# Author Archives: Keerthi Vasan

A Physics and Mathematics fanatic | Author of Fuck Yeah Physics! ( fuckyeahphysica.tumblr.com)

# Aviation 101 : Flight Dynamics

To most people, the sky is the limit.

To those who love aviation, the sky is home.

All vehicles are free to operate in three dimensions i.e the longitudinal, vertical and horizontal axes. But while cars are limited in that they can’t really take off from the streets, airplanes can really take advantage of all axes.

In an aircraft, movements are known by Pitch, Yaw, and Roll, respectively.

## Pitch

Motion about the lateral axis is called pitch. This is a measure of how far an airplane’s nose is tilted up or down and is controlled by the elevator.

## Yaw

Motion about the perpendicular axes is called yaw. It determines which way the nose of the aircraft is pointing.

This is controlled by the movement of the rudder.

### Roll

Motion about the longitudinal axis is called roll and in aircraft determines how much the wings are banked.

This is controlled by the movement of the aileron.

The position of the Aileron, Elevator and the Rudder on an airplane

## Where do you use it?

Although usually, plane flights are quite monotonous and employ just one type of movement at a time, there is a wide variety of times where all the three have to be employed, like in the crosswind landing. Crosswind landing is a landing maneuver in which a significant component of the prevailing wind is perpendicular to the runway center line.

The above maneuver is known as Crabbing.

The nose points towards the wind so that the aircraft approaches the runway slightly skewed with respect to the runway centerline ( depends on the direction of the wind ). Upon approaching the runway threshold, moments before landing, the pilot aligns the aircraft with the centerline.

And this is easier said than done as it involves the meticulous control of the pitching, yawing and the rolling of the aircraft in order to stick the landing ( as is seen in the animation )

## Some more examples

One need not restrict the usage of these terms merely to aircrafts, but can extend it other objects of interest as well.

Cars also experience pitch, roll, and yaw, but the amounts are relatively small and are usually the result of the suspension reacting to turns, accelerations, and road conditions.

Source

For a human- Pitch is like saying Yes. Yaw is when you say No! And roll is when you just wave your head.

Pitch, Yaw and Roll and that’s all there is to it.

# How to make diamond rings at Mach speed

Source

I am almost every time put in a trance whilst spectating an aircraft/jet takeoff. There is always something interesting in the occurrence that makes me go nuts!

And this time around, there are these series of rings that one can see in the exhaust plume of a jet engine when it takes off (usually when the afterburner is on).

I had no clue about the phenomenon nor did I know how to express it in ‘search engine’ terms to find a match.

But upon discussion with some of friends, I was shown this video of a space shuttle launch that seemed to produce a similar pattern.

Hmm.. Interesting

### Shock Diamonds

These set of rings/disks that are formed in the exhaust plume are known as Shock Diamonds or Mach discs (and by many more names).

These usually form at low altitudes when the pressure of the exhaust plume is lower than the atmospheric pressure.

### How does it form ?

Since the atmospheric pressure is higher than the exhaust, it will squeeze it inward. This compresses the exhaust increasing its pressure.

The increased pressure also instills an increase in temperature.

As a result, this ignites any excess fuel present in the exhaust making it burn. It is this burning that makes the shock diamond glow.

Source

The pressure is now more than the atmospheric pressure, and the exhaust gases start to expand out.

Over time, the process of compression and expansion repeats itself until the exhaust pressure becomes the same as the ambient atmospheric pressure.

In other words, the flow will repeatedly contract and expand while gradually equalizing the pressure difference between the exhaust and the atmosphere.

The same occurs in rocket engines as well.

## What if?

What if the atmospheric pressure is less than the exhaust plume ( like at higher altitudes ), would we still see shock diamonds?

Yup, we would! And here’s a picture of it too (The Bell X-1 at speeds close to Mach 1).

The same phenomenon as discussed above occurs except that the cycle starts with the exhaust gases expanding to atmospheric pressure first.

Did you enjoy this post?

There is an extensive explanation of shock diamonds given by shock waves which this post does not cover.

And this beckons the start of Supersonic Fluid Dynamics – a marvelous field of its own. If this captivated you, it is definitely worth a google search.

Cheers!

