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- Making sense out of the abstract...
"When we see it for the first time, it looks so abstract that it seems impossible something like this could have any real-world applications" - Edward Frenkel As the under-graduation curtains close, confronting and comprehending the ideas of abstract algebra seem barbaric to my mind. Right from high school, one can view the negligence or rather the amount of incompetence people exhibit while teaching this area. As a consequence, leaving students, like myself, blindsided to the essence and worth of the field. This, and the following few post are meant to encapsulate my thoughts on this area of the subject and it's significance in human civilization. Much of abstract algebra involves properties of integers and sets. In this post, let's specifically look at the concept of modular arithmetic. It is an application of the well known division algorithm, it is basically an abstraction of a method of counting. For instance, if it is now September, what month will it be 25 months from now? of course, the answer is October, but the interesting fact is that you didn’t arrive at the answer by starting with September and counting off 25 months. Instead, without even thinking about it, you simply observers that 25 = 2.12 + 1, and you added one month to September. Similarly, if it is now Wednesday, you know that in 23 days it will be Friday. This time, you arrived at your answer by noting that 23 = 7.3 +2, so you added two days to Wednesday instead of counting off 23 days. If your electricit’s is off for 26 hours, you must advance your clock 2 hours, Since 26=2.12 + 2. Surprisingl, this simple idea has numerous important applications in mathematics and computer science. In general, when a = qn + r, where a is the quotient and r is the remainder upon dividing a by n, we get, a mod n = r. As A consequence, 6 mod 2 = 0, since 6= 3.2 + 0. If a and b are integers and n is a positive integer, then, a mod n = b mod n, if n divided a-b. This very modular arithmetic is often used in assigning an extra digit to identification numbers for the purpose of detecting forgery or errors. In the next post, we‘ll discuss a few examples and delve deeper into abstract algebra in general. Image credits : Institute for Advanced study

- Meritorious Mach
The Science Of Mechanics In essence the composition of physics resembles a palace. Whose exterior form is apparent to the unceremonious visitor, but whose inner life, it's conventions, traditions and rituals which give a special outlook and kinship to it's occupants, requires time and effort to comprehend. Initiation into this special knowledge is the goal of our endeavor, low and behold the endeavor isn't as simple, since this palace is of an ancient origin. The momentum generated by Newton's discoveries gave physics an impetus which is still prevalent. The eighteenth and nineteenth centuries saw a flowering of science as great minds such as Euler, Lagrange, Laplace, Faraday, and Maxwell extended our knowledge of the physical world. However, their efforts were directed at upward extension of the palace, Newton's account of the fundamental laws of physics was so overwhelming, and so successful, that not until the last quarter of the nineteenth century was there a serious attempt to investigate the foundations. It was an Austrian mind, who first successfully challenged the Newtonian thought, It was Ernst Mach. Although Mach's work left Newtonian physics more or less intact, his thinking was crucial in the revolution shortly to come. In 1883, Mach published his texts "The Science of Mechanics," which incorporated a critique of Newtonian physics, the first incisive criticism of Newton's theory of dynamics. In addition to presenting a lucid account of Newtonian mechanics the text incorporates several significant contributions to the fundamentals of mechanics. Mach clarified Newtonian dynamics by carefully analyzing Newton's explanation of the dynamical laws, taking care to distinguish between definitions, derived results, and statements of physical law. Mach's approach is now widely accepted. "The Science of Mechanics" raised the question of the distinction between absolute and relative motion. Mach pointed out Newton's ambivalence on this subject, although he went on to show that the question was irrelevant to the application of Newtonian dynamics. In the process he dwelt on the problem of inertia and enunciated the principle that now bears his name, inertia is not an intrinsic property of matter or space but depends on the existence of all matter in the universe. The problem of inertia was not the crucial difficulty with Newtonian mechanics. The fundamental weakness in Newtonian dynamics, as Mach pointed out, centers on Newton's conception of space and time. In a preface to his dynamical theory, Newton avowed that he would forgo abstract speculation and deal only with observable facts. Although such a point of view is now commonplace, at the time it represented a tremendous intellectual leap. Before Newton, the business of natural philosophy was to explain the reasons for things, to find a rational account for the workings of nature, rather than to describe natural phenomena quantitatively. Newton essentially reversed the priorities. Against the criticism that his theory of universal gravitation merely described gravity without accounting for it's origin, Newton said "I do not make hypotheses".(from STEVEN WEINBERG's The Revolution That Didn't Happen ) Unfortunately, Newton was not completely faithful to his resolve to avoid abstract speculation and to deal only with demonstrable facts. In particular, consider the following description of time. that appears in the Principia. "Absolute, true and mathematical time, of itself and by its own true nature, flows uniformly on, without regard to anything external. Relative, apparent and common time is some sensible and external measure of absolute time estimated by the motions of bodies, whether. accurate or in-equable, and is commonly employed in place of true time; as an hour, a day, a month, a year.." Mach comments, "it would appear as though Newton in the remarks cited here still stood under the influence of medieval philosophy, as though he had grown unfaithful to his resolve to investigate only actual facts." Mach goes on to point out that since time is necessarily measured by the repetitive motion of some physical system, for instance the pendulum of a clock or the revolution of the earth about the sun, then the properties of time must be connected with the laws which describe the motions of physical systems. Simply put, Newton's idea of time without clocks is metaphysical. To understand the properties of time we must observe the properties of clocks. A simple idea? Yes, indeed, were it not for the fact that the idea of absolute time is so natural that the eventual consequences of Mach's position, the relativistic description of time, still come as something of a