Differential Calculus

A drop of water falling off a leaf in black and white

The course of a water droplet running across a leaf, making its way through the veins, and finally edging off to fall off it – can be expressed as a relation between three-dimensional coordinates of the system where the leaf resides. It doesn’t lose its beauty just because it can be described in more mathematical terms. It enhances it, really.

The discipline of Calculus is about movement and change. And this change is measured awfully accurately. In infinitesimals, a size so small it cannot be measured. And, infinity, a size so large it cannot be measured. So, calculus uses the immeasurable and incalculable to describe and define the real, observable, measurable world. Most people are very dismissive of calculus, complaining that they see no point in it. Every movement, every pattern, or object is a mathematical relation, and calculus studies it all. The real world wouldn’t function without calculus.

Relations, like the trajectory of the water droplet falling off a leaf mentioned before, operate calculus. And calculus expands on relations and paints a picture of their behaviour. Every instant of that trajectory contains important information about its movement. Differential calculus differentiates this movement – and finds out the rate at which this relation changes, at every infinitesimal point. And then combines them all together to create a relation of the rate of change of a relation. This relation can be further differentiated and so on.

Since each point is infinitesimal, there will be infinitely many such points between any two numbers. So what differential calculus really does is put together infinite infinitesimal numbers to find out the rate of change of a finite relation.

And I find that bafflingly fantastic.

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How Do We Know The Universe is Expanding?


The center of the Milky Way Galaxy from the mountains of West Virginia

It was common knowledge in the 1920s that the universe was static, infinite, and without a foreseeable beginning or an end. The universe was this absolute, unshakeable being. No modifications to the staticity of the universe were acceptable. It would seem as if scientists themselves were avoiding the truth, molding theories around a static universe and refusing to see it any other way.

Gravitational force deemed it impossible. All bodies with mass attract each other under an invisible gravitational force. Everybody knew it to be true. A static universe is doomed to collapse under the influence of gravitational force of its masses. Yet, everybody chose to believe in a static universe. Reason is a feeble thing. Without an advocate, reason would be surpassed by intuition every time.

Some proposed that gravitational force must become repulsive at really large distances, and this, coupled with the short-range attractiveness of gravitational force, would stable out the universe and there would be no movement. With close inspection, it is easy to see that the universe is extremely unstable this way. Even if a tiny speck of mass wavered in its position, the attractive or the repulsive force would beat the other and the universe would either collapse or expand into infinity. The universe just couldn’t be static.

The Doppler’s Effect

The relative motion of a wave-producing source and a receiver changes the frequency of the wave. When the light source or the observer are relatively moving away from each other, the frequency of the wave decreases (red-shift), and if they’re moving relatively towards each other, the frequency of the wave increases (blue-shift). That means, a light source  moving away from us will appear to be red-shifted, that is, it’ll appear redder that it actually is (if it lies in the visible spectrum). The Doppler’s Effect served as a tool for figuring out that the universe is expanding.


Edwin Hubble, 1931

In 1922, Edwin Hubble had set on letting the facts prove or disprove theories. He was already in the process of measuring the distances of many stars from us based on information about their luminosity, and discovered that they lay beyond the limits of our Milky Way Galaxy. It was a popular belief at the time that ours was the only galaxy. He also compared the apparent light spectrum of stars from other galaxies and compared them to similar stars from our own galaxy, and discovered that most of them were red-shifted. In a static universe, there should have been as many red-shifted stars as there were blue-shifted ones – to balance them out. Also, the speed with which the galaxies were moving away from us wasn’t random – it depended on their distance from us.

It was tempting to believe at this point that we lied at the centre of the universe and all other bodies were moving away from us. This was soon ruled out. In an expanding universe, every point would move away from every other point. There was nothing special about us.

By 1929, Edwin Hubble shook the scientific world with his discovery, but in their hearts they knew it was true. They knew that it was only probable explanation. Even though the reason for this expansion remains unclear, everything would now fit perfectly into the otherwise undeterminable and uncontainable universe.

The answers lie in the cosmos.

Images: Forest Wanderer, USA/ Wikimedia Commons ; Public Domain


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Alchemy: More than Turning Base Metals into Gold


A typical alchemical laboratory, as depicted by Pieter Bruegel the Elder,1558.

Alchemy came into existence like a winged creature touched by the lips of God, bringing with it the promise of wealth and happiness to the poverty-stricken. Little did they realize that despair was upon them and they were driving themselves into the depths of more poverty. Surely, alchemy was evil. It was sorcery. It was not to be trusted.

