemingly primitive, forms the base for other important ideas that can essentially help us in unravelling the mysteries behind the cognition of human thoughts and mental processes. The varied influence of memories formed during infancy as compared to the ones formed during adolescence or adulthood; impact of good childhood 08

memories as compared to grim ones; or the effects of bad parenting are some questions whose answers are continuously being searched for. Another set of questions deal with the influence of our subconscious thoughts on our belief systems and how the conscious decisions taken by us throughout our lives affect our subconscious behaviour. Most of us don’t recall the memories of incidents that have taken place in our lives during the periods of infancy. For most people, their earliest memories are from around the same time when they start learning their first language. This indicates that only after a certain age, the capabilities of our brain become strong enough to store detailed biographical memories. Does this mean that the experiences that we have in our ages of infancy (ages of 0-2 years) do not play any role in the structure of our behaviour? Though recalling memories of our early childhood is difficult for most of us, some psychologists believe that it might be a problem of accessibility rather than availability. The chances of people preserving their episodic memories from ages as early as 2 years increases considerably when it is associated with emotional events such as birth of a sibling or hospitalization. Just as we might not remember learning how to walk, even though the process gets ingrained in Geek Gazette


our subconscious, psychologists predict that memories of our childhood can affect the structure of our thought process in various ways. While the effects of memories from infancy on the character traits of an individual is something which forms the grounds of ongoing studies, it is distinctly understood that experiences from late childhood and adolescence are very instrumental. Children subjected to bad parenting or negligence during their childhood might have adverse effects on their behaviour which might even get carried to adulthood. An interesting pedagogical experiment carried out by a Hungarian psychologist Laszlo Polgár on his daughters Judit, Susan and Sophia explored the possibilities of exposing children to controlled environments in order to facilitate character and brain development in a determined way. Along with his wife Klara, Laszlo trained the girls in chess. It was the perfect activity for their experiment—it was an art, a science, and like competitive athletics, yielded objective results that could be measured over time. The girls were homeschooled by him and Klara in languages and advanced mathematics, and of course, several hours of chess lessons, with minimal focus on activities deemed ‘wasteful’, such as television. They spent their leisure time playing blitz chess against each other—while blindfolded. The results, as it turned out, were beyond astounding. The oldest sibling, Susan Polgàr, won the Serbian under-11 girls chess tournament with a perfect score at the age of 4. She went on to earn the title of a Grandmaster at the age of 21, becoming one of the only 35 female grandmasters in a mostly male-dominated community (their number being 1507). The youngest of the family, Judit Polgár, in many ways ended up as the most successful result of Laszlo’s experiment, achieving the title of Grandmaster at the age of 15 years. She is regarded as the strongest female chess player in the world, and remains the only woman to defeat a reigning world number one player, Garry Kasparov, in 2002. With the above example, it has become obvious that the environment that a human grows in and the experiences that come with it have a great impact on how his/her brain is ‘coded’. The ‘inputs’ that humans receive from their environment and experiences can mould their personalities to levels that may be considered superhuman. This means that given a Autumn 2017

proper environment for the upbringing of an infant, one can ‘code’ him/her in any way that they want to, going as far as erasing fundamental emotional concepts, such as joy, sorrow, rage or even affection. Since we can ‘code’’ both the cognitive and the emotional aspect of the human mind, one can essentially make whatever sort of being he/she wants if they were granted permission to govern the way in which the infant is brought up. Hence, the human mind is indeed ‘programmable’! The words ‘code’, ‘input’ and ‘program’ are all common terms in the field of computers. But based on the above explanation, they also fit in very well when we try to explain human behaviour and psychology. So could there possibly be a connection? Artificial Intelligence serves as the missing link. Certain correlations can be derived between AI and the programmable aspects of the human psyche. After all, the ultimate goal of AI researchers is replicating a humanesque intelligence within a machine. Of course, the practice of controlled environments and inputs is not strictly ethical, as far as human psychology is concerned. The Polgár sisters turned out to be a unique coincidence, as after all what are the chances that three girls destined for stellar achievement would be born to a man convinced that geniuses are made? Also, the chances of achieving a perfect control environment for humans is grim, if at all possible. The number of stimuli that humans receive and analyse as a precursor to their every decision is truly enormous. Determining all of them, in itself, is a very difficult task; controlling each of them takes things to the next level altogether. Conscience, beliefs and societal pressures, all play their little roles in our decisions. Isn’t this also the thing that makes humans interesting and provides them with their unpredictability? A game of Chess against a human is always more fun than the one against a computer; even though the challenge associated with beating a computer would be considerably high. Regardless of all this, human brain is an excessively beautiful and sophisticated entity—a biological genius. Attempts like those of the psychologists to understand infantile amnesia or the experiment conducted by Laszlo Polgár are only, at their best, tiny little steps towards understanding the processes of this superstructure. 09


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4811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 3724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118 4807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940 1320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713 9960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617176545 4159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812 4811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 37245 87006646315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611 egarded as one of the five most important Archimedes, who provided a theoretical calculation of π 79310 51185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767 52384674818467669405 mathematical constants, π can be strictly defined by using a series of polygons inscribed in a circle and 41592 65358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172 as a geometric quantity—the ratio of the circumference another circumscribing it. From there on, many prolific 5359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607 6024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261 of a circle to its diameter. This definition explains its mathematicians like Ptolemy, Euler, Leibniz, and 7931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748 ubiquitous nature in geometrical and trigonometrical Ramanujan have moved beyond polygons and 8467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813 2977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142 formulae. However, π is also an irrational, discovered numerous formulations of π which calculate 6171745256 transcendental number, which means that it cannot be it as accurately as upto first trillion digits. 4159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812 4811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 expressed as the ratio of two integers and can never be 3724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118 4807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940 the solution of a polynomial with rational coefficients. As it turns out, when mathematicians studied the first 1320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713 However, in modern mathematics π is calculated by trillion digits of π on a computer, they found that the 9960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617175645 4811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 using the spectral properties of the real number system digits were statistically random in the sense that the 3724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118 which has no relation to geometry. probability of each digit occurring appears to be 4807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940 1320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713 independent of what digits came just before it. 9960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617172535 The efforts to calculate π as accurately as possible, and Furthermore, each digit appears to occur essentially 4159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812 4811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 the history of mathematics, both are almost the same one-tenth of the time, as would be expected if the digits 3724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118 4807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940 age—the origin of which dates back to 1600 B.C. when had been generated uniformly at random. Apart from 1320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713 the Egyptians and Babylonians approximated the value this, π plays a significant role in the field of science. It 9960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617176545 4159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812 of π to one percent of the true value. Then came along often occurs in equations involving complex numbers, 4811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412 3724587006646315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118 4807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940 10 Geek Gazette 4807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940 1320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713

IS THERE ANY GREAT MATHEMATICAL TRUTH?

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vector calculus, Fourier transformation, Riemann Zeta functions and many more. This appearance of π in areas which are not related to geometry gives it sort of a magical connotation among mathematicians. There are many other mysterious instances of an appearance of π in seemingly unrelated equations of the physical world, and many results involving infinite series which converge to π. It is theoretically possible for numerical coincidences to occur in mathematics. But, is π one of such coincidences? Or maybe, there is not any mystery involved in it. Fundamentally, it’s all resulting from the well-understood connection to circles.

It is theoretically possible for numerical coincidences to occur in mathematics. But, is π one of such coincidences? The infinite series began to play a role in Mathematics only in the second half of the seventeenth century. This led to the development of the series formula of π. The arctan series was one of the first equation that was developed in the study of infinite series and, in fact, before the methods and algorithms of calculus were fully developed. It was derived in the West in 1671 by James Gregory from the formula for arctan(x ) and slightly later and independently by Gottfried Leibniz. However, the same formula was discovered long before in the 1300s by the great Indian mathematician whose identity is not definitely known. The results exactly illustrate the extraordinary way that maths can link patterns together. It connects two different concepts of mathematics, the geometry linked to the number π and the simplicity of the odd numbers. Leibniz found the relationship between arithmetic formula of a quadrature and infinite series of odd numbers while he was trying to find a process of finding the area under a curve. When he applied his ideas on the quadrature of a circle, he discovered the famous arctan formula to calculate the area of the circle, that is π/4. This experience gained by him in his calculations also led him to discover fundamental rules of infinitesimal calculus. Thus, whenever we prove that a certain Autumn 2017

expression equals to π, then, by definition, we have discovered and explicitly laid down the connection between the new expression and the already well-known definition of π. And whenever we face a scenario where the occurrence of π is unexplained or mysterious, it might be due to our limited understanding of a subject or our inability to explore the underlying mathematics of the equation. However, there are other cases when values of empirical coefficients are calculated and found to be equal to other stuff for no immediately apparent reason. People then toil to prove that the relationship is indeed correct, which often reveals the underlying reason for it to be true. Perhaps the most famous recent instance of this is Monstrous Moonshine, the amazing conjecture that there exist relationships between the monster group which comes from group theory, and the j-function which is one of the most powerful tools in number theory—two areas in mathematics which previously had absolutely nothing to do with each other. This was first observed numerically, and almost 15 years passed before Richard Borcherds explained why this happens and won a Fields medal for this achievement. Relations and deep-rooted connections exist everywhere in mathematics. Fundamentally, there is not any real mystery involved. Whenever there is π, there is a circle. It’s only that for most of us, π is the first irrational number that one learns about. Quite prosaic as it may sound, but the reason why mathematicians are obsessed with π is not that of its mysterious appearances, but because of its simple representation in geometry and its persistent nature in common calculations. The randomness in the infinite digits of π exists in other irrational numbers as well, and maybe, any other number can also have a large number of formulations. One can say that there is a great truth behind π. But that truth is like any other truth in mathematics—simple and inevitable.