# What are gravitational waves (an explanation for dummies)

The existence of gravitational waves has been confirmed. But you probably heard that already. In this post, we will break down this profound discovery into more understandable chunks.

This is going to be an amazing journey. Ready?

## Redefining Gravity

When we usually talk of gravitation we are bound to think like Newton, where objects are assumed to exert a force upon each other.

Like imaginary arrows of force in space. But this picture, although enough to explain the workings of the universe on a large scale, crumbled with the advent of Einstein’s Theory of Relativity.

## What is the Space-Time Fabric?

Think of space-time fabric as an actual cloth of fabric..

When you place an object on the fabric, the cloth curves. This is exactly what happens in the solar system as well.

The sun with such a huge mass bends the space-time fabric. And the earth and all the planets are kept in orbit by following this curvature that has been made by the sun. Depending on the various masses of objects, the way they bend this fabric also varies.

## What are Gravitational Waves?

If you drop an object in a medium such as water, the collision produces ripples that propagate as waves through the medium.

Similarly, Gravitational waves are ripples in space-time fabric produced when you drag heavy objects through space time.

And the nature of these waves is that they don’t require a medium to propagate.

## How do you make one?

Everything with mass/energy can create these waves.

Source

Two persons dancing around each other in space too can create gravitational waves. But the waves would be extremely faint.

You need something big and massive accelerating through space-time in order to even detect them.

And orbiting binary stars/black holes are valuable in this retrospect.

## How can you detect them?

Let’s turn to the problem of detecting them assuming you do find binary stars/black-holes in the wondrous space to suite your needs.

Well, for starters you cannot use rocks/ rulers to measure them because as the space expands and contracts, so do the rocks. ( the distances will remain same in both the cases )

PC: PHDComics

Here’s where the high school fact that the speed of Light is a constant no matter what plays an important and pivotal role.

If the space expands, the time taken for light to reach from A to B would be longer. And if it contracts, the time taken for it to reach from A to B would be smaller.

PC: PHDComics

By allowing the light waves from the contraction and expansion to interfere with each other, as it happens in any interferometry experiment, we can detect the expansion or contraction. Voila!

And this is exactly what they did! ( on a macroscopic level ) at LIGO (Laser Interferometer Gravitational-Wave Observatory)

## 14 September 2015

Two Black Holes with masses of 29 and 36 solar masses merged together some 1.3 Billion light years away.

The merger of these two black holes results in the emission of energy equivalent to 3 solar masses as Gravitational Waves.

This signal was seen by both LIGO detectors, in Livingston and Hanford, with a time difference of 7 milliseconds.

And with the measurement of this time difference, physicists have pronounced the existence of Gravitational Waves.

Source

All this is most certainly easily said than done and requires meticulous and extensive research, not to mention highly sensitive instruments.

Had they not have measured this time difference, we might have had to wait for the merger for more massive black holes to collide and maybe even build more sensitive instruments to detect these waves..

## Why is this discovery a Big Deal ?

Gravitational waves gives us another way to observe celestial phenomenon. These waves also form when supernovae explode, when black holes collide and during many other space activities.

Detecting them might give us a new perspective into the cosmic events. There is hell of a lot of space that is left unexplored or lies beyond human exuberance and this discovery might shed some light on it. ( like the big bang per se )

The ultimate goal is to understand the fundamental laws of the universe. It is a quest through the oblivion towards a theory of everything.

Although it is unknown how many years/decades it might take to get us there, but these discoveries are markers to getting there.

## And Einstein predicted this a 100 years back!

Mind Blown!

Note: Hope you are able to understand and appreciate the profundity of the discovery done by mankind from this post. It is truly an amazing moment to be alive.

# This material can heat to 2200 F, but it’s safe to hold in your bare hands

Designing an orbiter that is able to endure the brutal -250 F in the outer stretches of space, as well as the bewildering 3000 F during the reentry is a ridiculously challenging task.

Space shuttle is the name for the entire setup, whereas the orbiter is the ‘planeattached

## The Thermal Armor

Ergo, after churning the minds of the elite scientists and engineers, we now have the TPS (thermal protection system) that protects the orbiter from this harsh temperature difference.