It might have been evil, but alchemy was the scaffolding of modern chemistry. Unlike the pseudosciences of astrology and phrenology, alchemy wasn’t a fragment that diverged from true science early on, but rather what evolved into it. Alchemy did observe a fair share of charlatans claiming they could bring easy wealth, but it also had genuinely inquisitive minds who wanted to look inside matter and discover its secrets. Alchemists were no less than modern day particle physicists.

It all began with matter.

The first inquisitions about matter came well after mankind had used them for thousands of years. Iron, bronze, and obsidian were used to make weapons, gold was used in jewelry, glass was used to make beads and jars. It was also observed that matter could be changed – water evaporated, and fruits fermented to make alcohol.


Empedocles, line engraving, 1580

Greek philosopher Empedocles proposed (and incorrectly so) that the universe was made of four elements – fire, earth, water, and air. The world was not in possession of appropriate technology to put this theory to test. Empedocles’ contemporary, Democritus, suggested that matter was made up of indivisible particles which were in constant motion, and that these particles only differed from one another in shape and arrangement and could combine with each other. Democritus’ theory was not too far removed from modern atomic theory. He was 24 centuries ahead of his time. However, his theory was rejected and attacked, and fell into a downward spiral when Aristotle validated the four element theory. It is believed that Aristotle was responsible for inhibiting the advancement of chemistry. But alchemy, which is based on the four element theory, does go on to pave the way for modern chemistry.

Fire, Earth, Water and Air.

The world, as alchemists believed, was made up of fire, earth, water, and air. All matter was made up of these elements, and their form could be transformed without changing the elements. Ofcourse this theory is wrong and the world isn’t made up of 4 elements, but it is made up of four states – solid, gas, liquid and energy which can transform one of these states into another. There is some truth to this theory after all.


The Alchemist by Cornelis Pietersz Bega, 1663

Later, alchemists began to believe that essentially, matter had unity. That all matter had a common origin or they possessed a common “soul” housed in a particular form – the outward forms could be changed. In their pictorial representations, this ‘soul’ of matter was often depicted as a white bird flying away from a substance when heat was applied and re-entering it after its form was effectively changed. This changing of forms was known as, in alchemical terms, a transmutation. This is where the idea of transmuting lead into gold came from. The gold scrounging folks saw their chance and became self-proclaimed alchemists.

Nevertheless, there were many alchemists with a nobler objective – the quest for knowledge. They tested virtually every substance known to man at the time, from commonplace materials like wood and stone, to the rare. They performed countless transmutations, and failed, but succeeded in discovering many chemical compounds, acids, salts and alkalis. They are attributed with the discovery of elements like antimony, arsenic, bismuth, phosphorus, and zinc. Metallurgists and doctors frequented them for their chemicals, tools and techniques. They were essentially, chemists.

Science owes a lot to alchemy. When the basic principle behind alchemy was finally replaced with its correct alternative, scientists had a head start. They had a written record of all the properties of every material known. They had the tools, the experimental techniques, and the idea of the laboratory itself.

Alchemists dug deep and uncovered secrets we wouldn’t have known. They purged the darkness for us. They were the last of scientific wizards.

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An Introduction to Non-Euclidean Geometry

We live in an imperfect world where the ground isn’t close to a reproduction of a flawless flat surface. The land is heavily uneven, only speckled in some places with evenness and geometrical figures when drawn on such surfaces are broken and distorted.

In other words, conventional geometry fails. It fails for anything that isn’t flat, practically the entire universe.

For years following Euclid’s Elements, mathematicians were baffled by the parallel postulate. Its mere complexity in comparison to the other postulates was perplexing. It didn’t seem simple enough to be considered intuitive. And on several occasions, geometries of the universe didn’t obey the parallel postulate. So, mathematicians tried to formulate a proof. However, they stumbled upon geometries that didn’t satisfy the parallel postulate and worked in all the instances ‘ordinary’ geometry failed. Such were the cases of spherical or hyperbolic surfaces.

A triangle on a spherical surface whose sum of angles exceeds the 180 degrees predicted by Euclidean Geometry.

In the 19th century, Janos Bolyoi and Nikolai Ivanovich Lobachevsky independently formulated and published works on hyperbolic geometry, negating the parallel postulate. And it became clear that the dilemma was very real.

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The Fourth Dimension

Klocka,_T_Tompion,_ca_1680_-_Livrustkammaren_-_47602.tif - CopyThe first three dimensions are the measurable quantities that describe every physical object in the universe. They are visible, recognizable and measurable. The fourth dimension, however, is an entirely different thing.