surprise. There are similar weaknesses in the Newtonian view of space. Mach argued that since position in space is determined with measuring rods, the properties of space can be understood only by investigating the properties of meter sticks. We must look to nature to understand space, not to platonic ideals. Mach's special contribution was to examine the most elemental aspects of Newtonian thought, to look critically at matters which seem too simple to discuss, and to insist that we turn to experience to understand the properties of nature rather than to rely on abstractions of the mind. From this point of view, Newton's assumptions about space and time must be regarded merely as postulates. Mach's review had little immediate effect, but its influence was eventually profound. In particular, the youthful Einstein, while a student at the Polytechnic Institute in Zurich in the period 1897-1900, was much attracted by Mach's ideas on the foundations of Newtonian physics and by Mach's insistence that physical concepts be defined in terms of observations. However, the immediate cause for the overthrow of Newtonian physics was not Mach's criticisms of Newtonian thought. The difficulties lay with Maxwell's electromagnetic theory, the utmost achievement of classical physics, which could be an intriguing topic for discussion in a future post! In hindsight, Ernst Mach's contribution through his Science of mechanics, unclogged unaware minds, and gave rise to tremendous achievements with regards to understanding space, time and motion. Image source Further reading: The What, the How, and the Why: The Explanation of Ernst Mach https://physicstoday.scitation.org/doi/10.1063/PT.3.2214 https://plato.stanford.edu/entries/ernst-mach/

- Speciality of unconventionality
A week ago, on 22nd of July (22/7) was pi approximation day and this post, is aimed towards looking at how brilliant minds progressed towards earning bragging rights. In the early days, the methods and ideas to calculate the value of pi was meticulously slow and in a sense exhausting. However, as we have talked a lot about one guy on this blog, he makes his way through in this post too. Isaac Newton as usual with his class act of coming up with intuitive ideas, outwitted his brethren. This is Pascal's triangle and the basic idea behind it is, if you look at the coefficients of x, x squared and so on. They are actually just the numbers in pascal's triangle. The power of (1+x) corresponds to the row in pascal's triangle. It's something that has been known for a considerable number of years. Whenever you have a row, you just add the two neighbors and that gives you the value of the row below it, as a consequence you can probably calculate the coefficients to the power ten, instead of doing all the algebra. Over the years, people unraveled a general formula for the numbers in Pascal's triangle and that my friends is, the binomial theorem. Binomial, as the name suggests it gives us an expansion for any two terms and theorem because you could put in values in place of 'x' and show that the coefficients match the values in Pascal's triangle. This is where Newton demonstrated his brilliance for probably the ten time now, from all the endeavours that he had already demonstrated. The conventional requisites of the theorem suggests that you plug in only positives integers of 'n'. But as usual breakthroughs most likely happen when you try to go out and commit to unconventional tasks. And so, Newton plugged in all sorts of values, from negatives to fractions. He starts with n equals a negative one (it gives 1/(1+x) on the left-hand side), what happens? if you plug a negative one in the above equations, the signs of the coefficients alternate or shift between positive and negative respectively, on the right-hand side to infinity. As a consequence, if you don't have a positive integer, the binomial theorem just gives an infinite series. And you could check this for yourself by plugging in a positive integer, the series at the value of '(n-n)' becomes zero, as a result leaving with a finite value in the case of a positive integer. What Newton did (you could do this too and gain satisfaction towards the result) was basically multiply both the sides by (1+x), which essentially cancelled out most of the terms on the left-hand side, and left the value '1'(right hand side - 1/(1+x) * (1+x) leaving '1' as the end result, lhs=rhs). That's how Newton figured it out that it's reasonable that the formula works with his substitutions. As a result, implying that there is more to the pascal's triangle than just the positive part. Anyway, he doesn't stop there, he goes ahead and plugs in a half. Leaving us with an infinite series of the sort, Essentially the idea behind this is that, can be thought if like layers in the triangle which add up to give more layer or a say a plane. At this point you could basically plug in anything into the theorem and calculate the values of various square roots in this case, to a greater precision. Since, Newton wanted to calculate the value of pi, he was interested in the value where n equalled half, why? The equation of a unit circle is, x² + y² = 1 => y= √(1-x²) This is essentially the same equation as above by substituting '-x²' in place of 'x'. He represents 'y' in two ways and basically uses calculus, he integrates the above equation from 0 to 1, since its a unit circle and he knows that the area of a unit circle is just pi because the radius is equal to one. And since he is integrating the above equation from 0 to 1, he is only looking for a quarter of the area of the circle that is, pi over four. He just takes the four to the other side and substitutes the limits to calculate the value of pi. Newton doesn't stop there, instead of integrating 0 to 1, he integrates from 0 to 1/2. The idea behind this is, when you have an infinite series, you want the values to keep decreasing as quickly, which has a sole purpose of calculating things with utmost precision. So, Newton comes to comprehend that when he replaces the limit ti 1/2, he reduces each term by an addition factor of x squared. Now, the area that he'll be calculating with the new limit is, the portion AOB plus BOC, Now BOC is a thirty-degree sector of the circle which has an area of pi over twelve and AOC is a right triangle with a base of half and height of root three on two, from the unit circle and hence you get the left-hand side given below. After some tedious rearrangement... You end up with, Before Isaac Newton people inscribed unit circles into polygons and calculated the side lengths, which made the process very exhausting and none looked at what Newton showed in the coming decades. Moral of the story : Always strive for the most unconventional methods while trying out new things, even if it demands more efforts, because the results are going to be worth that extra effort. And so, conventionally it is the most unconventional ideas that most likely dictate our understanding of various aspects of life.