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Understanding planetary scienceS WITH

DR NACHIKETA RAI ASSISTANT PROFESSOR EARTH SCIENCES Planetary Science is a field that looks to unravel the mysteries behind the structure and formation of planets and other astronomical bodies. Dr. Nachiketa Rai, who recently joined the Earth Science Department at IIT Roorkee, says that it is the study of these celestial bodies that is helping us understand the formation and evolution of our Earth better. Though busy working on setting up his research lab, Dr. Nachiketa took time out to talk to Geek Gazette and introduced us to the mesmerising field of planetary sciences. GG: Hello Sir! Thanks for talking to GG. Tell us something about your field of specialization and the research that you do in it. NR: Basically, my field of specialization is planetary sciences and my research focuses on understanding how different rocky bodies in the solar system are formed and evolved. Our solar system, as we know, started with a spinning disc of gas and dust around the sun. We started from a fairly homogeneous composition in the solar disc and then, ultimately, ended up with these beautifully layered planets with the central iron-rich core surrounded by a rocky silicate mantle and a crust. I combine an experimental approach with analytical geochemistry to try and constrain these processes. I complement this with the analysis of extraterrestrial rocks such as lunar samples, meteorites from other planetary bodies or asteroids as they contain records of the processes that took place in the early days of the solar system. GG: How did your journey begin in this field? NR: I got really interested in this field while I was doing my Masters Dissertation. I was doing an experimental project on creating synthetic diamonds in laboratory for industrial use. The aim of the project was to improve the yield by putting graphite into diamond using different techniques. I was also working on improving the quality of the defective diamonds in the lab. You have diamonds that have defects that cause them to be straw coloured or dull. So, I was working on 12

experimental methods to convert the low value diamonds to canary yellow and pink colours which would enhance their market value, and it was during these experimentations that I was drawn to the field. GG: Is planetary science a very new field? How has it progressed over the past few years? Are we seeing any radical or groundbreaking results? NR: Human beings have always looked at the moon and wondered about the mysteries that lie in there. Under a more formal consideration, I would say that from the time of the first accurate measurement of the Earth’s age, 4.55 billion years, in the mid-1950s, which enabled to establish a timescale for the Earth, for life, and for the universe, as well. Ever since, there has been a consistent growth in the field. The defining moments include events such as the validation of the plate tectonic theory or the landing of man on the moon for the first time. Obtaining samples from the moon provided us a glimpse of a rocky body quite different from the Earth. The previous decade, which has been called as the “race for space�, has been particularly encouraging for the planetary science community as more and more countries including India are planning and sending planetary missions. GG: For people interested to work in the field of Planetary Sciences, what are some opportunities that are available in the industry and the academia? NR: There are plenty of opportunities present for Geek Gazette


Planetary Scientists in both the domains. Assuming you want to send out an interplanetary mission, you must decide the materials based on the conditions present on that particular planet. We can take the example of the Venera Missions to Venus where a few probes landed on Venus but were not able to sustain for more than few minutes just because of the adverse conditions present there. This is where planetary scientists become important. Material designing, system designing, technological developments do provide a lot of scope in the industry, as well. Good Partnership in R&D between the industry, academia, government agencies and research labs, is required to improve these exploration possibilities, and is essential for increasing the success rate of planetary missions right from the launch to maintaining the mission, collection, interpretation and processing of data, so there definitely is plenty of scope in the industry as well. GG: Do we have a good research community for Planetary/Earth Scientists here in India and how do they compare to the other countries that you’ve lived in? NR: We are still in a very nascent stage, working between groups or people with different expertise. Outside, however, there are very active groups of both Earth and Planetary Scientists who collaborate between different countries and disciplines. We still need to work on that here. For example, take the planetary science community in Europe/US, they have biannual/annual conferences where researchers from different institutes share their research and receive crucial feedback. These conferences are often sponsored by the respective Space Agencies. Such a culture needs to be developed here, in India as well. GG: Tell us about your experience at the National History Museum, London. What was your work there? NR: The NHM, along with being a museum, is also a very well acclaimed research institution. It houses some of the best, state-of-the-art laboratories that are used to pursue research about the solar system, planet formation, earth’s geology and also life sciences. I was working on a project involving ureilites which are one of the most complex and enigmatic meteorites. They are believed to be the only samples (in addition to terrestrial mantle rocks) in our collections that are from the mantle of a differentiated planetary body in our solar system. I Autumn 2017

was working on several aspects of these meteorites right from isotopes, trace elements to the 3D analysis of the samples investigating the distribution of micro diamonds within these rocks. GG: Can you tell us about the specific research that you are pursuing right now? NR: It has almost been a year since I arrived here and I’m trying to secure funds to develop my lab which would be primarily for analytical geochemistry. The lab would be used to analyze meteorites and extraterrestrial rocks. Now, I have also started research projects in the field of aqueous geochemistry, investigating groundwater contamination, etc. I have also obtained some new meteorite samples on which some of my students will be working on. One of my Ph.D. students is working on the problems related to differentiation in small rocky bodies in the solar system. We’re also doing some modelling work based on computer simulations investigating the observed diversity in the compositions of rocky bodies in the solar system, which will be followed by high temperature–high pressure experiments and geochemical analysis. GG: Which are some of the best universities in India and abroad in which students from Earth Science or students specifically interested in Planetary Sciences can look forward to join? NR: There are Indian institutes or research organizations, such as the Physical Research Laboratory in Ahmedabad, which are doing quite a lot of good planetary science research. They have some very sophisticated analytical instruments. TIFR is doing research on the solar system. The US, Europe as well as Japan have a large number of advanced research centers with high temperature and pressure labs, as well as geochemistry labs. It depends which area of research within Earth and Planetary Sciences one is interested in. The VU University at Amsterdam is one of the leading experimental centers for planetary research in mainland Europe, there are several UK Universities–Bristol, Oxford that have top rated Earth Sciences departments. There are several leading research labs in the US and Japan also. There are plenty of opportunities to explore, provided that the intense passion and strong motivation exist.

13


WHY do WE DO MATH? “It is a melancholy experience for a professional mathematician to ind himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done.”

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hese are the opening lines to Godfrey H. Hardy’s noteworthy 1940 essay, titled A Mathematician’s Apology. Throughout his life, Hardy held an unbridled affinity and passion towards pure mathematics, and these lines convey the frustration he felt at being forced to write a book on his career rather than actually doing mathematics. The essay served as a justification for the years of life Hardy had devoted to mathematics. At the age of 62, he had realised that he was nearing the end of his career, as the aptitude he had for mathematics in his younger years was simply gone. The questions he seemed to answer have surely occurred to everyone at some point or the other—Why do we pursue mathematics? And is mathematics worth the efforts of pursuing? His answer to the first was surprisingly ordinary—Firstly, a man does what he does best simply because he is good at it. And secondly, there is nothing a man can do extraordinarily well, and he simply does what comes his way. As for the second question, the answer is much more complex. Mathematics, to Hardy, was a perfectly harmless, unprofitable, and innocent 14

occupation. He was convinced that true mathematics is worth pursuing for the beauty it possesses, not for its applications in the real world. This beauty, which cannot quite be expressed in words, was something that Hardy compared to painting or poetry. “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. A painter makes patterns with shapes and colours, a poet with words.” Hardy repeatedly expresses how the most beautiful mathematics is that which has no practical application, for example, the simple fact that 8712 and 9801 are the only numbers which are multiples of their respective reversals (8712 = 4 � 2178, and 9801 = 9 � 1089) would probably never have a significant impact in the field of mathematical study, yet it is simple and aesthetic. It is almost analogous to a literary device, which in its own way, beautifies a piece of literature. This infallible beauty is often not apparent to everyone to the same extent. Where one might see a garble of numbers, a few unique minds may see something dazzling. In fact, the innate intuition of a pure mathematician in his goal towards the pursuit of beauty has been scientifically proven to generate justifiable Geek Gazette