Source

The thermal protection system is like an armor that maintains the outer skin of the orbiter within acceptable temperatures. This is achieved by employing various materials on the outer structural skin.

## Wait, what kind of materials?

The tile’s material is an insulator. These materials do not exchange heat easily.

Cardboard, being an insulator protects your hand from the hot coffee. (PC: Nirzar)

Conductors on the other hand are the exact opposite. They love to give away their heat.

This is the reason why touching a hot aluminum / stainless steel ( Conductors ) pan at a moderate 100 C would cause burns, but touching the Space shuttle tile (An amazing insulator) at 2200 C is probably not a bad idea!

## Let’s cut to the chase: What are they?

A used tile from Atlantis

Those small white cubes are LI-900. It is a type of low-density insulating material which is composed almost entirely of silica glass fibers.

Purest quartz sand with 94% air by volume constitutes the LI-900.

It’s sort of like foam/ a sponge, if you think about the huge amount of air that it contains.

And also, Air and silica are both extremely poor conductors of heat and thus great insulators.

As is evident from the animation provided above, they can be heated to 2200°F, and even after being subjected to that temperature can be picked up almost immediately.

Surprisingly, these tiles are not mechanically attached to the aircraft, but instead glued. And many of them are replaced after each flight.

White tiles (known as LRSI) were used mainly on the upper surface and have higher thermal reflectivity. These are therefore pointed towards the sun in order to minimize solar gain.

Black tiles (known as HRSI) are optimized for maximum emissivity, which means they lose heat faster than white tiles. This property is required in order to maximise heat rejection during re-entry.

### Screwing up the TPS is a recipe for disaster

Due to a damaged heat shield, the space shuttle Columbia disintegrated upon reentry in 2003, killing all crew members. Designing such an integral component of the space shuttle requires utmost meticulousness.

## Perceiving Temperature

Source

When something feels hot to you, it’s really because there is a large amount of heat transferred between the object and your skin.

And when there is very less heat transfer, we perceive it as cold!

In the case of the space tile, since it’s a good insulator it is conducting (transferring) energy at a remarkably low rate.

Ergo, if we were to touch it, it will fell the same as a quotidian household object.

Cool eh ?

# Hang on a second, Mr. Aristotle!

Either you run the day, or the day runs you

Time has this wonderful way of showing us what really matters. This so called ‘Time’ plays a pivotal role in our day to day life, but what exactly is time and how does one measure it?

## What is Time?

Defining time per se is tricky, but without loss of generality we can define it as:

A measure to distinguish the past, the present and the future. And also to measure the duration of events and intervals between them.

## How to measure time?

One could basically use anything that is periodic to define/measure time. Even a pendulum for that matter.

You could say -’Bro, it would take 3500 pendulum cycles to get there.’

And that’s perfectly fine!

But, as pendulums lose energy as they swing, they describe smaller and smaller arcs and aren’t exactly consistent as a timekeeping tool. Plus, whose pendulum are we referring to?

Since It’s very hard to reproduce the very same conditions of the pendulum’s oscillation, a communication barrier arises.

## Astronomical Time

Well, astronomical phenomenon seem to be consistent around the world, so why not define it in those terms. And ergo it was, for quite some time..

The second, was defined originally as the fraction 1/86 400 of the mean solar day. Great!

But there this one problem persists:

The rotation of the earth is not stable — it does not spin at a constant rate, it wobbles and does lots of other crazy stuff.

And this imparts irregularities in the measurement of time (small, but it adds up, ironically, as time passes by.)

## Atomic Time Keeping

We needed something that ticks consistently — so we turned to atoms. Let me first define the second and then take you through it:

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

If you impart a specific amount of energy to an atom, it will jump from the present state (ground state) to an excited state i.e it will make a transition.

By defining a second as the number of these transitions time can be measured so accurately that only a tiny portion of a second is lost every million years.

Pretty cool eh?

Food for thought:

Why is a second, a second long ? (or)

Why do 9192631770 periods constitute a second ?

Why choose Cesium?

# Food-science Sunday : The geometry of a Pringle

Mathematics is not about equations, numbers, computation or algorithms: It is about Understanding!

There are many ways to understand it – the one that this post is based on is real life visualization.