- Brachistochrone
I am in my fourth semester of going about analyzing proofs and various results arising from those incredible proofs, and the Brachistochrone is a part of one of those problems, essentially giving rise to calculus of variations. The Brachistochrone a geek word meaning shortest time, initially posed by one of the Bernoulli brothers, in this case Johann Bernoulli and is by far one of the most famous problems in the history of mathematics. Turns out that it was presented as a challenge “To the most acute mathematicians of the entire world.” (Which included Isaac Newton, who was old by then but hadn't lost his esoteric nature.) Before, we get to the problem itself, I would like to share my thought as to why this problem and its solution is or might be profound. Well, one of the main reasons in my opinion being that it provides us with such a simple and elegant way of finding a connection between simple analytical geometry and essentially a law of nature, as in a physical a aspect present in the framework of our nature here on earth and the universe as a whole, if this seems confusing it'll be explicit to you, once you go through the problem and the solution. Here's what Johann Bernoulli put out for his brethren, "Given two points on a plane at different heights, what is the shape of the wire down which a bead will slide (without friction) under the influence of gravity so as to pass from the upper point to the lower point in the shortest amount of time?" Now, well before such a problem was exclusively asserted, Galileo Galilei also had the same question pop in his head well before Bernoulli, in the 1600's. And to his credit he did do some relevant work to come to a conclusion to this problem but, it wasn't enough. He studied an objects motion under gravity, depicting that a body falling in space will traverse a distance proportional to the square of the time of descent. Using this law, he was able to compute the time of descent of an object falling along an inclined plane from point a A to point B, assuming that there is no effect of friction. Then the shortest path from A to B is a straight line, but is this path truly the one that will take the shortest time? For instance, you could imagine the points A and B joined by two vectors from A to C, C to B having some angle between them, by its nature the body would fall following a straight line from A to C, and then a straight line from C to B, it would do so in less time than if it traversed a straight-line path from A to B? Galileo believed that the answer to the brachistochrone problem was an arc of a circle, i.e., that the path A→C→B should be replaced by a circle passing through points A and B. Well, not going to discuss this piece of math here but only keep the result in mind, so that we can move on to the other piece of math shown by Bernoulli himself, It turns out we found a better way, which actually minimizes time more than this arc of a circle. Before we get to that, one of the first things Johann thought about was the peculiar nature of the way in which light travels. In the 1600's, Fermat, showed us the principle of least time, in which he stated the way that light travels, whether bouncing off of a mirror or refracting from one medium or going through a lens, where light essentially bends. All the motion of light could be understood by saying that light takes the path that gets it from one medium to the other in the shortest time. Which definitely seems mystical, but as I said in the beginning, this is probably one of those big physical aspects embedded in the framework of nature, it could have been anything out of many things, could think of it as just one outcome from many and not get too philosophical about it. Fermat's principle : "All the paths that light might choose to get from one point to another, it always chooses the path which takes the least amount of time. This principle is more general since it equally applies to both uniform and non-uniform media." Johann, very intuitively assumed that instead of a sliding bead, it was light traveling through media of different index of refraction. Meaning that the light would go at different speeds as it successively went down like the bead. Light bends when it goes from one medium into another, where its speed changes, consider the angle that it makes with a line perpendicular to the boundary between those two mediums. The sine of that angle divided by the speed of light stays constant as you move from one medium to the next, this is known as Snell's Law. Johann Bernoulli proficiently used Snell's law, i.e., sin𝜃 over 𝑣 stays constant, and uses it to solve the Brachistochrone problem. When he thinks about what the bead, about what it might be going through as it's sliding down, he detects, that by the law of conservation of energy, as the bead slides down it covers some vertical distance and as a result has a transition of energy from potential energy to kinetic energy. As a consequence, the velocity that the bead will be proportional to the square root of the distance from the top, shown in the image below. The loss in potential energy is its mass times the gravitational constant times h, i.e., the distance from the top. When you set that equal to the kinetic energy, one-half times 𝑚𝑣 squared, and you simplify, the velocity 𝑣 will as a matter of fact end up being proportional to the square root of h. As a result, considering the trajectory of light as it moves from one medium to the other from above, while instantaneously obeying Snell’s law, i.e., the ratio of sin𝜃 and v is always a constant, as we move from one medium to the next, then what will be the path, such that the tangent lines as shown above, are always instantaneously obeying Snell’s law? At the end of it, the deduction that Johann came to conceive, was that the time-minimizing curve had the sine of the angle between the tangent line at that point and the vertical divided by the square root of the vertical distance between that point and the start of the curve, as shown in the figure above (Snell's law), will be some constant independent of the point that you chose. Johann Bernoulli saw this and recognized it as the differential equation for a cycloid. (Which at first glance isn't certain at all) Above, is a desmos graph describing a cycloid, which is essentially the path traced by the point on the cicumference of a rolling wheel. This is the part where things get interesting, Mark levi a mathematician who is renounced for his intuitive ways, through which he uses principles of mechanics and, more generally, physics to solve all kinds of math problems. As he points out if you examine the geometry of a cycloid, through a few modifications at the right places, the principle of velocity over sin 𝜃 being constant ,i.e. Snell's law is embedded into the motion of the cycloid itself. Which in my opinion, is just splendid. Here's the solution from his original article, a very simply and elegant explanation to how things theoretical/analytically can relate to actual physical aspects of nature.