G. H. Hardy (1877-1947)

results. In a study conducted in 2014, researchers employed functional magnetic resonance imaging (fMRI) to display the activity of brains of 16 mathematicians, at a postgraduate or postdoctoral level, as they viewed formulas that they had previously judged as beautiful, mediocre or ugly. The results of this analysis showed that beautiful formulas stimulated activity in field A1 of the medial orbitofrontal cortex (mOFC), which researchers have identified as the seat of experience of beauty from other sources, namely art or music. Legendary French mathematician and polymath Jules Henri Poincaré, in an essay titled Mathematical Creation, has eloquently explained how “sudden illumination” is a valuable route to ideation in the field. Turning one’s mind away from a problem, only to be stricken with a whole new perspective for its solution, is the manifestation of beauty and creativity amongst scientists, and perhaps it is the elation on the emergence of these epiphanies that drive mathematicians to pursue it. As Poincaré said, “These sudden inspirations never happen, except after some days of voluntary efort which has appeared absolutely fruitless and whence nothing good seems to have come, where the way taken seems totally astray. These eforts then have not been as sterile as one thinks; they have set agoing the unconscious machine and without them it would not have moved and would have produced nothing.” Ineffable beauty is not the only characteristic which empowers mathematics. It opens an entirely new school of philosophy dealing with its fundamentals. Schools such as Platonism, which views mathematics as a composition of independent, abstract entities and its contrast, Empiricism, which outrightly denies that mathematics can be known a priori at all, have stemmed from the field, with the goal of finding the purpose of mathematics in our lives. The logical and structural Autumn 2017

nature of mathematics itself makes these studies unique and interesting among its philosophical counterparts. And of course, the way mathematics works out—physicist Max Tegmark argues that the reason that mathematics works so well, and so elegantly, in physics is because the universe (or, more properly, the multiverse) is, ultimately, just mathematics—mathematical structures and the relations that connect them constitute the ultimate irreducible “stuff” of which our world is made. This phenomenon is not limited to the field of physics, but every field of study imaginable. Mathematics helps us comprehend that which lay beyond our grasp—in other words, it allows us to make sense of the world.

The universe is, ultimately, just mathematics—mathematical structures and the relations that connect them constitute the ultimate irreducible “stuff” of which our world is made. It might seem at this point that one needs to be born a prodigy to perceive this notion of “beauty” that has been discussed continually by the mathematicians and philosophers. This, albeit true to some extent, should not be seen as a hindrance. The simple thrill of correlating mathematical models to reality is enough to provide a brief glimpse into what mathematics has to offer the world. “Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is”. —Paul Erd�s

15


the quest for elegance

Keats, in one of his poems—Ode on a Grecian J ohn Um, claimed that beauty and truth are the same. To

mathematicians, great theorems and great proofs, such as Euclid’s proof of the infinitude of primes or proof to the Pythagorean theorem, have about them what Russell described as “a beauty cold and austere”, akin to the work of a poet or a musician. But, beauty is a slippery concept, much more subjective, and it is difficult to frame its applicability within mathematics or in any other field for that matter. Despite all this, mathematicians do not abandon their talk of beauty. Whatever it may be, for most of them—beauty in their discipline is elegance and simplicity. It lies in the connections one can draw between seemingly separate ideas, and an elegant proof highlights this beauty as well as proves the result. An honest proof is one which seems to hold true in every aspect, and in this way ensures truth. It is precisely these qualities that delight mathematicians in a proof; surprise and inevitability. Elegant proofs are a rare genius, and we are compelled to seek them, for their beauty and value. But, do we have a reason to believe them to be ubiquitous? Or, as it 16

seems to be the case, whether there exists a wonderfully simple and elegant proof to every theorem in Mathematics. To cite an example, Fermat’s Last Theorem is a problem that had been lingering on since the 17th century. It is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The theorem states that it is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. In the scribbled note, Fermat claimed to have discovered a proof to one of the most beautiful and simple mathematical equation. The full text of his statement read: “I have discovered a truly marvellous demonstration of this proposition that this margin is too narrow to contain.” It was called a theorem, despite the fact that no other mathematician was able to prove it for hundreds of years. In 1993, Andrew Wiles was able to prove the theorem after giving his ten years to the problem. However, several holes were discovered in the proof Geek Gazette


shortly thereafter, and it took two more years for Wiles to publish the complete proof. Since all of the tools which were eventually brought to use on the problem were yet to be invented in the time of Fermat, it is interesting to speculate on whether he actually was in possession of an elementary proof of the theorem. While some believe that Fermat, being an amateur mathematician trying to understand number theory, himself did not have a proof of the theorem, others believe that there exists a much simpler mathematical proof which is beyond our reach. There is no significant evidence for either of the cases. However, if the ABC conjecture is believed to be true, then the proof of Fermat’s last theorem would reduce to a few, much simpler lines. On the other hand, the two papers that Wiles published, together are 129 pages long and too complicated to be called beautiful. Maybe, there is no simpler proof for this theorem. Of course, mathematicians can always hope and believe otherwise.

Paul Erdős (1913-1996)

Paul Erd�s, one of the most prolific mathematicians of the twentieth century, was famous for his eccentric ways. In his own idiosyncratic vocabulary, music meant noise, and divorce was liberation. Before his death, he had managed to think about more problems than any other mathematician in history. He had no wife or children, no jobs, no hobbies, not even a home, or any other factor that could have diverted his attention from mathematics. Despite being an atheist himself, he believed in the existence of “The God’s Book” of math proofs, which contained the most elegant proofs to every mathematical theorem. According to him, there exists a solution to every problem, though some of them are yet to be discovered. In his own words, “You don’t have to believe in God, but you should believe in The Book.”

Autumn 2017

Elegant proofs are a rare genius, and we are compelled to seek them, for their beauty and value. But, do we have a reason to believe them to be ubiquitous? The fact that there are unprovable statements in mathematics is no news. Kurt Gödel’s famous incompleteness theorem has established its existence as early as 1931. It states that in every mathematical theory, no matter how extensive, there will always be statements which can’t be proven to be true or false with the current set of axioms. There are a number of such simple and natural theorems in number theory which are known to be not provable. Goldbach’s conjecture which states that every even number can be written as a sum of two primes is one such example which is true but not proven. But, it strains our credibility to call the statement undecidable just because it has not been proven or disproven. As with many other conjectures dealing with prime numbers, one can’t say for sure, since there is not enough understanding of prime numbers. There might be an easy and obvious proof once there is a better understanding of prime numbers. Mathematics is infinitely complex, and consequently any finite theory can only ever capture a small part of it. Crucially, this means that there are mathematical statements that cannot be proven. Thus, if there exists no solution, the possibility of the existence of an elegant solution to every theorem also does not exist. It could be otherwise that there are mathematical proofs which are beyond our current state of mathematical knowledge. Or maybe, mathematics is too abstract to have a finite set of solutions. Probably, Erd�s also had similar beliefs and his book tale could just be a fancy way to describe our search for elegance.

17


Cypherpunks and the Surveillance Dystopia

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n the late eighteenth century, British philosopher and social theorist Jeremy Bentham came up with a design for an institutional building that, he believed, would bring prodigious changes in the way correctional facilities work. Called as the Panopticon, the structure would enable just a single, centrally placed guard to observe all the inmates living in a circular building. The unique proposition offered by this design was that while the guard could keep a watch on only a couple of inmates at a time, the inmates did not have any way of identifying if they were being observed, thus, living under the constant consciousness of being watched. Even though the idea of Panopticon could never materialise in its true physical sense, the concept became very popular for its unusual approach towards exercising total surveillance. With the coming of the age of the Internet, the model is again finding its metaphorical similarities in the world we inhabit today. Markedly, the Panopticon scheme has now moved beyond prisons and workspaces and is trying to engulf the society as a whole. It is no longer a guard, sitting in a large watchtower, scrutinising every activity taking place in a circular building. It is a complex setup of devices, algorithms and data trawlers that are capable of carefully intercepting, analysing and storing any and every data that we generate over the Internet. Such a perfect surveillance dystopia seems very far-fetched and fiction-like to an average internet user. The nature of work of the surveillance agencies demands it to be. Until before the leak of CIA documents by Edward Snowden that exposed the vast surveillance infrastructure set-up by various 18

governments around the world, the idea of anything like this existing was not even present in the minds of the general people. Till then, the presence of both the guard and the prison were unknown. Snowden’s revelations from early June 2013 exposed various activities being undertaken by the NSA and the US government whose ethicalities can, at least, be criticised at length, if not openly bashed. From having unrestricted access to almost every phone call made in the US to openly asking companies like Google and Facebook for personal data of its users, a long list of monitoring activities taken up by the NSA in the name of national security was disclosed to the general public. The role of NSA’s British counterpart Government Communications Headquarters (GCHQ) in tapping the fibre optic cables around the world to intercept the data flowing through the global internet was also revealed in one of the subsequent releases. It was also brought to notice that the NSA and GCHQ worked in close association with each other sharing data and intelligence. With all these revelations, a debate found its place in the popular media which was something already being discussed at length by cryptographers and cyber geeks—the question of privacy in the digital world. A group of individuals who associate themselves with the name ‘Cypherpunks’ has been noticeably vocal, over the past few years, on the same issue. Under a broader definition, a Cypherpunk is someone who advocates the use of cryptography as a means of promoting online privacy. The Cypherpunk movement started out in the late 1990s in the form of a mailing list where general discussions took place on varied topics Geek Gazette


such as information security, computer science, and sometimes, even politics. Members of the Cypherpunk mailing list include prominent cryptographers, programmers and computer scientists such as John Gilmore, Eric Hughes, and Julian Assange, to name a few. The various ideas of the Cypherpunk movement were popularised in a book co-authored by Julian Assange (along with Jacob Appelbaum, Andy Müller-Maguhn, and Jérémie Zimmermann). The book, published in the year 2012, almost a year before the leak of documents by Edward Snowden, talks about the possibilities of the existence of a ‘Big Brother’ intercepting our phone calls, snooping our internet connection, and storing our data.

