Math ANd Science
Terence Tao:
The Mozart of Maths
-Shravan h
When we think about famous mathematicians and scientists, we often think about those that lived in the 20th century, like Einstein or Feynman. Maybe some of us think about the mathematicians of the 17th and 18th century, like Euler and Newton; or even more ancient ones like Pythagoras. But one thing’s for sure - many of us are oblivious to the undoubted scientific geniuses trying to answer the most significant questions in the modern world. However, they are contributing just as much to their fields as their classical counterparts, and their work can never be overlooked.
The best demonstration of this is the famed Australian mathematician and International Math Olympiad winner, Terence Tao. Dr Tao has made great strides towards our understanding of mathematical progressions and sequences and has succeeded in capturing a lot of the complex nuances behind the most basic mathematical concepts like prime numbers. He has often been described as one of the smartest people living in the 21st century, along with Grandmaster Magnus Carlsen and Stephen Hawking. He has a confirmed IQ of 230, more than twice the average person’s IQ.
Tao is a child prodigy, having learned how to read at the mere age of two. His parents used to find him in his room as a young child of six years old doing complex algebra textbooks. He also managed to score a 760 out of 800 on the SAT math section while just eight. His age never stopped him from pursuing his intense love and appreciation for math, and neither did society. He won one of the most prestigious awards known to students - the Gold Prize at the International Mathematics Olympiad - when he was no more than twelve years old.
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After that, he went on to write various papers, pioneering in fields ranging from geometry and calculus to algebra. He is internationally recognized by universities and other scientists as one of the smartest mathematicians alive today, and his work shows it. His research papers are some of the most celebrated among the math community, and his colleagues and students hold him in extremely high esteem.
But why do we need to know about his work outside a mathematical context? Is it essential? Almost all mathematicians would answer that with a resounding YES. His work is a direct demonstration that math never ends - it is infinite, and it goes on forever. Much like in every other subject, once a discovery is made, there tend to be millions more that build on that discovery, and that cycle keeps repeating. Tao has shown us this by taking things we thought we
Terence Tao (right) as a 12-year-old boy winning the 1988 IMO
knew everything about, and finding out so many more things that define them that help us represent them in infinitely many ways.
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Tons of discoveries have been made in the fields of prime numbers, calculus and geometry, but Tao expanded on those and delved deep into the core of these concepts and was able to find new properties that helped boost our understanding of them. Tao has also pioneered in the field of combinatorics, a portion of math that has to do with something we’ve all learnt in our kindergarten years - counting.
Every day, we discover new things about the most basic, simple math concepts that may revolutionize the way we view them. Tao is regarded as one of the greats when it comes to this - taking a primary, well-known idea, magnifying it under a microscope, and identifying more of its properties that we don’t usually consider when looking at this idea. Doing this leads into new branches of math and science, and it often astounds us just how we will never end in our pursuit of understanding all of mathematics.
Tao may be a prodigy, but he has reaffirmed his belief in hard work various times. He believes that anyone, with the right amount of patience, perseverance and energy can study any subject they’d like. According to Tao himself, “Education is a complex, multifaceted, and painstaking process, and being gifted does not make this less so.”
Quantum Entanglement and Communication
-Aryan Mahesh
On the 15th of June 2020, China made a breakthrough in the field of quantum mechanics. They demonstrated quantum entanglement - a method of intertwining two particles so that measuring properties of one can tell you about the other. This has been done in the past. However, what makes China’s achievement significant, is that they achieved this between a particle on earth and one in orbit.
What is Quantum Entanglement?
Quantum entanglement is the phenomenon where two particles are connected in such a way that when you observe one of them, you can get information about the other. The easiest of the particles that can be entangled are photons (i.e particle form of light). When observing, the state of one particle is the aspect that is measured.
When light travels it oscillates. It behaves like a wave and a particle at the same time.
Project QUESS - Quantum Experiments at Space Scale
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In August 2016, China launched a satellite named Micius named after the ancient philosopher. Its aim was to create quantum entanglement and observe its working through long distances.
Earlier last month, the satellite sent a stream of pairs of entangled photons to two observatories around 1200 kilometers apart. The stream was sent so that there can be a direct communication link between the two observatories. This kind of communication is done by manipulating the polarisation of one in order to signal the other to do the same. A change in polarisation in one direction can indicate a 0 and a change in the other direction can be 1.