- ALL-SILVER-TEA-CUPS Explained
One of many blunders that high school teachers commit (at-least in my case) is that the very first thing they would teach you about trigonometry, is the fact that trigonometric functions have certain sign conventions when we move from one quadrant to the other in a Cartesian co-ordinate system. In addition to that, they give you an "easy" way to memorize how this sign convention changes, i.e., "ALL-SILVER-TEA-CUPS". ALL - Implying that all the functions are positive in the first quadrant. SILVER - Implying that only the sine function is positive in the second quadrant. TEA- Implying that only the tangent function is positive in the third quadrant. CUPS- Implying that only the cosine function is positive in the fourth quadrant. {NOTE: Bullet points are inclusive of reciprocals} And that is it, that's pretty much the stuff that is presented to you, which is quite a shame, because it definitely doesn't do a rightful justice to the intuition of how these functions have different signs in different quadrants. Intuition: Another standard introduction to trig is through triangles which persists in school and in reality, trigonometry is more about circles. Also, that intuition of trigonometry in terms of circles (more specifically unit circles, since sine and cosine cycle between 1 and -1) is more usefully while using stuff like Fourier transforms and problem solving in physics, problems involving harmonic motions, problems from electrodynamics and so on. Imagine moving around along the circumference of a unit circle, Sin(x), 'x' is the input in this case and it is the measure of how far you have moved along the circle. As you can see from the desmos graph present above, as we move from the right most side of the circle which has a unit radius, and your journey around the circle is at a constant rate. As function of how far you have moved, sine graphs out the height at a certain input along the y-axis. The y-axis as depicted in the graph, is the distance between you or a point on the unit circle and the x-axis. As a result, as you transverse around the circumference the wave starts to oscillate with a constant period. From the picture it is quite clear the reason why a sine graph starts from zero, it is because by definition sine is giving you the height (y co-ordinates) and when it starts off from the right side, it starts with zero, since, the height or the output is zero and it gradually increases. 2.Cos(x), the cosine is defined very similarly, but this time it gives you the x co-ordinate values or the distance to the vertical line as you moved around that unit circle. It starts off at 1 and as you move around the circumference, recording the distance from the horizontal line, which gets lower, reaching to the value of "-1" before it starts increasing again. As a consequence of the two illustrations above, we can say that sine and cosine can be thought of as distances from the two co-ordinates, resulting from the various inputs, i.e., 'x'. Let's take an example, for the input values, look at the cosine graph. At the input pi, we have traveled halfway around the circle to the value "-1", resulting in the negative value of that function in the second quadrant, for any input the output of cosine after the first quadrant lies on the negative x-axis. Similarly, look at the sine graph for the input Pi, the output is zero and for any other input value lesser than pi, the vertical distance or the distance to the y-axis co-ordinate is always going to be positive between inputs of zero and pi. One more thing, the point where the two distances meet, if that point is joined to the origin a right triangle in formed and so some the connection between sine and cosine functions, with triangles. The intuition behind the sign conventions can enlighten us, when we think of sine and cosine as functions that help us to find the position of points along a unit circle, which in many people's sense is more important than merely knowing how to compute these values. since, we have computers to do that for us.