Such a perfect surveillance dystopia seems very far-fetched and fiction-like to an average Internet user. The nature of work of the surveillance agencies demands it to be. The Cypherpunks identify that our concerns about privacy and protection of data would become more prominent as the impact of Internet percolates through the deeper layers of our personal lives. The limited abilities of people to comprehend the problem and their technical incompetencies to affect the process make the situation a little more complicated. The ideas of Cypherpunks say that the only way for us escape the scenario of perfect dystopia is to understand our systems better. The increased influx of proprietary technologies in the market make this process a little more difficult. In contrast to proprietary technologies, open-source technologies allow the interested ones to peek through the layers and check what is going inside. Even though open-source software has become considerably popular over the last few year, applications of the same principles in hardware is a fairly alien concept even today. Another issue with the present system pointed out by the Cypherpunks, lies with the prominence of Autumn 2017

centralized computer systems and architectures. A centralised system such as a data centre or a cloud computing infrastructure provides a single interception point for all the data flowing through that network. Though these technologies have made both data storage and computing considerably cheaper, they pose serious threats when the concerns of mass surveillance are discussed. Distributed peer-to-peer connected networks provide a workaround for this problem as here every connection in the network needs to be intercepted independently which substantially increases the complexity of the process. One decentralised network technology that has become fairly popular over the past few years are blockchains. Bitcoin, a blockchain based crypto-currency, is a very close realisation of the ideas of Cypherpunks as far as economic transactions are concerned. In any Bitcoin transaction, the basic information about the transaction is publically stored which is accessible to everyone while no other personal information is exposed to the network. The greatest advantage of Bitcoins, though, is not limited to this. For any Bitcoin transaction that takes place, there is no mediator or facilitator (like Visa or MasterCard are mediators for credit card transactions). Since the transaction is out of the control of corporates or any other central authority, it essentially means that it will also be out of the influence of governments or those in power. Thus, facilitating freedom in economic interactions, one of the very important aspects of digital utopia. Whenever the questions of privacy will be raised, someone will surely associate the question of security with it. Authoritarian regimes will continue to justify their acts of mass surveillance as ways of countering societal problems such as organised crime, child pornography, or terrorism. Regardless of all these considerations, the greatest issue in the present scheme of things is people’s comfort with disclosing their sensitive data on the Internet. It is important to understand that everyone in this world of Internet could be a malefactor, a victim or a suspect. It is something that we all could keep in mind before we push that ‘Submit’, ‘Upload’ or ‘Send’ button.

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the shape of things to come


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et us imagine a two-dimensional world, where two-dimensional beings exist. These creatures would have a single eye on their faces and would see the world as a single line. This is the setting of Edwin Abbott’s renowned novella, Flatland. For these individuals, the very existence of a third dimension would be absurd because two dimensions are all that they have known and understood. One day, an intriguing sequence of events lead to a higher three-dimensional world mixing with the universe of Flatland. A sphere arrives in Flatland and as it passes through the two-dimensional world, the inhabitants could only see a continuously changing and distorting shape. They see a circle that first increases in size and then decreases. If one of those beings, somehow, is lifted out of the two-dimensional world into the third dimension and is provided with three-dimensional vision, then, it would be able to see the world from a completely different viewpoint. A similar analogy can be drawn for three-dimensional beings as well when dealing with the fourth and other higher dimensions. A fourth-dimensional object when passing through our world would appear to us as a constant shape changing three-dimensional object. By giving the parable of Flatland theory, one can say that there is a possibility that the world we live in has more dimensions than we see. There is no experimental evidence for the existence of more than three spatial dimensions so far. However, since special relativity is true, time should be referred to as the fourth dimension under any sensible definition of dimensions. Trying to imagine the fourth dimension like the first three would be difficult because our mind comprehends things in a three-dimensional space and as such anything with higher dimensions appears distorted. However, our inability to visualize higher dimensions in our everyday, three-dimensional imaginations, has not restricted us mathematically to work on ideas involving the fourth and other extra-dimensional spaces. Mathematically, the number of dimensions can be defined as the number of coordinates required to specify a point. A very simple example of this is the mth number of an n-dimension tuple. Despite the intricacies of working with extra dimensions, mathematicians and physicists have regularly explored higher dimensional

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spaces in the past. One of the most famous examples of this is string theory. The most remarkable fact about string theory, that no other theory in physics entails, is that it predicts the number of dimensions in space-time, unlike other theories that tactically assume them. For the mathematics of string theory to be consistent, the number of dimensions has to be ten—the extra dimensions curl up into small shapes, existing everywhere but too small to observe. And to predict that we are actually used to living in a curved, multidimensional universe is a profound thought. Due to the high number of dimensions predicted by string theory, there are various ways to compactify the extra dimensions to get down to the usual three dimensions of space, and one dimension of time. Because of the many possible ways in which one can compactify, we are not yet able to produce laws that govern our universe using the string theoretic framework. For years, science fiction writers have also contemplated about the possibilities of higher dimensional spaces. According to the early 20th-century horror writer H.P. Lovecraft, these higher dimensions do indeed exist and are home to all manner of evil creatures. Lovecraft had some interest in mathematics and used ideas such as hyperbolic geometry to lend extra strangeness to the fictional beings in his stories. Lovecraft’s use of mathematics has been explored in many papers. In particular, Hull’s “H.P.Lovecraft: A Horror in Higher Dimensions” which points out the existence of exotic four-dimensional spheres that Lovecraft describes in three-dimensional universe as “a congeries of iridescent globes, stupendous in its malign suggestiveness”. A simple experiment with a set of spheres can improve our understanding of higher dimensions. The maximum radius of a sphere that can be placed at the center of the void created by circumscribing spheres is √N-1, where N represents the number of dimensions. For two and three dimensional spheres, this value is always less than one. However, as we move to the fourth dimension, the radius of centrally located sphere becomes one, implying that the size of the void is similar to the size of spheres circumscribing it. In a 21


on a graph than trying to understand it just by looking and solving the equations. However, when we concern ourselves with more than three parameters then we have to take other higher dimensions into consideration and that is where our understanding of them comes in use. This use also extends to topology and defines the need for understanding shapes in higher dimensions. nine-dimensional space, the sphere in the void even touches the outer boundary containing all other spheres without disturbing the other spheres in the space. As hard as it is to imagine, it actually happens in the higher dimensions and if we are able to visualise it, then we would see proper spheres but in our limited three-dimensional imagination, we would see these spheres completely distorted and spiky, something like a sea urchin. An obvious and predictable question would be that if we cannot physically imagine higher dimensions and there is no physical proof that they even exist then why are we concerning ourselves with them? Higher dimensions have been used for over a century by both mathematicians and physicists in different contexts. It is often far more convenient to plot an equation or a result

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A lot of phenomena can be also explained using higher dimensions. As mentioned before, 10 dimensions are used/required in string theory to describe the world, 11 can describe supergravity and M-theory, and the state-space of quantum mechanics is an infinite-dimensional function space. Our understanding of the higher dimensional world has improved significantly in recent years. Thus, the question is no longer about the visualisation of higher dimensions, or how to intuit them, or how unintuitive they are. Rather, to speculate about the kind of problems that might lend themselves to analysis more easily if higher dimensions come into play. While imperceptible to us, from as far as our senses are concerned, there is also a possibility that these extra dimensions have governed the formation of the universe from the very beginning itself.