Time. The river of perception through which events flow. The imaginary sequence. You can measure it with a clock, but you cannot move it or change it any way. Time isn’t even a thing on its own unless we use it to measure changes in distances or states of other objects.

Socially and philosophically, time is considered to be very important. All living forms mature and age with time, and eventually die. But the universe doesn’t concern itself with the aesthetic value of human life. Life spans mean nothing. Seconds, minutes, weeks, months, years mean nothing. In fact, the universe can and does exist in multiple times, completely unbiased towards all.

I don’t mean to say that timekeeping should be discarded. It shouldn’t. It has value, and it has a story.

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A Leap Into Conventional Geometry

Conventional Geometry would have been nothing but a series of broken, non-intersecting concepts without Euclid, the mathematician who laid down the framework of geometry for us. The genius. The author of a book with a compilation of this information. A book which was to become one of the most powerful works of mathematics.



It was very clever of him to put down this information as postulates, providing logical reasoning while leaving deductions to the user. He refused to solve our problems for us, he would only teach us how to solve them.

He began Elements by describing the objects of his postulates, and then proceeded to state them while labelling them as ‘intuitive.’

Euclid’s Postulates:

  1. There is a straight line that joins any two points.
  2. Any straight line segment can be extended indefinitely.
  3. A circle can be constructed with a line segment (radius) and a point (centre).
  4. All right angles are equal.
  5. (The Parallel Postulate) If a straight line intersects two other straight lines and forms interior angles less than right angles, then if original lines are extended indefinitely, they will intersect on that side.

Euclid derived these postulates based on observations, and anybody can proceed to confirm the truth of his theorems by constructing these geometrical objects for themselves. Despite the lack of a proof, the logic behind these postulates does seem intuitive.

However, if there’s anything science has since taught us, it’s that intuition knows the tricks of deception. These theorems might seem correct to the extent of what is observable, but they fail in many instances. In fact, they are only true for ideal, well-behaved surfaces. The world we live in is much more imperfect.

More on this later!

Citations: Weisstein, Eric W. “Euclid’s Postulates.” From http://mathworld.wolfram.com/EuclidsPostulates.html

Image: Hector Zenil/ Wikimedia Commons/ CC-by-SA

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Theory: Matter and Antimatter are Gravitationally Repulsive

“But that’s not how gravitational force is supposed to work!”

An experiment under reliable conditions testing the gravitational interaction between the two hasn’t been successfully conducted yet. I am entitled to this claim.

This hypothesis solves one of the major flaws in the big bang theory – according to which matter and antimatter were produced in equal quantities.

But, most of the observable universe contains matter and shows little to no signs of antimatter. So they must have unmixed somehow – which is unlikely because they annihilate in each other’s presence. But if they did annihilate, there should have been nothing left (except photons, which have no mass). The explanation offered was that there must have been just a tiny bit more matter, so there could be some remaining matter after all the matter-antimatter pairs were exhausted.

This is where the theory comes in. Matter and antimatter repel each other gravitationally. That gravity isn’t only an attractive force.

If the masses of antimatter and matter produced were suffiently large, it’s possible that the gravitational force between the two exceeded their tendency to annihilate. And we’re talking about repulsive gravitational force, so they separated and went opposite ways.

Maybe the universe is a long stretch of space with two ends, one made entirely of matter, and the other made entirely of antimatter. Maybe there’s another ‘world’ made entirely of antimatter, housed by antimatter beings. Maybe an anti-apple with fall on an anti-Newton’s head and inspire him.

I hope the universe is large enough to accommodate all this speculation.

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Reasoning with Chaos

Galileo_Galilei_showing_medicean_planets bigger sizeSince the earliest beginnings of humankind, there was a yearn for order in things. It was a rather tragic discovery that there wasn’t any. The unkemptness was unsettling. And humanity immediately set on creating an absolute, impregnable order. Or several of them.

Then, the scientists arrived. The eccentric theorists with wild hair and even wilder ideas. It was no surprise that they were greeted with disapproval by the well-polished and orderly people who had risen above mankind and its happenings. They offered an unconventional approach to the world and threatened its stability.

All disciplines of science seek to untangle the natural chaos of the world and make it understandable. Science believes that chaos isn’t randomness. The fact that it is understood doesn’t imply that it can be predicted, and its unpredictability doesn’t imply that it’s merely a hypothesis or theory. Chaos can be reasoned with.

Yes, science isn’t absolute. It’s chaos. It’s nature. It’s us.


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