Traditional communication involves radio waves from satellites. However through this method, there is no link to the satellite. The satellite merely serves as a method to create the entangled photons. The communication happens directly from one location to the other. This also implies that the signal will be untraceable by current methods. It is the most secure type of communication so far, since the communication cannot be intercepted while travelling from point to point as the communication happens instantaneously. The message will be transmitted from one side and received by the other side at the same time.
However, the beams of photons could only be detected at night. The sun’s light would have drowned out the beams before it could be detected, even if the experiment was successful. To counter this problem, Jian Wei Pan, the head of the Chinese National Space Science Center plans to launch many more satellites over the next five years. The satellites will send more concentrated beams so that the detection process becomes easier and efficient.
Now this oscillation can be either in one plane or many planes. When it happens in one plane, it is called polarised light and when it happens in many planes, it is termed as unpolarised.
Making the light oscillate in only one plane is called polarisation. If two photons are entangled, their polarisations are linked.
Take two photons, one that oscillates up and down and another that oscillates left and right.
If these two photons are entangled, that means that if you changed the direction in which one of them is oscillating, the other would change its direction by the same amount.
This happens regardless of however far apart these particles may be. Albert Einstein called this ‘spooky action at a distance’.
NASA has drawn up plans to rival the Chinese in this field by using the International Space Station’s laser system to relay quantum information between two areas. It is known as the National Space Quantum Laboratory program. Dr. Earl of Qubitekk says that Beijing seems to be much further ahead in the quantum race. However he cannot be sure as the progress is confidential and thus cannot evaluate the country’s progress.
Many experts also say that this is the basis for a high speed quantum internet. A network of satellites could one day connect millions of computers that may also be working based on quantum computing principles.
The Quantum Race and competition
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The launch of Micius though being significant, it also has political and military implications that are very hard to ignore. Any country could theoretically trust Micius to provide entangled photons to secure its communications. It is a strategic resource that other countries are likely to replicate. All countries would now try to reach the standards set by this experiment. This creates competition which is both good and bad for progress. One way, the countries are encouraged by each other to lurch forward, while another way, countries might try to impede on another’s progress through malicious methods.
Conclusion
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Micius has proven that we are only in the beginning of an era of fast, secure and efficient communication. Dr. Pan’s paper says that the future of the internet will be based on quantum principles.
Bifurcation Diagram:
Seeing Chaos
-Devansh Mishra
Counting the population of rabbits has been one of the most widespread (and jobless) things mathematicians and physicists have ever done. Discovering the Fibonacci sequence also came from studying a herd of rabbits. Sorry for this, but let us do it again, hoping to look at something new. It might be a rollercoaster of a ride for many people, so most of it may just blow over your head (I’m in that category too). You’re not the only one. Suppose you want to model a function, which maps the population of a herd of rabbits in a period. It makes sense that the equation would look something like this,
Xn+1 = r(Xn)(1-Xn)
with 'r' being the growth rate, and 'Xn' and 'Xn+1' being the populations this year and next year, respectively. The term Xn represents the birth rate, and 1-Xn controls the population representing the death rate of the herd.
The equation looks like an elementary logistic map, making complete sense at first glance. It may seem
The bifurcation diagram
logical to assume that it is a one-to-one map, that is, any ‘r’ value will relate to one and only one equilibrium population. On further investigation, we will see that this theory is terribly wrong. As the ‘r’ values seem to increase, there is a visible chaos that occurs in the diagram. One ‘r’ value relates to a wide range of equilibrium populations, i.e, the population never settles on one singular value. It oscillates between multiple values and never stops at one. Why does this happen? Is there any order behind this seeming chaos?
When the ‘r’ value is less than 1, the population slowly reaches 0 and goes extinct.
When the ‘r’ value is around 2, the population stabilises on one value.
When ‘r’ hits 3, the population behaves like this, never settling on one singular value
In 1978, Mitchell J. Feigenbaum had an epiphany while studying the bifurcation diagram. Being 99% sure that his approach would lead to a dead-end, he tried doing what is shown in the diagram below. Upon doing the same with other bifurcations, he got the same constant - 4.669. Mathematicians have tried to figure out the significance of this constant, but with not much result. They have tried to reason that it may be the fundamental constant which explains the entire universe, which I’m pretty sure is 42, but whatever floats your boat.