The Pringle shape is what is known in mathematics / calculus as a hyperbolic paraboloid.

## Why are Pringles a hyperbolic paraboloid?

The saddle shape allowed for easier stacking of chips. This also minimized the possibility of broken chips during transport.

Since it is a saddle, there is no predictable way to break it up. This increases the crunchy feeling and hence that weird satisfaction.

It is relatively more feasible to manufacture the press block compared to other shapes.

The concave U-shaped part is stretched in tension (shown in black) while the convex arch-shaped part is squeezed in compression (shown in red).

Through double curvature, this shape strikes a delicate balance between these push and pull forces, allowing it to remain thin yet surprisingly strong.

All of this, and also to make some cool ring structures.

Lays is a parabolic cylinder, not as interesting as a Pringles but worth knowing for the sake of completeness.

You can replicate one with a piece of paper, but you can’t do that with Pringles without cutting the paper and actually adding more paper to it. This makes it more mathematically desirable.

## Flavor is subjective. Math is irrefutable.

This is what a mathematician had to say:

They’ve got these Lays Stax right next to the Pringles as though they are equivalent. How can they do that?

One is a positive semi-definite quadratic form and the other is an indefinite quadratic form – they’re not even the same definiteness!

Sources and more:

How it’s made- Stacked potato chips

Best way to hold a pizza slice

Best snack shape

# Gyroscopic wired animals

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Chickens, Cats, Owls and some more animals have this ability to keep their heads fixed on a spot, regardless of which way you hold their bodies. It is as though they have an in-built gyroscope in their heads.

They are able to do this, courtesy of the Vestibulo-ocular Reflex. (a reflex where the head and eyes remain fixed on a point, as long as doing so puts no undue stress on the rest of the body.)

This allows them to feel the force generated in an attempt to change their orientation and automatically rearrange their head’s position.

## Humans have it too

Humans are wired with a similar feature too. But ours is specifically geared towards keeping our eyes on a fixed point.

We wouldn’t be able to read words while moving our heads, or even take walks without some loss of vision otherwise.

Nature is fascinating, isn’t it?

In every walk in nature, one receives far more than one seeks – John Muir

# You can never truly kiss anyone!

I am sorry to break it to you, but you can never truly kiss anyone. But hey, It’s not your fault! This is a consequence of the fact that two atoms can never really touch each other.

## Wait, what?

Push them together as hard as you want, and they will resist.

Two like charges always repel each other. And also no two objects can have the same exact properties ( courtesy of the Pauli’s exclusion principle ). This curtails any possibility of two objects coming together.

This also means that from an atomic perspective. you can never really touch anything and that chair you think you are sitting on, you are actually hovering over it!

This goes beyond our intuition because we know it when we have kissed someone. That adrenaline rush and the dopamine surge is just inevitable! You feel it, yet not touch?

## Atoms love to Share.

You might have heard about bonds in Chemistry. Well, as it turns although atoms don’t like getting close to one other, they are completely okay with sharing! Hmm..

## The Grandeur of touch

If we can’t really touch anything, how can we explain the perception of touch?

This is where it gets astonishing. The answer boils down to how our brains interpret the physical world.

What your brain is perceiving as touch is merely the electron’s repulsive force.

A legion of atoms and molecules collectively known as your skin are interacting with a surface. The repulsive force they experience is being sent to the brain which then interprets the data as touch!

The sensation of touch is arguably a Grand Illusion, created as a way of interpreting interactions between electrons and electromagnetic fields

## The profundity of nature

It is of quintessence to realize that these are constraints of nature and not some man-made voodoo. This is how the laws of nature have been laid.

We are all merely spectators to Nature’s endeavors.

# Why don’t rain drops kill you?

Pardon me for being melodramatic. Think about it, rain drops fall from thousands of feet in the air and yet we hardly twitch when one falls on us.This defies all logic, if you drop a penny from the top of a building and let it fall on you, we know that it hurts!

Have the angels cast a magical spell on the rain drops to spare us from the pain?

Photo Credit

## Terminal Velocity

When you drop something in air, it does not accelerate forever. Molecules in air constantly bombard with the object, exerting an upward force. This is known as air resistance or drag.