There is a huge list of feature films where mathematics is central to the plot of the movie. Can you guess the movie which each of these represents? Send in your entries to mail@geekgazette.org

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1729

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11 12

Log(N)/Log(D)

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FIGURING OUT

COMPETITIVE CODING Competitive coding is something that everyone is recommended to try out during his/her initial years at college. Its close affinity to logical reasoning and mathematics makes it one of the most followed hobbies at campus. While a lot of the starters give up once the initial motivation dies, a few take it to an all new level garnering laurels not only at the national but at the international level, as well. Over a gloomy Friday afternoon, we hung out with Adarsh Kumar, Saharsh Luthra, and Vaibhav Gosain (from left to right) who easily qualify as one among the best competitive coders of the campus, and discussed with them about their journeys in competitive coding, ICPC and a lot more.

GG: The three of you have participated in various competitions and excelled in most of them. What is your most profound memory from any of these competitions? Saharsh: It was during my second year that I took part in a contest organised by IIIT Hyderabad. The contest focussed mainly on mathematical problems. A lot of people participated and amongst them was a guy, Gennady Korotkevich, one of the best competitive coders in the world—a prodigy. The great thing was that I gave him tough competition and was beaten by just one problem. I came first in India and that too with a great margin. That was one of the best moments for me. Vaibhav: For me, the exceptional moments are when I perform very poorly in a contest or if something funny happens. One such instance was during the 2015–16 Chennai Regionals. After a series of wrong solutions, we had lost all our hopes of cracking one of the easiest problems in the contest. In the last minute, we made one final change to our solution but didn’t bother to submit it. Luckily for us, one of our teammates stayed back and pushed the submit button. That was one of the most unexpected correct answers that we had got and the incident still remains fresh in my memory. GG: How did your journey in competitive coding begin and did you face any major difficulties? Vaibhav: When I came to IIT Roorkee, I used to

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interact with my seniors in search of things to explore in Computer Science. In our first year, there was a great hype around competitive coding. Going with the wind, I too started doing it. Initially, it took me ten-fifteen attempts to solve even the basic problems. But soon, I developed an interest in it. I started out by solving problems on SPOJ, then I moved onto CodeChef and CodeForces. GG: What is the importance of having a good community when we talk in context of competitive programming? Adarsh: A coding community plays a very important role. It gives you the liberty to discuss new problems, unknown concepts and much more. For me, I knew Saharsh, who had deep interest in mathematics and I could ask him any doubt. Apart from that, it also creates a healthy competitive environment which motivates you to perform. One can always find people to look up to. China and Russia are the strongest countries when it comes to competitive programming. Many of their high school students perform better than us. This, essentially, is because of a different coding culture prevalent in these countries which is somewhat unrelatable to us. Competing with coders from these countries is generally difficult because of the same reason.

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GG: Tell us about one course that you have taken in IITR and which should be taken by every mathematics and competitive coding enthusiast? Saharsh (instantly): Discrete Mathematics. Adarsh: I did not have a course on Discrete Mathematics in Mechanical. I had to study it from different sources and master the concepts by solving questions on different online judges. Discrete mathematics forms a base for many important mathematical concepts. GG: You all must have come across many algorithms, theorems, and data structures. So, which one appeals to you the most? Adarsh: The Fourier Transform. It solves a naive problem with a very different approach. The Fourier Transform finds a beautiful way of multiplying two polynomials using complex numbers which is pretty cool. That’s quite intriguing for me. Saharsh: For me, it would be a data structure called Segment Tree. It is simple but at the same time very interesting. The speciality of this data structure is that it has a wide range of applications, and it is very feasible to use. Vaibhav: Well, I have a list of favourite algorithms. Centroid Decomposition is one of them. It became famous in competitive programming very recently, thus currently it’s at the top of my list. GG: Moving on to the ICPC, you guys nearly made it to the world finals last year. So how was your experience there? Adarsh: There were seven problems in the online round. The first five problems were relatively easy and we solved them like many others, but we got penalties that made us miss the top positions. However, the last two problems were comparatively difficult, and only a handful of people were able to solve them. We started thinking about them and a crack came with around half an hour left in the contest. In the final ten minutes, we submitted the solutions but the CodeChef server was very busy and our codes were in the queue. We were not getting any verdict. Even if we had got a wrong answer or TLE, we could have tried correcting it, but we were helpless. Then, all of sudden, our sixth problem got accepted and in a few minutes, our last submission turned out to be correct as well. Only five teams in India could solve all the problems and we were one among Autumn 2017

them. Now, Vaibhav will tell you about our experience in the Regionals. Vaibhav: In one word, the entire experience was frustrating. Kharagpur Regionals, which was the first one, turned out to be one of our worst performances. We ended up at the twelfth position and it was quite disappointing for us. The interesting part, though, was that the team which was on the top of the leaderboard gave one of the best performances in the history of ICPC. The next in line was the Kolkata Regionals which went well and we ended up ranking third. Still, we had no hopes of qualifying for the world finals. We gave the India final and we were ranked sixth. We made it to the verge of qualifying for the world finals but we couldn’t make it through. GG: The three of you complement the skills of each other and have come out as a very strong team. What suggestion you have for young competitive coders who are looking to form that perfect team? Saharsh: One of the mistakes that beginners make is that they try to team up with the best players without considering team coordination. Coordination is the key factor. Sometimes while solving a problem, half of the solution is done by one while the other half by someone else. So one should try teaming up with people one is comfortable with and not just run behind the big names. Vaibhav: Practicing together also matters. You get to know how the other person approaches a problem, in which type of problem the other is more comfortable and it also strengthens team bonding. GG: Finally what’s your message for the beginners in competitive programming, basically the freshers and the sophomores on how to have a head start and develop a proper approach towards problem-solving? Saharsh: For the first years, I would suggest to get well equipped with the language you are using, and get familiar with some of the basic elements of competitive programming like how to provide input, debug programs, etc. Learning Discrete Mathematics can help a lot. After that take part in contests, try to learn from the codes that other people have written even if you get the correct answer using your approach. They might code the same thing in much shorter or cleaner way. Persistence is key here.

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hacker emblem

he current generation tends to see hacking as a foolproof “method” to obtain secured information from a computer. This is, however, a very crude interpretation, as hacking encompasses in itself much more than simply the idea of obtaining information. The blame cannot be attributed to a single entity—be it the protagonist of a high-budget Hollywood movie, typing furiously over a flurry of flashing screens or a video game character who, with a few miraculous keystrokes, manages to bypass a highly complex security system. These predetermined notions are romantic enough to sway audiences into a bittersweet love for hacking, but they also widen the gap between the popular interpretation of the art and the true definition of it. The pioneers of hacking view it as completely different from its popular interpretation. In fact, the very definition of the term “hacking” is engaging in activities such as programming and cybersecurity in the spirit of exploration and playfulness. This particular definition is also adopted by the hacker culture, which refers to a subculture of individuals who seek to creatively overcome software limitations, and hence achieve new and clever outcomes, while enjoying the intellectual challenge hacking proposes. The origin of the term dates back to the original hacker’s movement—when the technology of computing attracted the world’s best and brightest into discovering what they could do in the metaphorical software playground. As a matter of fact, it was actually the Signals and Power committee of the MIT’s Tech Model Railroad Club which serves as the inception point of the hacker culture as we know it today. The committee took a keen interest in the MIT’s first Programmed Data Processor-1 (PDP-1) computer, through which they invented a variety of programming tools, slang, and the fundamental blocks of hacker culture. The steadily blossoming influence of hacker culture was rapidly advanced with the advent of ARPAnet, the predecessor of the Internet as we know of today. This country-wide computer network brought together small, independent communities across universities and research facilities. ARPAnet accelerated the hacker revolution, with the development of the first iteration of The Jargon File, better known as “The Hacker’s Dictionary”—a collection of slang terms used by various subgroups of the culture. This file also documents how the hacker culture is disappointed with the mass media’s conflicting association of the word “hacker” with a security breaker, which they refer to as crackers. These individuals are concerned with exploiting weaknesses in computer security, be it for academic or malevolent purposes. The difference between a hacker and cracker is apparent as the

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Geek Gazette


hacker culture adheres to a set of unspoken rules or code of conduct known collectively as the Hacker Ethic. These tenets promote the ideas of free and open-source software, collaborative efforts and freedom, and eloquently summarised in Steven Levy’s book Hackers: Heroes of the Computer Revolution, listed as: Access to computers—and anything which might teach you something about the way the world works—should be unlimited and total. Always yield to the Hands-On imperative! All information should be free. Mistrust authority—promote decentralization. Hackers should be judged by their hacking, not bogus criteria such as degrees, age, race, or position. You can create art and beauty on a computer. Computers can change your life for the better. According to Eric S. Raymond, author of the widely popular The Cathedral and the Bazaar, it is important to unify the hacker community under a symbol that shows what hacker culture stands for. And this is what led to the unofficial symbol which represents the entirety of hacker culture and what it stands for—The Glider. This symbol is as almost as old as the hacker culture itself, and ever since its inception, it has fascinated computer enthusiasts and mathematicians alike. The key to understanding this symbol lies with Conway’s Game of Life, which is a cellular automata devised by John Conway in 1970. Basically, it is a zero-player game which consists of an infinite grid of square cells. Each of these cells has a state, given by their colour—black (alive) or white (dead). The state of these cells depend on a set of rules, which depend on the state of the eight neighbours of each cell. These rules are: 1. Any live cell with fewer than two live neighbours dies, as if caused by underpopulation. 2. Any live cell with two or three live neighbours lives on to the next generation. 3. Any live cell with more than three live neighbours dies, as if by overpopulation. 4. Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction

Autumn 2017

The only input required is the initial pattern with which the system seeds itself, after which the system undergoes evolution. The game was originally designed as a mathematical model as an answer to mathematician John von Neumann’s problem of designing a hypothetical machine that could build copies of itself. With its development, a whole new field of research in cellular automata opened up. This is because it has the power of a universal Turing machine: that is, anything that can be computed algorithmically can be computed within Conway’s Game of Life. It could be used to study emergence and self-organisation of patterns, finding applications in not just computer science, but physics, mathematics, philosophy and economics. It is fascinating to see how a simple set of rules can result in highly organised repeating patterns, seemingly out of nowhere. Conway himself believed that it was impossible to generate a repeating pattern, and set out a reward for any individual who could disprove this conjecture. This led to the creation of the Gosper Glider Gun, which was invented by Bill Gosper and his team from MIT. The gun produced a repeating pattern known as a “glider”, on the 15th generation, and every 30th generation from that. After this initial discovery, many other types of “guns” were discovered, along with new devices known as “puffers” and “rakes”. The glider seeks to represent to some extent what the hacker culture stands for—according to Raymond, the Game of Life was born at the same time as the Internet and Unix, and the rules of the game, albeit simple, can lead to unexpected complexities, as a parallel to how hacker culture itself generated novel philosophies and innovations. An emblem to some hackers may seem completely unnecessary, but the truth remains that the hacker community is a product of an innovative and enthusiastic group of people, and is closely knit together by their common goal of achieving the improbable. In the same way, a simple configuration of cells can result in highly complex patterns in the Game of Life, the elements of a computer can also be broken down into few simple rules. By taking these rules and tinkering with them, hackers can create something novel, complex and beautiful. This is what hacking stands for, realising that the world around us is a mere playground to unleash one’s creativity.

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YOUTUBE

recommends

T

he birth of YouTube can be traced back to a room above a pizza shop in California where three Paypal employees, Chad Hurley, Steve Chen, and Jawed Karim started a dating website, albeit an unsuccessful one. The failure of the website, combined with their inability to send large videos via email, compelled them to turn YouTube into a video streaming platform. While it morphed into an all-conquering video portal from a failed dating website, YouTube has affected the world in numerous ways. Its potential as a medium of undertaking promotional activities was soon recognised, and businesses, as well as other media bodies, started collaborating with YouTube. In the 2008 US Presidential Elections, candidates announced their campaigns via YouTube and questions asked by the users drove the debates. Parallels can be drawn between this and the Nixon-Kennedy debate of 1960 which was the first-ever televised presidential debate. The importance of the debate being televised was evident from the statement made by Kennedy shortly after winning, “It was the TV more than anything else that turned the tide.” Despite the similarities, YouTube is fundamentally different from television. It isn’t a megaphone for a single message, but a platform for anyone with a camera to voice an opinion. In addition to the effects like launching people into stardom or fueling nation-wide protests, YouTube has also had innumerable micro-level effects on its users. It has provided breeding grounds for many communities within the website ranging from enormous gaming 28

communities to the small groups of jazz aficionados, RC toy enthusiasts, and even conspiracy theorists. It has helped people turn their hobbies into careers and fostered fame for many YouTube channels like nigahiga, Pewdiepie, and Epic Rap Battles of History, whose content was deemed ‘unfit’ for traditional media platforms. Many musicians, film-makers, comedians, and educational channels have thrived on YouTube and made the platform more and more diverse. The ever-increasing content came off as a boon as well as a curse. The task of managing a large video corpus became too daunting to be performed manually, and hence, machine aid was required to manage the search results and suggestions. So, engineers at YouTube came up with a very basic algorithm that ranked videos by the number of views. Good videos that garnered viewership organically were aided by the algorithm, were featured on the website and ranked higher on search results and suggestions, making them increasingly popular. Such videos started going viral. Initially, the algorithm worked well, but soon, some channels acquired a large number of subscribers because of one or two viral videos. The videos that followed the popular ones were watched by most of the subscribers and eventually got promoted on the website. This quickly triggered a positive feedback loop that resulted in some very highly subscribed channels and each one of their videos going viral regardless of the quality. Another major problem with the number of views Geek Gazette


being the primary parameter for ranking the videos was click-bait. Misleading thumbnails and titles spread on the website like the plague and people started exploiting this loophole to generate more and more views on low-quality clickbaity videos. As a solution to these problems, YouTube underwent a paradigm shift as it changed its primary parameter to watch time.

As a solution to these problems, YouTube underwent a paradigm shift as it changed its primary parameter to watch time. The use of watch time as a driving parameter is beneficial for both users as well as YouTube. To maximise the amount of time a user spends on YouTube, the suggested videos must be the ones that the user wants to watch, and if the algorithm succeeds, the optimum user experience is ensured. Also, users spending more time on the website implies higher ad revenue generation, so it’s a win-win situation. This change in YouTube’s algorithm led to the exponential growth of gamers, vloggers, tech review channels and traditional entertainment companies on the website. Channels like Pewdiepie, Unbox Therapy, and TheEllenShow blew up because of the high upload frequency and binge-able video format. YouTube uses deep learning to power its algorithm. The use of neural networks makes it one of the largest and most sophisticated industrial recommendation systems in existence, and rightfully so because of the enormous size of the video library and the myriad of factors it takes into consideration while recommending videos. Watch time in itself isn’t just the number of minutes for which a particular video has been watched, but a fairly complicated parameter. The hidden factors involved are the number of sessions that start and end at a particular video, the length of the sessions that includes that video and the upload frequency of the channel. The algorithm takes data from users (browsing data) as well as videos (demographic data and click through rate) to find the videos that users are most likely to watch. Autumn 2017

The video recommendation process involves two steps, candidate generation and ranking. The first layer is to filter relevant videos from the enormous library, then the videos are scored by neural networks and ranked accordingly. The algorithm also introduces a bias towards fresh content which makes it easier for new content to surface which would otherwise be unlikely due to the lack of data about new videos. That is why some videos are marked new instead of the number of views being mentioned. The algorithm favours longer videos to maximise watch time. All these biases that creep into the algorithm make YouTube a very different platform than it used to be. The involvement of traditional media companies has increased manifolds since their format fits the ideal long and frequent videos that can trigger binge watch sessions. Also, videos get old in a couple of days and are replaced by newer videos that the algorithm considers relevant. This makes the algorithm biased towards trends and news and makes it harder for educational videos, short films, case studies, video essays, etc. to be discovered. YouTube and other online video on demand services are increasingly being used as a substitute for television, and a part of that is certainly the long-form content, but the overwhelming emphasis on longer videos is undermining the ability to make shorter content. In essence, YouTube is controlling what gets created by how they reward that content and it’s putting the squeeze on short-form content. The best example of this glitch in the system is the unreasonably high number of views on nursery rhyme videos. “Wheels on the Bus” video currently has an astounding ~2 billion views and the fact that it’s an hour long makes it perhaps the most watched video on the website, in terms of watch time. Compilation videos fall into the same category of videos that encourage passive viewing and tap into YouTube’s bias for longer videos to gain popularity. So the content that gets promoted isn’t necessarily the best content on YouTube and the algorithm may not conform to everyone’s definition of ‘perfect’, but despite its quirks, the fact that it is very successful at getting people to spend more time on YouTube is undeniable.

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A MAGICAL TRAGEDY Pan’s Labyrinth (2006) Genre

: Drama/Fantasy

Starring : Ivana Baquero, Doug Jones, Sergi Lopez IMDB : 8.2

“Hi! Are you a fairy?”.

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felia, a young girl engrossed in the world of fairy tales, falls into a swirling pool of excitement and thrill when she finds out that these stories can come to her in real life. Like all other great fairy tales, Pan’s Labyrinth features a girl on the cusp of womanhood who embarks on a magical journey. Though filled with unimaginable, huge, and horrific creatures, her gothic fairy tale is kind of an escape and somehow, easier to swallow than reality. Ofelia had to move from the city to country when her pregnant mother married a sadistic army officer during the times of Spanish Civil War. While wandering through the woods she found a labyrinth. There, she met a faun (not a pan) who told her that she was actually a princess, and asked her to return to her kingdom where she needed to complete certain tasks. As she struggled to cope up with her new bitter lifestyle in the countryside between war and her mother’s bad health, she was presented with the hope of living her fantasies where her story can have a ‘happy ending’. She was caught between the harsh reality of life and her dreams. Thus, the film alternates between the world of Civil War Spain and the increasingly bizarre, dark and frightening world of the Pan’s Labyrinth.