- Many Worlds
We humans have come a long way with regards to experiments on various fronts, one experiment in particular was very fascinating to most of us, so much so that it transcended our ways of understanding reality, it was the famous double slit experiment. But, before we talk about the ways in which the double slit experiment changed our views, it's necessary to point out one key factor regarding atoms, which has a very different explanation when looked at, through the lenses of quantum mechanical aspects. Me, being an undergraduate student, I have a rough time trying to comprehend any of it, nevertheless it is a huge leap for me from something like Newtonian mechanics. The classical model of the atom, as most of us know it, brought forth, showings us that the atoms consists of electrons that revolve around the nucleus, just like planets revolve around the sun in the solar system. This very model also tells us that most of the space in the atom is just empty, with most of the mass being concentrated at the center of the atom. But, it's a completely different interpretation of the atom when it comes to quantum mechanics. The quantum physicists tells me that, no, most of the atom is basically a cloud and it's described with the help of a wave function, since electrons are basically waves. Now, again the wave function has a weird feature to it, when you are not looking at it, it's a wave its all spread out or it is localized somewhere and it obeys the Schrodinger equation and all of it fits like the classical ways of predicting things, the problem comes when you measure it, because there is some level of you interacting with the electron. That was one of the outcomes of the Copenhagen interpretation, fields i.e., waves is what we are all made of and particles are what we see. Coming to the famous double slit experiment, if you have two slits and you let marbles go through it, you get two lines on a screen placed right behind the slits. Whereas, if you let a wave go through two slits you get an interference pattern, one wave, from one of the slits cancels out the other and you get a band of lines depending on the regions where the two waves cancel each other and respectively where the crests are formed. And so what happens when you let an electron go through those very slits, the answer is you get an interference pattern , inferring that the electron is more like wave than a particle. But, the real weird thing is, if you let an electron go through two slits and you place little detectors on the slits, in order for you to predict or say which slit did the electron passes through. Then the experiment always says that the electron goes through one or the other, and never goes through both, as a consequence the interference pattern on the other side disappears. You only see two lines that you would have seen if the electrons were marble like, so the point is when you're not looking the electron is acting like a wave and when you look at it the electron acts like a particle that's the conclusion of the double slit experiment. When you put the detectors on the slits, you have interacted with the electron and you have in some ways localized it, There's no such thing as the position of electron, there's no answer to the question, did electron go through one slit or the other?, there's only a cloud, there's only a wave going through, but you affected it with the detector, it affected the electron when looking through the slits to see, whether it i.e., the electron went through one or the other slit and that reason transposed the electron from going through both slits, to only going through one of the two. That phenomenon is called quantum entanglement, the detector gets entangled with the electron and this is where physicists came up with a version of quantum mechanics, which is the many world's interpretation. The right way of thinking about that electron was that cloud like atom which is a wave going through the slits, that's the natural representation by quantum mechanics. The weird part in this view would be that, when you look at the different slits you see the electron go through one or the other and it acts like a particle. In other words, our natural intuitive way of thinking about electrons is as particles, little marbles and quantum mechanics says that's not the precise picture, the electron is naturally a wave, the gizmo in this case is when it acts like a particle. The many world's theory would say, when you look, did the electron go through one slit or the other, you and the detector have become entangled with the electron. What that means is that the wave function of the whole universe, the wave function of both you and the detector and the stars and galaxies and so forth, splits in two and so there's now one branch of the wave function that acts like its own separate world which says the electron went through the left slit and your detector saw the electron go through the left slit and as a result, it made a little band on one side, there's another branch of the wave function, which says the electron went through the right slit and it made a band or line on the other side and so they're both still there, the world split in two and you're in one of them, Both results take place, the only reason you don't see both is because you don't see the whole world anymore.

- History Of Change - I
There have always been those last questions which remain unanswerable. Right from knowing whether certain axioms in geometry are true to how does the curvature of continuum actually work, making them the barrier between the frontier of our understanding and ignorance. One such frontier was understanding change. The earliest accounts of why understanding change was important was to decode the mystery behind motion. Above all, motion was more or less a change in speed (acceleration), a change in momentum (force) and a change in location, a mixture of the three. In the fifth century BC, Zeno of Elea a philosopher, a brilliant mind, formulated a ton of paradoxes, described by some as "subtle and profound". But at the end of the day he was a philosopher, in today's world philosophy is close to being obsolete. In the fifth century though it was of great interest. One of Zeno's paradoxes with regards to motion was according to his reasoning, when an arrow shot from a bow cannot actually move in reality. If one observes the arrow during its flight at particular point in time, it also occupies a very specific place in the air. It must be at rest in this place, otherwise it wouldn't be in a place. What is valid for one point in time must be valid for all points in time during the flight, which must signify that the arrow is constantly at rest, implying motion is paradoxical. As Zeno nicely put it, “What is in motion moves neither in the place it is nor in one in which it is not.” Now, obviously that doesn't seem precise. Summoning motion to be impossible is not only vague, it also doesn't agree with our daily life experiences. Today we understand change in a very precise manner, thanks to calculus. But why would Zeno, given that he can think very well, make such a point. Take this for instance, I drive a car on a track of about eight kilometers, I take about a minute to cover that distance. So, that is an average speed of about 37 kilometers per hour, but that evidently is not going to be a constant 37 kilometers per hour, considering all the curves and slopes and air resistance on the track. Further, it's not difficult to reckon the fitting average speeds for each individual kilometer. Or for all eight-hundred-meter intervals. Or even for every individual meter, centimeter, or millimeter. Nevertheless, these are still average values that tell me how quickly I completed a certain distance. But what if I want to know how quick I am at a very specific moment? Then we are back with Zeno's contradictory story of the flying arrow. In a single moment, I do not cover any distance, there is no line along which I have moved with a tangible speed. From a mathematical perspective, it ushers to the question of infinity. The distance for which we want to calculate an average speed smaller and smaller and simply need to divide this distance by the time it takes us to cover it. As long as the length of the distance is not zero, that's a straightforward task. At some point between “zero” and “arbitrarily small,” we meet infinity and this was mathematically difficult to comprehend in the seventeenth century. In order to calculate the speed of an object during a very specific moment, the distance has to be made infinitely small. But how? The answers as we know today came out in the following century with the advent of Newton and Leibniz.