As the object gains velocity there comes a time when the force of the air resistance is enough to balance the force of gravity, so the acceleration stops and the raindrop attains terminal velocity.

Terminal Velocity is the maximum velocity an object can travel in air! For a more detailed explanation, complete with equations visited NASA’s page on terminal velocity.

## The Angel’s mystical spell

Rain drops are only 0.5 mm-4 mm in diameter. Their terminal velocity is only about 10 m/s ( 20 mph ) That’s the maximum speed that they can travel in air irrespective of their initial height. Also the mass of the rain drop is about a few milligrams.

Hence, the force that it exerts on the body is really small, small enough that we find the experience pleasurable and soothing.

Photo Credit

# Crown Shyness – Trees can shy away too!

Crown shyness is a phenomenon observed in some tree species, in which the crowns of fully stocked trees do not touch each other, forming a canopy with channel-like gaps.

## Why?

Well, scientists are not certain what causes these remarkable patterns. But some theories have been proposed since the 1900’s to explain the phenomenon.

Photo Credit

Since most of the trees are tall, slender and typically found in high wind areas, crown shyness is thought to prevent them from bumping into one another and abrading each other. The leading shoots get dispatched as an aftermath of the abrasion.

This was proven experimentally too. Scientists artificially prevented the trees from colliding in the wind and found out that they inturn fill the canopy gaps!

Photo Credit

But studies done on the Camphor tree found no evidence of abrasions. Instead, it was suggested that the leading tips were sensitive to light. Ergo, fewer buds developed in regions that were already dense or where the crowns of different trees met. This curbed the development of shoots in regions that already were populous.

## Species

Where would you find these species? Well, Crown shyness is not an exclusive phenomenon that occurs only in a country/region, it’s universal and  has been reported in various parts around the world.

Species of Dryobalanops( including Dryobalanops lanceolata and Dryobalanops aromatica ), eucalypt, Pinus contorta or lodgepole pine, Avicennia germinans or black mangrove, Didymopanax pittieri, Clusia alata,  Celtis spinosa and Pterocymbium beccarii are some well-known ones that exhibit Crown Shyness.

Photo Credit

With over 100 years of research into this phenomenon, we are yet to truely uncover the mechanism of Crown Shyness. Although it is not for the lack of trying, we seem to be missing a conjoining piece that connects all the pieces together.

Look deep into Nature, and you will understand everything

As we further our understanding about the world that we dwell in, we will hopefully be able to unravel the mysteries that underly baffling phenomena of nature such as Crown Shyness and appreciate nature better.

# A Boy and his Atom

“A Boy and His Atom” is a 2012 stop-motion animated short film released on YouTube by IBM Research. The movie tells the story of a boy and a wayward atom who meet and become friends. It depicts a boy playing with an atom that takes various forms.

And these sphere shaped beads that you see are indeed atoms! How cool is that?

# How does an eraser work?

### What happens when you write?

Although we call the black stuff in pencils “lead,” it’s not the real metal known as lead. It’s actually a mineral called “graphite,” which is made up of carbon. When you write with a pencil, graphite particles from the pencil rub off and stick to the fibers of the paper you’re writing on.

### What do erasers do?

When you rub an eraser across a pencil mark, the abrasives in the eraser gently scratch the surface fibers of the paper to loosen the graphite particles. The softeners in the eraser help to prevent the paper from tearing. The sticky rubber in the eraser grabs and holds on to the graphite particles.

### The physics

Erasers work because of friction.

As the abrasives in your eraser are rubbed against paper, friction produces heat, which helps the rubber become sticky enough to hold onto the graphite particles. As the rubber grabs the graphite particles, small pieces of combined rubber and graphite get left behind. That’s the “stuff” you brush off of your paper when you’re finished erasing.

[Source]

# Rockets 101 – How to turn during flight

Outstanding control is what distinguishes a toy rocket from a real one. And it is of quintessence to be able to channel the rocket’s direction. In the case of a NASA launch, failure can mean hundreds of millions and years of work down the drain. In the most extreme, it can mean the difference between life and death. To be able to fly is cool, but what’s cooler is being able to pinpoint the destination and the trajectory of a rocket or shuttle.

In most modern rockets, this is accomplished by a system known as Gimbaled Thrust.