Rotten Tomatoes : 95%

deaths everywhere. However, her choices only worsened the situations that she was previously living in. The chances of her having a perfect life in the Labyrinth also appeared to be very thin. There are several other darker undertones that are explored in the movie. The Mexican writer-director, Guillermo del Toro has clearly portrayed that life in a combat zone is not pretty. Apart from that, there is an interesting contrast between Ofelia’s so-called fantasy and the rational thinking and actions of other characters dealing with war, love and trust. Also, the movie is filled with references from similar, but cuter fantasy worlds. Ofelia’s smock is similar to the one in Alice in the Wonderland, her magic book has been swiped from Harry Potter, and her faun from Narnia. Adding to the eeriness created by the story, the soundtrack is hauntingly emotional, and the art direction is simply dark, uncompromising and brutal. The film received a twenty-two minute long standing ovation in Cannes, and justly so. The idea of a dark fantasy fairy tale with a war background seems absolutely absurd but does not feel so at any point during the movie. Del Toro has managed to tie all the three genres seamlessly. And what makes the tragedy even more tragic, is the inevitability of it all.

The idea of the movie was to show that we would always prefer fantasy over reality. One could say that Ofelia chooses the fantasy, she lives in, due to her world being such a nightmare, a time of war and danger with 30

Geek Gazette


PHILOSOPHY OF FATE Stories of Your Life and Others (2002) Author : Ted Chiang

Goodreads : 4.3

“Now mathematics has absolutely nothing to do with reality. Never mind concepts like ininites or ininitesimals. Now goddamn integer addition has nothing to do with counting on your ingers. One and one will always get you two on your ingers, but on paper I can give you an ininite number of answers, and they’re all equally valid, which means they’re all equally invalid.”

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here are two parallel narratives in the story Division by Zero. The first tells the story of a mathematical prodigy Renee, who was disturbed after formulating an honest and a totally natural theorem, albeit one which contradicts all existing principles of mathematics. Without any logical inconsistencies or any other loopholes, she had demonstrated that mathematics is inconsistent and that all its beauty was just an illusion. For her, the contradiction that she herself established meant an end to everything; mathematics, her academic career, and even her life. The other narrative is of her husband Carl, who was stuck in an emotional turmoil. He was empathetic towards her wife as she had been recovering from depression. He understood her since he had endured something similar in past. However, in the meantime, he had also realized that he no longer loved his wife, even though he never thought he would be the kind of person to leave his marriage in difficult times. In the end, their storyline converged as they both deeply and implicitly believed in something, but proved it to be untrue. The story marvellous intertwines two absolutely unrelated areas, mathematics and the complexities of human emotion. Autumn 2017

Another story, Hell is the Absence of God was set in a world where divine interventions were a regular part of everyday life and affected each individual differently. While they meant good fortune and miraculous cures for some people, for others they brought misery, injuries and even death. Hell and Heaven existed in the storyline and the characters could sometimes witness them. It is told through lives of three people, who cross each other’s path in a particularly interesting way. The story follows the actions of a man who was bereft of devotion for God but was married to a devout woman. After the death of his wife during one of the angelic visitation, the man is left to make a difficult choice. He could either become actively resentful of God or truly love the entity which caused the death of his beloved wife, Sarah. The tale describes his painful journey to truly devote himself to God with a hope that it might reunite him with Sarah, who had ascended to Heaven. The story has a traumatic, but a fantastically realistic end which thoroughly explores religion, its purpose and possibilities. The book is a collection of eight such short stories which are weaved around a common theme: fantasy, horror and science fiction. One of them, The Story of Your Life has been adapted as a major motion picture Arrival. Other stories in the book are critically acclaimed and have won both the Nebula and the Hugo award. Fate, free will, language, ideas, thoughts, all are beautifully tied together in these stories, which actually make one question their own philosophy.

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nature of truth in literature

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he introduction of fiction to the literary landscape wasn’t quite like the story Ricky Gervais’ character wrote on pizza boxes in The Invention of Lying. Humans have always had the ability to manipulate their account of historical events and distort the truth. But fiction isn’t the same as lying or making up something that deviates from history. The development of the concept of fiction is an interesting tale of how we understood art and began to consume the products of storytelling. Before Plato arrived at the scene with his theories on poetry, critics were mindful of some of the make-believe in epic poems such as Homer’s depiction of The Trojan War in his Odyssey, but they didn’t quite know what to say about it. Plato drew a distinction between a poet possessed by the Muses and the technical poet, and introduced the Theory of Mimesis. He said that non-mimetic poetry was divinely inspired and hence it was the equivalent of philosophy. It delivered the higher truth that benefitted the society. Here, the poet was possessed by the Muses and couldn’t be held individually responsible for the factual 32

correctness of what he wrote. This theory has many problems—how do we know who is ‘divinely inspired’ and why should his poetry be superior? Perhaps, Plato realized that himself when he excluded the concept of non-mimetic poetry from his later works. But that didn’t stop people from placing poets such as Aeschylus in this category. The theory Plato did hold onto was that of mimetic nature of poetry. He believed that the arts dealt with illusion and were imitations of imitations, hence ‘twice removed from reality’. Art was inferior to philosophy. Plato banned art (drama and poetry) from his ideal world. Let’s take an example to understand this better. Any real life object like a chair or a human can be called an idea. Philosophy dealt directly with that idea. Art didn’t. A painter would draw an image, a poet would produce a personal interpretation. For Plato, art was the equivalent of a photoshopped picture of a supermodel. It appeals to our baser instincts but is not the reality. The mimetic poet focused on his own personal beliefs and emotions, using techniques in producing poetry that people derive pleasure from. His poetry was his own enterprise and he was solely responsible for it. Dramatic and epic poets were placed in this category. Geek Gazette


Plato trashed art and Aristotle, his greatest student, came to art’s rescue. Now we should appreciate the fact that Plato is the poet and Aristotle, the critic. Aristotle countered the Theory of Mimesis with his Theory of Aesthetics. He said that while Plato is correct to state that poetry is an imitation of what’s real and that likeness is always inferior to reality but what he failed to see was that art offers something more which is absent in reality. Art isn’t a mere photograph, it is a well thought-out representation of certain ideas that are inspired by reality but enhanced in importance by the poet’s imagination. This unique artistry is what gives literature and art a new meaning and beauty. Also, the value of an artist’s work should be judged by his purpose behind the construction of the art, not the mere truth and falsity of the account. Understanding the writer’s perspective gives us new knowledge and yields aesthetic satisfaction. This is wisdom and, for Aristotle, at par with philosophical truth.

Art isn’t a mere a photograph, it is a well thought-out representation of certain ideas that are inspired by reality but enhanced in importance by the poet’s imagination. This unique artistry is what gives literature and art a new meaning and beauty. Aristotle provided, perhaps, the greatest developments in the concept of fiction. Firstly, he completely disregarded non-mimetic poetry. Modern poets since then have taken full ownership of their work unlike the primitive poet who considered himself a mere mouthpiece for a divine power. Yes, authors do mention being inspired to write certain things and tell certain stories but as E. J. Pratt says, “To be inspired is not to abandon responsibility”. Secondly, he asserted that art should not be judged by the amount of truth in what is said but by the poet’s purpose in saying it. This changed how we perceived art, especially poetry. Today, the reader is well aware that a poem could be partially or completely fictitious. Even if a writer retells history as it happened but adds some fictional Autumn 2017

characters, we know which category to place it in. Tolstoy’s War and Peace is a historical fiction but Anne Frank’s The Diary of a Young Girl is not. The purpose of writing the two are entirely different. With this, the third development was made evident when readers and critics could distinguish between fiction and history. This was especially crucial in a time when the narrative genre was becoming more and more common and writers could be found in every household, in every part of the world. The rise of the novel as the most popular and widely used form of literature introduced sub-genres and redefined the purpose and method of storytelling. People are rewarded for their imagination and unique perspective as much as for their literary competence and technique. Science fiction is a brilliant example of how we consume fantastical and bizarre stories today. In the more recent times, the narrative seems to have shifted again towards shorter, radical and more personal storytelling. Young writers take to social media to tell stories or simply voice their opinions; writing novels isn’t on their to-do list. Young readers don’t have the patience for consuming Seneca’s aphorisms or Hegel’s lengthy verbose theories. How this has affected the languages and the verbal environment is a separate story but it might be useful to look at its effects on what we choose to believe in. Most stories that are highlighted are ones dealing with polarizing issues and while the medium of modern networking sites was meant for enhanced communication, somehow this might have blurred the line between fiction, an artistic pursuit, and opinion, the whit of modern thought that has the power to offend and divide people. Because when we start considering the amount of writing that is produced everyday and the number of people consuming it, there is no system for checks and balances. We want the better story irrespective of what the context is. This sort of propaganda isn’t a recent invention but might be more dangerous today than ever before.

