The Great Circle

Something as elementary and intuitive as ‘the shortest distance between two points is a straight line’ is completely thrown off if we change the kind of surface we’re dealing with. This element of Euclidean Geometry fails for Non-Euclidean surfaces such as a sphere. If we change our surface to a sphere, the shortest distance between two points is no longer a straight line, it’s a curved line that contains the diameter.
It may be argued here that the straight line joining two points of a sphere in space is the shortest distance, even it goes through the sphere. But the space inside of a sphere doesn’t exist if we’re on 2 dimensional sphere analogous of a plane surface. The sphere is our surface and space inside it or the space outside isn’t on this surface and doesn’t exist.

The shortest distance on the sphere between points A and B is the arc of the great circle that passes through them.

A great circle is an equator. It cuts a sphere in half and is the largest possible circle on the sphere. It is possible to draw as many such circles as you want. Between any two points on a sphere, a segment of a great circle can be drawn. And this segment is the shortest distance between those points on that sphere. This is called a great circle distance or orthodromic distance.
Even though earth isn’t entirely spherical, it can be estimated as such and a great circle route is a close enough approximation for the shortest distance between two places. This fact is heavily used in navigation. Routes taken by airplanes or ships should stay close to the orthodromic distance or it might mean a waste of fuel and time. Navigation on a sphere cannot be done on its plane surface analogue as a sphere is a non-euclidean surface and has rules of its own that cannot be recreated on a map.

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What is Summer Solstice? The Science Behind Summer Solstice

On this day of Summer Solstice, as eager polar bears watch on for the sun at its highest in the arctic sky, the rest of the northern world will bask in the extended daylight hours and celebrate the official beginning of summer. At the same time, the south pole will be engulfed in moonlit darkness and the southern world will enjoy a longer night and the beginning of winter.

The earth rotates around an imaginary line called an axis and revolves around the sun on its orbit and completes a rotational cycle daily and a revolution cycle yearly (which define the length of a day and length of a year). This axis of rotation is at a 23.4 degrees tilt relative to its orbit around the sun. Throughout the revolution cycle, earth’s inclination relative to the sun keeps changing as the sun looks upon it from different directions.

A solstice is a day of the greatest inclination of the earth towards the sun. There are two such events annually, when the north is inclined towards the sun and when the south is inclined towards the sun.

Summer Solstice over northern hemisphereThe part inclined towards the sun receives more sunlight compared to the part away from it and observes a summer solstice with the longest daylight hours of the year. At the same time, the part of the earth away from the sun observes a winter solstice with the shortest amount of daylight received in the year. This is what marks the seasons as well. The part of earth inclined towards the sun has a summer and the other, due to less sunlight, has a winter.

The solstices are an announcement of the seasons to come for the northern and the southern dwellers of this happy planet.


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Johannes Kepler and The Ellipse

Scientific discoveries and their explanations are often received page by page in a book full of unwritten pages. Each page is a part of the story without quite ending it. Copernicus housed one such page in the book of planetary motion. He described the solar system as heliocentric – that the planets run in circles around the sun which lies at the centre. He wasn’t quite right but wasn’t wrong either. The planets run in ellipses, which are flattened circles, instead of perfect circles. This made for another page in the book, written by Johannes Kepler.

The remarkable thing about this was that he formulated this theory of elliptical orbits entirely without the knowledge of gravitation . Planets follow an elliptical path around the sun as the direct consequence of gravitation. This leads to the question – why elliptical? It could be attributed to the fact that there are other gravitational forces acting on planets besides the sun which could result in a deviation. Also, the sun isn’t stationary. The sun orbits around the centre of the Milky Way Galaxy and this, coupled with the movement of the earth around the sun, could result in a slightly elongated circle.

Just like a circle is centered at a point and every point on the circle is the same distance (radius) from that point – an ellipse is centered at two points. An ellipse is derived in such a way that the sum of it’s distances from the two points it’s centered at, the foci, is the same.

The two points from which the red and the blue lines originate are the foci of the ellipse.


The orbits of planets are similar, with the sun positioned at a focus.

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Isaac Newton and Philosophiae Naturalis Principia Mathematicia

Isaac Newton, painting by Godfrey Kneller in 1689

Isaac Newton, painting by Godfrey Kneller in 1689

In 1664, as The Great Plague swept through London in waves of fatalities, one man returned to his hometown in the country, shielding himself from the epidemic and formulating the biggest theories of physics in his backyard. Supposedly, an apple fell onto his head and he wondered why. Could there be imaginary threads of force binding every piece of matter in this universe? The answer is yes. This ‘gravitational’ force ties us to this planet and the planets to their orbits around the sun, and binds this cosmos together.

Abandoned by his mother at a very young age, Isaac Newton was cared for by his grandparents who educated him and recognized his interests in the academics. He went on to study Mathematics at the University of Cambridge, where he didn’t excel but obtained a Bachelor of Arts degree. This is attributed to the fact that he read heavily from modern philosophers during the time in addition to balancing a part-time job to supplement his education.

Woolsthorpe Manor - Isaac Newton

Woolsthorpe Manor, where Isaac Newton spent his childhood and returned when The Great Plague hit Cambridge. Now open to public access.


When The Great Plague came to Cambridge, universities were forced to shut down and Isaac Newton retired back to Woolsthorpe manor and studied and carried out research in a variety of topics, independently. In these 18 months, he founded calculus, the theory of light, and his theories of motion. These breakthroughs remained unpublished for decades.