In a Gimbaled thrust system, the exhaust nozzle of the rocket can be swiveled from side to side. As the nozzle is moved, the direction of the thrust is changed relative to the center of gravity of the rocket and a torque is generated.

As a result, the rocket changes direction. After necessary corrections are made, the exhaust nozzle is brought back to its initial state.

The angle by which the rocket’s nozzle swivels is known as the Gimbaled Angle.

### Up, Up and Away!

PC: NASA, learnengineering, achingtentacles,campnavigator

# Remembering Edmond Halley!

November 8 – it is the birthday of the famous English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley’s Comet.

## What is Halley’s Comet?

Halley’s Comet or Comet Halley officially designated 1P/Halley, is a short-period comet visible from Earth every 75–76 years.

Halley is the only known short-period comet that is clearly visible to the naked eye from Earth, and the only naked-eye comet that might appear twice in a human lifetime

One rather pretentious question that comes up when talking about comets is how on earth did it start orbiting around the sun in the first place and how long has it been orbiting?

## Where do comets come from?

Comets come from two major areas of our solar system: the Kuiper Belt and the Oort  Cloud. Each of these regions contains billions of comets, but they’re so spaced apart that they get no closer to each other than Earth does to the sun.

## How long has it been orbiting?

It is thought that Halley has been in its current orbit for around 60,000 to 200,000 years. However, it is very difficult to calculate any comet’s previous orbit with accuracy due to the fact that the orbit is always altered once it passes close to the Sun.

## When will we see it again?

The last known sighting of the comet was in 1986 and the earliest it will come close enough to see is 2061!

## The Halley’s Lunar and Martian Crater.

Among other things named honoring Halley is the Halley’s crater on the moon and on Mars.

Halley’s crater. ( Upper left )

# Meteor strikes Thailand twice in 3 months

The first (seen in the first animation) took place on September 7 and the second one on the November 2. People though these were some planes crashing, but were later confirmed to be small meteor showers.

Just so you know, this happens all the time in our atmosphere and there is nothing to be alarmed about. Albeit, legion of these meteors enter the atmosphere, most of them are burned away in the outer strata of our atmosphere. But some produce streaks in the sky that can be visualized. And finally a few do manage to make it to the surface of the Earth, but those are a rarity.

# The Marangoni Effect – an affair with surface tension

The Marangoni Effect says that fluid will want to flow from areas of lower surface tension to areas of higher surface tension.

Soap has a lower surface tension than Water/ Milk. And as a result, when soap is placed on the surface of a fluid (as it is, in these animations), it wants to flow away to areas of higher surface tension. A more in-depth explanation of the Marangoni effect and surface tension in general can be found a previous ZME Science post.

And this propels the small boat, causes the pepper flakes to spread away, makes the string to expand, and the dye to fan out. It is also responsible for the Tears of Wine phenomenon that you might have already witnessed. : )

PC: Flow Visualization at UC Boulder, source video, MIT, Dan Quinn

# Looking at the universe naked – an Ontological Awakening

It was Stephen Hawking who said:

We are just an advanced breed of monkeys on a minor planet of a very average star. But we can understand the universe. That’s what makes us special.

The pictures showcase the universe in its cosmic brilliance. Spanning the entire electromagnetic spectrum, these images have been false- colored to help us perceive the universe that lies beyond our visual cognizance.

Breathtaking isn’t it ?

Courtesy: Chromoscope.

# Taming the sound from a shuttle using water

What purpose would a water tank have in the proximity of a space shuttle launch?

Well, believe it or not, it is used to suppress the acoustical energy (sound and rocket exhaust reflected from the flame trench and Mobile Launcher Platform ) during launch.

## Underlying Principle.

NASA came up with an ingenious way to suppress the sound – bubbles!

Bubbles are excellent at absorbing the sound. They absorb the acoustic energy and as a consequence of which get heated up. NASA exploited this and sprayed water molecules in the air surrounding the Mobile Launcher Platform. This reduced the sound from the firing of the rockets by almost a half!

## The Sound Suppression System

The Sound Suppression System includes an elevated water tank with a capacity of 300,000 gallons (1,135,620 liters). The tank is 290 feet (88 meters) high and is located adjacent to each pad.