- A Past To Revolutionize The Present - I
The following are some random historical events that have revolutionized today's life (some in indirect ways and some in no way, yet fun to know), I tried including events that are less known. If you're bored at home and have nothing to do, then probably this post might be a good way to pass your time. In around 600 BCE, Static Electricity was discovered by Thales of Miletus, when it was taken into account that rubbing amber against fur would attract small pieces of straw. In the absence of proper tools the phenomenon remained a mystery for almost more than 2,000 years. In around 2400 B.C., The Sumerians (One of the first civilization) accounted the sun's annual path cross the sky was approximately 360 days. In a requisite to track the sun's motion, they decided to divide the circle in 360 degrees. And so a circle has 360 degrees. During the plague of 1350, Giovanni Boccaccio wrote Decameron, one of the greatest collections of short stories. In a similar situation, during the plague of 1666, Newton discovered a subset of a big picture. In 1582, Pope Gregory XIII introduced the Gregorian calendar. Used by almost all of us today, stripping out the inaccuracy of the previously followed Julian calendar, making a very credible contribution to daily life. In 1781, William Herschel discovered "George", today known as Uranus. In the Consequent century, his son, John Herschel pioneered color photography. In 1883, Krakatoa's volcanic eruption left wide (wider then the wavelength of the color red) particles in the atmosphere, causing the scattering of blue color, leaving the moon blue for almost 2 years and giving rise to the phrase "once in a blue moon." In 1897, J.J. Thomson showed electrons are particles. And in 1937, his son G.P Thomson showed that electrons are waves. In 1911, Ernest Rutherford discovers that most of an atom is just empty space, leaving him terrified of getting off his bed the following morning, fearing the empty space he would step onto. In 1920, Lewis Richardson, an English mathematician, physicist, meteorologist, psychologist and pacifist, who pioneered modern mathematical techniques of weather forecasting, 6 weeks to predict the weather 6 hours in advance. In 1980, Chemist August Kekulé recounted how he had only discovered the structural formula for benzene, during a daydream, he had seen a snake biting its own tail, which he then later interpreted as the molecular bond between atoms. In 1968, Apollo 8 left earth to reach the moon's orbit. In the process we discovered ourselves through the picture termed earth rise. It left an immense hallmark on all of mankind and so two years later we celebrated the first ever earth day in 1970.

- Isaac Newton and the "Philosopher's stone"
Not quite Harry Potter, yet a true story. While digging through the internet I came across this interesting piece of history, and thought why not share the story and consider this a continuation to one of my previous posts Not so egocentric Issac. In the 17th and 18th century Alchemy was in a diminished state. One of its more widespread goals was to find the philosophers stone, which could presumably turn lead into gold and ensured an imperishable life. The Alchemist, however, weren't especially approved in the day. One of the reasons was their means of falsely claiming to have knowledge and their highly deceptive nature. In 1688 there was even a law in place called "act against multiplication", making it a punishable offense to "multiply gold and silver". Newton, however, couldn't care less, he conducted his experiments in secret. There are over a million words written by him on the very matter. In his perspective Alchemy was a metaphysical enterprise. The most peculiar characteristic of Newton was his way of interpreting things and his lack of interest in showing or exhibiting his findings. Most of his writing are difficult to understand not only because they are complicated, but because Newton never intended for anyone to comprehend his work. He deliberately made it difficult for the reader. No wonder, students did not attended his class and he ended up teaching to the walls of classrooms for most of his career. According to him, you truly have to be worthy to apprehending what he has discovered. The Alchemist in this period did not communicate with their real names, but used pseudonyms. Newton called himself "Jehova Sactus Unus" which translates to a holy god. It is important to know that this was the 1600's and religion was quite dominant with regards to giving "answers". In today's world Newton is found to have believed in god, but that information is highly misinterpreted. He did believe in an existence, but never approved of the interpretation that the conventional society gave to it, also he couldn't disagree publicly, because of the fact that he could have been punished by death in that time, for such beliefs. But, he did end up conducting his research on various ancient scriptures, for he believed that he could find an answer to god in them. As a consequence, he spent most of his life studying theological and alchemistical aspects, physics was merely a hobby for him, discovering gravity, developing optics and a new branch of mathematics was merely just a hobby. Woah! I find his story quite relatable (not exactly but with a few similarities) to that of Leonhard Euler. In 1736 Euler was grueling to develop a way to cross every single one of the seven bridges of the city of Königsberg at exactly the same time. He realized that this seemingly predictable problem was impossible to solve and in the process he created a new branch of mathematics, the graph theory. He never intentionally set out to discover something (or invent, the argument between discovering concepts and inventing them remains one big question in our conscious mind in my opinion), but yet he did end up discovering something big. In a similar instance, Newton did not intent to discover what he did discover. Backtracking to the "Philosopher's stone", In 1693 Newton outlined his work in alchemy in a text containing more than five thousand words. With him exclaiming, "amalgamating the stone with the mercury of 3 or more eagles and adding their weight of water..." Whatever it was that he had written, obviously did not work, this was one of those moments when he turn out to be wrong. A huge blow to him personally, as a result it turns out he went into depressions. And by the end of the 17th century and the beginning of the 18th century, alchemy was mostly dead. The depression was quite justifiable on Newton's part, the dude had just written over a million words on the subject, so that's quite a reason to be unhappy. Anyway he did come out of it and continued his work as an administrator.