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You might say, ”Okay sure, it creates cool patterns, but so does a two-year-old kid equipped with paints and colour pens on the carpet of my living room. What's so different about the bifurcation diagram?” The fantastic thing about it is that it comes in handy in some of the most astonishing places possible. From the oscillating temperature in a convection current to the dripping of water in a tap faucet, the bifurcation diagram comes up in many unlikely places other than mapping population. Scientists have studied the effect of flickering of light to the blinking of our eyes. They found that once the rate of flickering reaches a certain frequency, our eyes completely ignore one of the flashes and blink at every other flash. The beautiful emergence of the bifurcation diagram, even if it was somewhat fuzzy, in something so biologically dependent proves its universality.
One of the most famous examples of the bifurcation diagram coming up in something different is the Mandelbrot set. The beautiful diagram created when the Mandelbrot set is plotted on the complex plane hides the famous bifurcation diagram within its curves. When you look at the same thing in a 3D space instead, with the z-axis being the equilibrium population, the bifurcation diagram
A further bifurcation occurs at around 3.5, gaining a period 4 in the oscillation of the value of population
The Feigenbaum constant is calculated by dividing the distance between the first two white lines by the second and third white lines. As this process continues, the number approaches 4.669.
emerges. Some of you keen observers may notice that both the Mandelbrot set and the bifurcation diagram are composed of fractals. This is another way both of them are interrelated. I can't even begin to understand how this works, so I'm not even going to attempt to explain it to you. All we can do is stare at its brilliance.
Another use of the bifurcation diagram is to generate random numbers by a computer system. A computer can never generate truly random numbers like us. This is why those numbers are called pseudo-random numbers.
Upon further examination of the bifurcation diagram, you will see an orderly behaviour emerge at some moments, bifurcation and later return to its trademark chaotic behaviour. This order that emerges in between all the chaos could define the world much more elaborate than you think.
It shows that order and chaos are just two sides of the same coin, one cannot exist without the other.
Alan Turing and Enigma:
How math helped reduce the impact of WWII
-Ameya Kakade
The onset of the twentieth century saw nothing but wars and social hardships. Just as Europe seemed to recover from the First World War, it was only a matter of time until there was yet another one. In this gloom and chaos, there were some exceptional brains working to serve humanity. Alan Turing was one such man who saved millions of lives by cracking the “uncrackable” Enigma.
Left: The German Enigma.
Right: The Bombe
In my opinion, Turing was a man ahead of his time. He had created a machine that had the capability to function like a human brain, only a lot faster. The Bombe was an early example of what we call Artificial Intelligence today. As Turing thought, a computer was said to contain Artificial Intelligence if it could mimic human responses under certain conditions. The Bombe did exactly so by analyzing the various
The British Intelligence was always caught unaware of the strategically planned German attacks, incurring numerous casualties. It was then that the British decided to set up the Bletchley Park headquarters, where they brought Britain’s best cryptographers and mathematicians to crack the Enigma. One of them was the brilliant mathematician from Cambridge University, Alan Turing.
Having graduated with first-class honours in Mathematics, Turing was asked to lead the team of Britain’s best minds to crack the Enigma. But how did Turing crack the Enigma in 6 months when the odds of cracking it were 158,962,555,217,826,360,000-to-1?
The answer was the Bombe. The Bombe machine, which Turing designed in the early stages of World War 2, was crucial to crack the German communications encrypted by the Enigma machine. Turing’s machine, which in many ways was a precursor to a computer, was able to rapidly speed up the rate at which the intercepted messages were decoded, allowing the Allied Forces to react accordingly.
The Bombe was composed of the equivalent of 36 Enigma machines. When it was switched on, each of the Enigmas was allocated a pair of letters from the obtained crib text ( a known piece of German plaintext, at a familiar point in the message ). As the Bombe worked, each of the three rotors moved at a rate mimicking the Enigma itself, checking approximately 17,500 possible positions until it found a match. Furthermore, the machine stopped only when each of the Enigma machines found what it believed to be the correct pair of letters.
During the course of the war, Bletchley Park had 200 different Bombe machines made to break the cipher messages transmitted by the German forces. Weighing almost a ton, they had about 12 miles of wiring and 97000 different parts.
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combinations of the Enigma, which helped reduce the World War by almost two years.
