The Golden Ratio
Most of you must have seen this picture
And chances are that you have encountered some video/article explaining how the ratio is linked to aesthetics, architecture or the human body.
Let’s explore theorigins and the wonders of this seemingly ubiquitous ratio.
Where does it come from?
Imagine we have a rectangle with a length to width ratio of ϕ.
Now if we cut this rectangle such, that the new rectangle also has the length to width ratios equal to ϕ. Then:
Imagine we have two lines of unequal lengths
Then what would happen if the ratio of the two lines was equal to the ratio of the sum of the two lines to the line with the larger length? Meaning
The under root 5 should not come as a surprise since the length of the line that makes a triangle in a pentagon (5 sides) is also the golden ratio
The ratio also appears in the pentagram. If the shortest of the lengths of the pentagram is 1 then each consecutive length is a higher power of Φ.
Golden ratio around us
If you are amazed by the awesomeness of this number, then you’re not alone.
500 years ago, the renaissance thinker, Luca Pacioli wrote a book on the number called “The divine proportion” with illustrations from Leonardo Da Vinci.
Now let’s look at the misconceptions with regards to the golden ratio. It is likely that you think the ratio is inherently linked to the Fibonacci series. That is
1 1 2 3 5 8 13 …
And as we divide the next number with the preceding number we approach the golden ratio
But what if I told you that this is not unique. Let’s try.
Pick any 2 numbers. Let’s say we choose 125 and 145.
Let’s add each preceding number, so
Myths regarding the golden ratio
Now that we have established that the golden ratio is awesome, we need to debunk some myths. Most of you would have seen some pictures like these
Well, I must pop your bubble. Aesthetics are more complicated that this. These pictures are nothing more than a spiral and a rectangle forcefully fitted on something beautiful. There is nothing intrinsically pleasing about the golden ratio. How do I know this?
Well, pick your favorite rectangle.
Now if you are like most people, it is likely that you found this rectangle the most pleasing.
But this rectangle is far from the golden ratio. The rectangles that have the golden ratio are:
But don’t be sad. There are a number of aesthetically pleasing things that use the golden ratio. An example is the famous Penrose tiling.
Another example is the way the sunflower seeds are packed.
The flower packs seeds such that more seeds are available in a smaller space and it so happens that the best way to fit this is the Fibonacci spiral.
By now I am sure you are mesmerized by the awesomeness of the golden ratio and will notice it when you come across it.
Waleed Bin Khalid
The Fermi Paradox
Have you ever been in a situation where you’re sitting with your friends, enjoying your lunch and one of you says something or passes a comment that just sticks with your group for an eternity?
One day, Enrico Fermi, an Italian physicist, was discussing the possibility of the existence of extra-terrestrial life with his friends when one of his friends mentioned that it was very likely that out of all the quintillions of planets existing in the universe and orbiting their respective stars, there was another planet, other than Earth itself, where conditions for the existence and evolution of life had struck the correct permutation and an intelligent life form such as us humans existed on its surface.
This is where Fermi was hit with a question that makes researchers scratch their heads away even today. He asked: if there really is such a massive probability that so many intelligent civilizations exist, well,
Where are they?
This apparent contradiction is known as the Fermi Paradox.
Scientists in NASA, SETI (Search for Extra Terrestrial Intelligence) and other organizations have gone to great lengths to estimate the number of alien-inhabited Earth-like planets that exist in the universe. The most conservative of estimates places this figure at a 100,000 intelligent civilizations in our own Milky Way galaxy.
Furthermore, compared to the cosmic scale of time, it would take a civilization with a modest ability of interstellar travel about 3.7 million years to colonize an entire galaxy. If this is the case, why have all of these civilizations gone completely unnoticed? Why is this galaxy, allegedly teeming with thousands of intelligent civilizations, so silent?
Researchers and scientists have continued to scratch their heads (probably why most of them are bald) and come up with theories to explain this mind-boggling paradox. Two sides have emerged, one that claims that the reason we have not seen any sign of an alien existence is that there is none. The other party does not believe that lack of evidence about existence implies non-existence and have proposed some logical explanations for our not finding any signs of non-Earth life. Let’s explore some of these reasons.