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all new particles, no new physics “All the superlatives are justiied. This is one of the cases where the hype is approximately accurate. [...] The LHC is this generation’s people’s, my generation’s people’s, only shot!” —Nima Arkani-Hamed (Theoretical Physicist, IAS Princeton)

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ince the birth of the modern science with Galileo, we have formed theories to explain our observations. Every time we succeeded in understanding one aspect of our world, there was another observation about some other piece of nature—waiting to be explained. But remarkably, in only about 500 years since Galileo, in the late 20th century, we ran out of these unexplained observations. We had a theory that explained everything that had been observed till then. Well, actually, there were more than one such theories but since all of them agreed with all the experiments conducted so far, a wait for more observations to decide which of these theories were true was inevitable. The competing theories differed in their predictions about the microscopic physics and thus, we needed to find experimental data about such interactions. A fantastic way of finding such experimental data is to bang highly energetic particles into each other and see what comes out. Particle accelerators like the SLAC and the LHC promise to do this for us. The higher the energy scale at which a particle accelerator operates, the 34

more ‘resolution’ it provides us with. The reason behind this relationship is contained in Heisenberg’s uncertainty principle and Einstein’s special theory of relativity. The Standard Model is a theory that explains each and every subatomic observation made by humans so far. It is a remarkable fact that no experiment ever conducted deviates from the standard model, not even a little bit. However gratifying this might seem, this is the root of all our troubles. We already know that the standard model is not the whole story. The most obvious reason is that it doesn’t describe gravity. In addition to this, the standard model also fails to do a lot of things that we want our final theory to do. We would not like our final theory to leave a whole bunch of constants to be determined through experiments—we would like them to be determined theoretically. The standard model fails to do so. Thus, we know that there lies new physics beyond the standard model and the LHC is our hope to get a glimpse of that. We have many theories about what the new physics might look like, for example, the ones with supersymmetry and those with extra dimensions. But only the experiments at the LHC can confirm which of these theories are actually true. LHC is currently working at the TeV (Tera electron Volts) scale. The scale, of which we are certain to see new physics at, is the Planck scale. No one really knows Geek Gazette


at what point between the TeV scale and the Planck scale will the new physics actually start appearing. We have theories that are consistent with the new physics appearing at pretty much anywhere between the two scales. When the LHC was turned on, people expected it to find out which of these theories are true by detecting new particles and observing their properties. The LHC has observed many previously unobserved particles but no truly ‘new’ particle that wasn’t expected by the standard model. As we keep on boosting the scale at which the LHC operates, even without finding new physics, we are putting more and more restrictions on the possible theories and its parameters. For example, since we have not detected any extra spatial dimensions up to the TeV scale, we can rule out many of the theories that entertain the idea of large extra dimensions. Since we have not found supersymmetric particles until now, and since the observed Higgs mass is not exactly same as that predicted by the complete supersymmetry, we now know that we can only have a weirder version of supersymmetry, probably something like the split-supersymmetry. Although we know that the standard model isn’t the whole story and thus, the absence of deviations from the standard model is frustrating, we should take a step back and see what an incredible piece of science we have been able to create with the standard model and the LHC. Since there are no disagreements with the standard model up to the TeV scale, we can be sure that despite our ignorance about the physics beyond that, we have a perfectly correct theory for all the phenomena at energy scales lower than the TeV scale. Put in other words, we already know the basic laws that, in principle, describe everything from the atomic nucleus to life itself. We can write these laws on a coffee mug if we wish!

Put in other words, we already know the basic laws that, in principle, describe everything from the atomic nucleus to life itself. We can write these laws on a coffee mug if we wish! We have been asking what the LHC has been up to but it is equally (if not more) important to ask as to what we Autumn 2017

have been up to due to the LHC. Apart from its direct and unprecedented impact on our experimental knowledge of particle physics, LHC has given us many spin-offs. But perhaps, the greatest of all the LHC spin-offs is an ongoing research program in Theoretical Physics known by the catchy name of `Amplitude Mini-Revolution’. This program is based on a truly new and game-changing theoretical observation made by some people while predicting the outcomes of the experiments that the LHC was going to conduct. They noticed an unbelievably simple and curious mathematical trick. The trick involved a simple one-line calculation and produced the exact correct answer as was produced by the actual Feynman diagram calculations that took hundreds of pages. Many theoretical physicists, including Nima Arkani-Hamed, have extensively studied the physics behind this trick in the past few years and have discovered that there is a deeper truth and reason as to why that trick works. It turns out that it is possible to describe the physics of interacting particles in a way that doesn’t fundamentally depend on the concepts of space and time! In fact, introducing the space and time unnecessarily hinders the actual physical content of the situation. In Feynman diagrams, the concepts of space and time are deeply ingrained. This is the reason why Feynman diagrams hindered the actual simplicity of the interactions and took hundreds of pages of calculations. The newly discovered theoretical object, called the Amplituhedron, inherits this new physical content. Just like the calculations done with Feynman diagrams, calculating the volume of the amplituhedron gives us the probability of a certain interaction. But the amplituhedron is not based on space and time, rather the space and time emerge from the physics of this amplituhedron. Ours is an absolutely unprecedented place in the history of science. We have demonstrably reached a scientific maturity where we have built our biggest machine, not for any commercial, military, or economic purpose, but for the purpose of understanding the most basic laws of nature. The LHC, with or without the new physics to come, is the icon of this unprecedented scientific commitment that has culminated in our generation. 35


position of points; two things are identical as long we don’t change the neighbourhood of any point.

FROM DONUTS TO

COFFEE CUPS T

he origins of mathematics can be traced back to the origins of human civilization itself. Mathematics was the language that allowed our early ancestors to comprehend the world around them, constructing their analysis from basic logical deductions and reasoning. Intuitively enough, the first mathematical objects that humans interacted with were not equations and not even numbers. They were real physical entities that, in mathematics, would be called as lengths, areas, and volumes. Geometry was one of the first branches of mathematics to come up with its own consistent set of axioms and propositions. Surprisingly, over the years, the ideas of geometry have no longer remained limited to the explanation of objects that have a fixed shape and size but have also expounded on those that essentially lack it. In Euclidean geometry, we can only move objects and rotate them without changing their properties. Two objects are said to be congruent if they can be superimposed on each other. A new kind of geometry—projective geometry—developed during Renaissance brings in the ideas of perspective drawing into the field. In this, two things are considered to be same if they are both the views of the same object. A cylinder appears to be a circle when viewed from the top, but looks like a rectangle from the side-view. But pure mathematics has much more than this in its lap. There exists a branch in which the properties of objects are preserved under continuous deformation, twisting, and stretching of objects, as long as there is no tearing. This branch is called Geometric Topology, also known as ‘Rubber Sheet Geometry’. In fact, in geometric topology, a circle, a triangle, a square, or even the shape of an amoeba are the same as they can be formed from each other by just pulling, stretching or moulding. Topology explains things on basis of relative 0336

The field of geometric topology, just like Euclidean topology, brings with itself a study of dimensions. In modern mathematics, a circle is a curved line and the area within that line is a disc. The disc is indeed two-dimensional, the circle isn’t. If we stand at a point on a circle, we can move only in two directions—either forward or backward, but, if we stand at a point on a disk, we can move in four directions—forward, backward, right or left. Topology isn’t as theoretical and abstract as it sounds, at all. Even so, we have played games in our childhood that fiddle with the fundamental concepts of topology. Be it playing with clay or challenging our friends to draw a certain figure without retracing any line and lifting the pen, topology was involved everywhere. The birth of topology traces back to a similar ‘retrace without lifting’ kind of problem—Seven Bridges of Königsberg. The problem was to devise a walk through the city that would cross each of the seven bridges once and only once. It was solved by Euler in negative and his argument shows that a necessary condition for the walk of the desired form is that the graph should be connected and have exactly zero or two nodes of odd degree, i.e., points where an odd number of lines meet. Topological studies find a lot of other applications too. It can prove things in geography like, given at least some wind on Earth, there must be, at all times, a cyclone or an anticyclone somewhere on it. Being a new and modern field of study, topology has myriads of possibilities—in research and as well as in applied sciences. Be it fractional dimensions, unconventional propositions, or strange surfaces, implications that this field might have in domains ranging from quantum field theory to the design of electrical insulators, are truly infinite. Topology also provides us with a new perspective for understanding things, brushing off the specifics and then looking at the generalities that remain. It is a way of appreciating things based on their interconnectedness and continuities while ignoring the moulds that they fit in.

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Geek Gazette Autumn 2017  

Geek Gazette brings to you its 18th Issue with the aim to reintroduce readers to the beauty and simplicity in Mathematics. Other articles co...

Geek Gazette Autumn 2017  

Geek Gazette brings to you its 18th Issue with the aim to reintroduce readers to the beauty and simplicity in Mathematics. Other articles co...

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