Unfortunately, Turing did not live the life any of us would have ever imagined. His invention was destroyed to keep the secret, but this did not stop him to continue his career. Between 1945 and 1947, Turing moved to Hampton, London, where he worked on the design of the ACE (Automatic Computing Engine) at the National Physical Laboratory. He presented a paper in 1947, which was the first detailed design of a stored-program computer. It was often said that Von Neumann’s first draft of the EDVAC computer was based on Turing’s ideas. Although ACE had a wonderful framework, the secrecy surrounding the work at Bletchley Park led to delays in the project. Regrettably, Turing did not live to see the ACE’s first prototype.
In 1952, Turing was prosecuted for homosexual acts. He accepted chemical castration treatment, as an alternative to prison. Turing died in 1954, only 16 days before his 42nd birthday, from cyanide poisoning. In 2009, following an Internet campaign, British PM Gordon Brown made an official public apology on behalf of the British government for “the appalling way he was treated”. Queen Elisabeth II granted Turing a posthumous pardon in 2013. The “Alan Turing Law” is now an informal term for a 2017 law in the UK that retroactively pardoned men convicted under historical legislation that outlawed homosexual acts. Despite all the hardships he faced, Turing and his contributions will always be etched in our hearts for years to come.
Alan Turing (1912-1954)
Quadratic ROots:
An Ingenious derivation of the quadratic formula
-Shravan H, Devansh mishra
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How many of you can confidently say that you have used the quadratic formula outside of school? I’m assuming none of you can say that definitively, unless you are a certain Pixar employee, about whom I will talk a little bit later. So why do we learn this sing-song formula in middle school, something that we most probably aren’t going to use for the rest of our lives? Why do we memorize a few symbols that supposedly don’t have any meaning? Well, you don’t. And if that’s all is needed for you to lose interest, you skip right over the next bit.
If you’re still here, I would like to introduce you to Po-Shen Loh, a mathematics professor at Carnegie Mellon University and the national coach of the United States’ International Mathematics Olympiad, under whom they have won awards in straight years recently. A few weeks back, Mr. Loh published a paper with his new proof of the quadratic formula with a simple and easy-to-follow derivation. This proof is incredibly intuitive and will help all middle schoolers who are struggling to understand why the quadratic formula is what it is to figure out their problems.
The derivation follows the thought that the average of the roots of a certain equation (AX^2 + BX + C = 0), say R and S, has to be -B/2A. Then, he states that there must be a common difference, z, which when added to or subtracted from the average will give both the roots. This equation is elegantly simplified into a difference of squares, a formula we’re all familiar with and is much easier to remember and understand. This is then demonstrated as an expression of z. This is then plugged into the original equation, and the quadratic formula emerges from within.
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This is a remarkably intuitive process, which gives us an idea of where the quadratic formula comes from. But there was just one issue. This derivation had already been familiar to the mathematics community, and there definitely was no distinct new insight seen in the paper. In fact, a plethora of content can be found using this different, easier method of quadratics as a way to teach middle-school and high-school students how to solve these equations. However, there is no record of any similar proof ever written down, so Mr. Loh has seemed to find a loophole. Any claims over this proof may not be withheld, but it doesn’t take away from the elegance this proof holds.
An excerpt from Prof. Loh’s paper, showing a part of his derivation of the quadratic formula.
There was a talk given in Pixar about things that are taught in schools, but are never used in real life. A mention towards the quadratic formula was given as an example, and everyone agreed. Well, almost everyone. There was an employee who waited till the end of the talk to tell the speaker that he uses the quadratic formula very frequently. Working as a graphic designer, he has created marvellous computer
From this graph, a quadratic function emerges, which is used to colour a certain pixel accordingly. In fact, the movie ‘Coco’ has used the quadratic formula over a trillion times over in the entire movie. So even though the formula is completely useless to most of us, some animated movie creators are quite thankful to it for making their lives easier. Not only do animators use the quadratic formula, financiers,
images just with graphics of balls and their shadows. He essentially maps out all the rays from the camera, and checks if they hit a ball to colour the pixel accordingly. To check whether a certain light ray hits a ball or not, a vector with a position-time graph is drawn. If the ray hits the ball at two distinct points, the graph goes below the x-axis. Glancing off will make it touch the x-axis, and not hitting a ball will make it not touch the x-axis at all.
This picture was computer-generated and it used the quadratic formula at least a 1000 times!
businesspersons, athletes and many other groups owe a part of their success to the quadratic equation. It’s incredibly neat to think about how a seemingly insignificant equation we were bored of in school turns out to be one of the most important formulas used in real life today!