- Simultaneity
Imagine you're iron man and you're traversing through air by flying (also it's at night). You're also moving with a definite velocity without a change in your direction. Now, there are two 747's on your path having the same velocity but are in opposite directions to each other. Since, the given description is at night, it is obvious that both the jet have their strobe lights turned on. Now, considering you as a rigid reference body you find that the light from both the jets doesn't reach you simultaneously, whereas, when i look up at the sky i find that the light from both the jets reaches you simultaneously (considering i am observing the event from a roof top at rest with regards to my co-ordinates). Consider, another instance and this is one of my favorite examples, its by Brain Greene from his book 'The elegant universe'. Picture that there are two nations that have long been at war. They are called forward and backward land. Now, the two nations have finally come to a peace treaty, but neither presidents of these countries wants to sign the treaty before the other. So, they need some scheme that they will each sign the treaty simultaneously. They consult the UN and come up with a plan. The two presidents will be seated at the opposite end of a long train and a bulb will be placed between them and the bulb will initially be in the off position. And then the secretary general will turn on the bulb, sending light heading left and right and since the speed of light is constant, it isn't effected by the direction in which it moves, also both presidents are equidistant from the bulb. So,the travel time of light to each presidents eye ball will be the same, when they see the light each president picks up the pen and signs the treaty. And what they add to the process is that the general says, when you sign the treaty let the inhabitants of the two nations that live on the opposite sides of a railroad track, let them watch the signing ceremony. As the train could go right along that track and both presidents agree to the idea, so that both parties could witness this historic moment. On the day of the ceremony, the two presidents are seated in the train, with the president of forward land facing forward and the president of the backward land facing backward. And when the train approaches the station where the inhabitants of both the nations are present, the general decides to turn the bulbs on. As he does so, the light reaches the two presidents on-board the train simultaneous with regards to the general and the two presidents sign the treaty. But, later the news arrives that a riot has broken out at the station, and the reason behind it is that the light did not reach the two presidents simultaneously. The light reached the president of the backward land first with regard to the people on the platform. The other example would be the one Einstein initially gave, visualizing a particular body of reference, like a railway embankment. Considering, a very long train traveling along the rails with a constant velocity and in the same direction. People on-board the train will consider the train itself as a rigid reference-body, they regard all the events in reference to the train. Now contemplate a lightning has struck the railway embankment at two places far distant from each other. Now, obviously the main question that arises is whether the two events, i.e., the lightning strikes, are they simultaneous with reference to the the railway embankment and also simultaneous relatively to the train? And the answer to that would be that the two lightning strokes are simultaneous with respect to the embankment, and the people would find that the two strikes occur at the same time. But, for a person on-board the train which moving in the direction of one of the strikes, would find that the two lightning strikes aren't simultaneous. He/she would find that the lightning strike toward which they are traveling will occur first. The three examples tells us that the events that seems to be simultaneous in view of one frame of reference is not the same in any other frame of reference, that is relativity of simultaneity. Every co-ordinate system has its own time, Further we know the rigid reference body to which the statement of time refers, then there is no sense in a statement of the time of an occurrence. In the three examples its evident that the light reached first to the reference body that was traveling in the direction of one of the sources in a lower period of time, as a consequence the two sources of light doesn't seem simultaneous to that reference body. This itself disregards the fact, which was known before the theory of relativity, that time is independent of the state of motion of the body of a reference. It gave time a completely different perception, time is more like the progression of events with regard to a particular reference or co-ordinates, which can change from place to place. Everyone of us have a different perspective of time, we all have our own time.