The water releases just prior to the ignition of the Shuttle engines, and flows through 7-foot-diameter (2.1-meter) pipes for about 20 seconds. Water pours from 16 nozzles atop the flame deflectors and from outlets in the main engines exhaust hole in the Mobile Launcher Platform, starting at T minus 6.6 seconds.

By the time the solid rocket boosters ignite, a torrent of water will be flowing onto the Mobile Launcher Platform from six large quench nozzles, or “rainbirds,” mounted on its surface.

The peak rate of flow from all sources is 900,000 gallons (3,406,860 liters) of water per minute at 9 seconds after liftoff.

Exquisite, isn’t it?

PC: NASA.

# Only time will tell – A complex number tribute

I was in high school when the notion of complex numbers was fed into my vocabulary. None of it made sense! One of my friends remarked “Why on earth did they have to invent a new Number System? Uhh.. Mathematicians!!”. And as distressing as it was, we weren’t able to comprehend why!

There are certain elegant aspects to complex numbers that are often overlooked but are pivotal to understanding them. Of the top of the chart – the events that led to the invention / discovery of complex numbers. To shed some light on these events is the crux of this post.

## A date with history.

There were quotidian equations such as x² + 1 =0 which people wanted to solve, but it was well-known that the equation had no solutions in the realms of real numbers. Why, you ask?Well, quite intuitively the addition of a square real number (always positive) and one was never going to yield 0.

And also, as is evident from the graph, the curve does the intersect the x- axis for a solution to persist.

For the ancient Greeks, Mathematics was synonymous with Geometry. And there were a legion geometrical problems which had no solutions, peculiar quadratics like  x² + 1 =0 were branded the same way.

“ Why make up new numbers for the sole purposes for being solutions to Quadratic equations? “. This was the rationale that people stuck with.

## The Real Challenge.

Quadratics, per se were easy to solve. A 16th century mathematician’s redemption was confronting a cubic equation. Unlike Quadratics, cubic equations pass through the x axis at least once, so the existence of a solution was guaranteed. To seek out for them was the challenge.

The general form of a cubic equation is as follows:

f (x) =  au³  + bu² + cu + d

If we divide throughout by “a”, it simplifies the equation and substituting x  = u – ( b / 3a )  gets rid of the squared term . Thus, we obtain:

x³ – 3px – 2q = 0

A mathematician named Cardano is attributed for coming up with the solution for the above equation as :

x = ³√( q + √ ( q²  – p³ ) )  +  ³√( q – √ ( q²  – p³ ) )

This equation is perfectly legit. But when p³ > q² it yields incomprehensible solutions.

## Bombelli’s “Wild Thought”.

The strangeness of the formula enticed Bombelli. He considered the equation x³  = 15x + 4. By virtue of inspection, he found out that x = 4 satisfied the equation. But, plugging values into the cardano’s equation, he obtained:

x = ³√ ( 2 + 11 i ) + ³√ ( 2  – 11 i )    where i = √-1

Wait a minute! The equation is hinting that there exists no solution in the real numbers domain, but in contraire x = 4 is a solution!!

In a desperate attempt to resolve the paradox, he had what he called it as a ‘wild thought’. What if the equation could be broken down as

x = ( 2 + n i ) + ( 2 – n i )

This would yield x = 4 and resolve the conflict. It might sound magical, but when he tested out his abstraction, he was indeed right. From calculations, he obtained the value for n as 1. i.e

2 ± i =  ³√( 2  ± 11 i )

This was the birth of Complex Numbers. By treating a quantity such as 2 + 11√-1, without regard for its meaning in just the same way as a natural number, Bombelli unlike no one before him had come up with a modus operandi for dealing with such intricate equations, which were previously thought to have no solutions.

While complex numbers per se still remained mysterious, Bombelli’s work on Cubic equations thus established that perfectly real problems required complex arithmetic for their solutions.This empowered people to venture into frontiers which were formerly unexplored.

And for this triumph Bombelli is regarded as the Inventor of Complex Numbers.

### Fun fact

A moon crater was named after Bombelli, honoring his accomplishments.

PC: NASA , Flickr

Sources : Mathematics and its history, Visual Complex Analysis.