After returning to University and achieving a Master of Arts degree – Isaac Newton became Lucasian Professor of Mathematics. Among his first lectures, he demonstrated a reflecting telescope of his design which attracted the interest of the Royal Society. He became a fellow of the same in 1672 and published his initial notes on optics that claimed that light was made up of particles rather than waves. Unable to handle the criticism his notes received, he had a nervous breakdown and isolated himself completely.

Philosophiae Naturalis Principia Mathematica, third edition, 1726 - written by Isaac Newton

Philosophiae Naturalis Principia Mathematica, third edition, 1726 – written by Isaac Newton

In 1684, Edmond Halley visited Isaac Newton and persuaded him to mathematically derive the theory of gravitation, offering to fund his research. This led to the publishing of Philosophiae Naturalis Principia Mathematicia (or Mathematical Principles of Natural Philosophy) in 1687. Written in Latin, this book came to be considered one of the most influential books of the scientific world. It laid out his famous three laws of motion: a body at rest remains at rest unless acted upon by an external force; force is mass times acceleration; and every action has an equal and opposite reaction. Along with these theories, he derived the equation for gravitational force and successfully explained planetary motion. And he used his theories of calculus to accomplish this, filling the gaps that remained in physics. Years later, he also published Opticks, dealing with his theory of light. It is no understatement to say that he revolutionized science. In addition to this, he also studied alchemy heavily and carried out independent research in the same. He is considered one of the last alchemists.

Although his life spiraled into frequent episodes of depression and aggression and he became tyrannical due to his new found power, he delved into the world of science and unearthed secrets of this universe. Secrets that got us a step further towards knowing everything, and knowing nothing.

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The night sky

The night sky

While its origins are unknown, I imagine the endless immensity of infinity being invoked while people gazed at the night sky. The number of stars speckling the sky might have been considered too large to be measured. Intertwined with the endlessness of the dark spaces between them: It’s infinite, people would say.

A mystifying concept – infinity is the amount that is greater than anything and everything. If we’re talking about numbers, infinity is that number which is greater than any number you could possibly think of. And it’s not even a number, it’s that quality of being greater than anything.

Unsurprisingly, infinity is a topic that has generated a lot of philosophical interest. Some have linked it to God, others find debating over the existence of an essentially non-existent quantity, quite absurd. Nonetheless, it’s impossible to reject the awe a topic such as infinity brings about. It’s something that exceeds every bound possible, it’s something whose inherent trait is to surpass anything. And yet, from calculus, we know that it’s embedded into the very molds of this universe.

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Earth is at the Centre of the Universe

Nicolaus Copernicus - painting by Jan Matejko

Nicolaus Copernicus – painting by Jan Matejko

The earth is a special place, it is home to complex creatures called human beings that are capable of anything – creating large buildings for its aristocrats to live in, invoking the non-existent, and brewing fear in the hearts of people for millenniums. Although, they do tragically fail to end systematic exploitation of  the poor among them, human beings must be special. The entire planet must be special. And to credit this specialness, it’s only fair that planet earth acquire a special position in the universe: naturally, its centre.

In the years preceding the 15th century, this was the only truth.

Nicolaus Copernicus (1473-1543) was a Polish scientist who challenged this view. He proposed that the sun was at the centre of the universe instead, and planets revolved around it in circles. A student of mathematics with a keen interest in the cosmos, he meticulously studied the night sky and perfected this planetary model. And in 1514, he completed a manuscript laying out his assumptions and offering to provide formal proof.

His theories weren’t entirely correct, however. The sun doesn’t house a special place in the universe either. It’s just a moderate-sized star taking up a random location in the universe. Also, planets revolve around the sun in elliptical paths rather than in perfect circles. Despite these errors in his judgment, his model was a great measure of progress.

Just before his death, De revolutionibus orbium coelestium, his life’s work, was published. It is believed that he died clutching the book in his hand, taking pride in the scientific progress he facilitated.

The book was banned posthumously, and the ban wasn’t lifted until centuries later. For the centuries that remained in dark about this development, the earth was and would always be at the centre of the universe.

Recommendations: Reasoning with Chaos

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Existence of Extraterrestrial Life

Arecibo Observatory, Puerto Rico. Used to conduct a sky survey in hopes of making contact with intelligent extraterrestrial life forms.

Arecibo Observatory, Puerto Rico. Used to conduct a sky survey in hopes of making contact with intelligent extraterrestrial life forms.

The expansion of the universe gave us clues to how large it actually is. The edge of the currently observable universe is about 46 billion light years away from us. That means, it would take light about 46 billion years from the edge of the universe to reach us. And as we’re debating its size, the universe would have already expanded further. Since the universe surpasses every definition of enormous, is it possible that extraterrestrial life exists? I wouldn’t rule out the possibility.

This is what drove scientists to pursue the search for extraterrestrial life. This was the driving motivation behind the SETI (Search for Extraterrestrial Intelligence) project. How could we be anything special? But any concrete evidence of such life hasn’t been found. Some people attribute it to the large size of the universe. They contest, the size of the universe is large enough to harbour life, but the distances between these life forms might make it impossible for them to ever make contact. Some people contest that the life forms close to us might not possess enough intelligence to make contact at all – maybe they’re microbial life forms. Or maybe both are true. The unlikeliest of all claims is that aliens have already made contact and governments have been withholding this information from the rest of us.

Despite the failed attempts, it’s unlikely that we’re anything special. What’s likely is that the technology we have in our possession isn’t enough. Life exists. And it cannot just exist here. Yes, the evidence at hand points to the contrary. Perhaps the universe is making a mistake?

Image: Arecibo Radio Telescope – Public Domain – NSF
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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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