- Not So Egocentric Isaac
This post is a continuation to my previous post egocentric Isaac. I admire Isaac Newton, not only for his groundbreaking work as a teenager, but for his fearless attitude towards those who were on a higher pedestal than him. One such person was Robert Hooke, who was at-least seven year's senior to Newton and was a renowned scientist of the time. In 1663, Hooke was accepted into the royal society and was its first curator, in-charge of all experiments. He was allotted the task of bringing portrayals of things he observed with his microscope to the weekly meetings they held at the society. He was possibly the first individual to have conducted ample use of the microscope to understand the world of small things. The royal society authorized an entire book in 1665, Micrographia, based on his impressive work on enlarged insects. He also included a theory of the nature of light and colors in the book. According to him light arose through movement and that everything that was shining was vibrating in some way. He also expressed that there were two colors - red and blue, When pulses of light come in contact with the eye, according to him these two color perceptions were created. He also stated that," It would take a little too long to explain all of that in detail and to prove what kind of movement is responsible. It would take too long to insert how i found out the characteristics of light." Now, considering that statement from a renowned scientist such as Robert Hooke, one would believe him. But, Newton wasn't in that category. His reaction to Hooke's view was hostile. And there was no reason you could disagree with Newton's behavior, the dude had stuck a needle through his pupil to figure out how light works and found its characteristics. He asserted that Hooke's theory was not correct, it was obvious for him to react to other people who hadn't conducted experiments and who lacked constructive evidence to prove what are the real characteristics of light. And so, there was the beginning one of many feuds in Newtons life time. To be honest, he was way ahead of Hooke from the scientific research perspective. And its probably wise to argue that in this case his anger would have transcended in the same manner if it was any other person. For instance, in 1679 Hooke shared his idea of explaining the motion of planets by a force of attraction whose strength changes with distances, well Newton had the same idea back in 1666 and he even ended up deducing it mathematically. That's how ahead Newton went, but his nature of not indulging with the public to discuss his ideas made it a little difficult for people to discover the intellect that he had, in a time where even the quantities such as mass, momentum, velocity... weren't yet objectively deduced. Even Newton's Principia starts off by exemplifying names to the different quantities in nature. Backtracking to the characteristic of light, Hooke asserted that he himself had already carried out the experiment that Newton had expressed in his work. He also made an accusation that Newton had falsely interpreted the findings from the experiments. Newton was angered by all of it, considering the man was ahead of his time, he had no patience to wait until people could grasp his findings. They would consider them as just hypotheses, which offended Newton by a great deal. Since, he had developed a lot of mathematics and physics apart from his theory on optics (which was and is mostly correct), it turns out he was outraged on various fronts with regard to how people interpreted his work. The conflict prolonged with Hooke exclaiming that light consisted of waves, while Newton thought it was a current of particles. Hooke stated that he had done an experiment that showed the wave nature of light. While, Newton explained that this experiment had been carried out by other scientist's much earlier and Hooke shouldn't call it as his experiment. Hooke accused Newton of having copied ideas from his articles and that he hadn't mentioned the source. The quarrel continued and the question of the true nature of light wasn't answered until a century later, when quantum mechanics was established, proving the duality of light. Anyway, the feud definitely left Hooke on the losing side, with Newton's mathematical approach to things being more valuable, he described them in his book Opticks in 1704. For what Newton was, he never could handle criticism and that is one of the reason why he did not reveal most of his thoughts on gravity, considering that the latter was still uncertain in that time, until about 230 years later with the advent of Einstein. Compared to the last post about his feud with Flamsteed, when put on a timeline this feud fall before that one. Where Newton had all the power with regards to him being the president of the royal society, giving him an ego boost. However, in this incident Newton never intended to be in the limelight. He was the king of nerds, it was only after his entry into the royal society that he began to indulge in other activities, such as his role in catching hold of William Chaloner. The feud with Hooke was when he was not so egocentric.

- Coronaverse
From the deep-seated crucibles of stars to your huge subsequent divergence, Set off by a supernova and trillions of random DNA mutations, leaving chemically rich elements throughout various galaxies and about three billion nucleotides in every one of your trillions of cells. On one hand giving rise to the chemistry of life and forming a prequel to the other, giving rise to a single precise sequence to make you, develop you, run all your physiology and even dictate your instincts. If your not unique then I don't know what is, baring that if you wish to preserve that uniqueness, then the best thing you could do is stay home. One of the others reasons to stay home would be the very nature of how nature works. For instance, considering the current research scenario of the transmission of the virus through droplets which can not stay in air for a longer time due to the effect of gravity and conditions such as temperature and atmospheric pressure ( also note that experts have figured out that the virus could stay airborne, except at a particular condition of temperature and pressure, considering that the jury is still yet to be out on it). So, that makes the chances of an encounter with the virus more likely when you touch surfaces of various kinds and through fellow humans as well. And so the notion of the nature of how nature works comes into play when you consider factors such as how the water you drink, the air you exhale, all the moist that comes out of one's body is spread throughout the surface of the earth depending on time, making you more susceptible to the virus. A bottle of water at your home will probably have more molecules of water, than there are bottles of water of that volume in all of the worlds oceans. Considering that if even a single molecule of water is infected, then the person carrying the infected molecule is more likely to spread the infection through contact with other surfaces or direct human contact by the means of sneeze or a cough. Establishing mutual distancing as the only viable option. And probably all of this information is familiar to most of us, but a lack of an intersection between principle and execution wouldn't help us lead the frontier between us and an invasion by a species no different than an alien.