Group 1, which believes that we’re the only life form in this universe, that we’re special refers to what is called the non-exclusivity problem. This problem basically says that however many reasons the blokes from group 2 come up with for the higher civilizations not contacting us, there are always going to be exceptions to each one. These exceptions, considering a 0.01% probability of their occurrence, would still number in the thousands within our own galaxy, and so eliminate any possibility that they would go unnoticed.
This brings us to the idea of the Great Filter.
What this means is that there were, as calculated, thousands of life-forms already existing in the universe. However there came a crucial stage, a critical step in the evolutionary process known as The Great Filter on which all but one civilization got wiped away from the fabric of the universe.
If you think about it, this has three implications for our species. Either we’ve already passed the Great Filter and are the extremely lucky ones, or we’re the first ones to develop into intelligent beings and all other forms are yet to catch up to us.
Worst of all, it could be that the Great Filter is pretty closely ahead of us, and we’re all doomed.
On the other side of the battlefield, scientists and researchers are exhausting their creative juices in order to come up with viable and logical explanations. They suggest that:
- Super-intelligent life forms already visited Earth but at a time when humans did not exist (They realized the dinosaurs were not very hospitable)
- We live in a rural, cut-off region of the universe and our existence has gone unnoticed
- We’re all just part of a computer simulation and other life forms have not been programmed into it (think of the implications if this turns out to be true)
- Just like certain devices only detect certain kinds of data, we just do not have the technological capability to detect whatever form of data the super-intelligent civilizations communicate with
- Or maybe instead of our devices being too primitive, we’re too primitive to be noticed (Just like that random anthill you didn’t notice on the side of a road yesterday)
- We’re animals inside a cosmic zoo, and the super-intelligent civilizations are silently observing us (look at that ignorant little monkey!)
We’ll let that battle go on, until some real answers are found. What we do extract out of this, regardless of the group we agree with, is a crushing sense of humility. We could be all alone in this universe, a few nanometers of life in the Tera-meters of the space we call the universe; or, we could be one of thousands, yet another civilization coming to grips with the reality and insignificance of its own existence.
Whatever the answer to the Fermi paradox is, once it is found, everything will change.
By Ishaq Ibrahim
P.S. If, in the process of reading this article, you had sudden bursts or scattered specks of ideas popping up in your head, and if you feel like you could make a nice sci-fi story out of them, do send it to us! We’d love to publish your work on our blog.
The Science of Imagination
Imagine yourself sitting in that one class you hate the most. You simply can’t pay attention to what the instructor is saying. Instead, you look out the window and lo behold! There it is, a colossal alien spaceship right above the ground. You keep staring until the door unfolds and guess what’s coming out of it? Cows. Purple cows. Wearing space suits. And snap! The instructor screams and the rest is history.
Now that we have your attention…
It is unlikely that you’ve been in a ridiculous situation like this. (And if you have, a therapist is the need of the hour.) Despite that, you were easily able to throw some, if not all, of the elements together to form that scene. If you’ve never seen it before, where were all those mental images popping in from?
Welcome to the fascinating world of imagination!
It’s a curious thing, imagination. It forms a great part of your everyday life. It’s what makes you keep seeing an ‘A’ on the test which hasn’t been checked yet, visiting Willy Wonka’s factory (that chocolate stream, though) and flying over the city with a pair of wings. For something we use so commonly, it is surprising how we know so little. So, today we thought we’d take you on the search for this Eureka moment with us! (And of course, ‘imagine’ that we’re doing a fair job at it)
A series of complex processes inside your brain ensure that your ‘imagination’ gears keep whirring. Everything, object or person that you’ve ever seen, your brain has broken down into characteristics. In our example, ‘cows’ would read as ‘animal’, ‘four-footed’, ‘hairy’, ‘big eyes’ ‘cute!?’ and definitely not ‘purple’. Each of these characteristics is encoded by different neurons in the posterior cortex of your brain. Every time you see a cow, all these neurons fire from different parts of the brain and link together in what is known as a ‘neuronal ensemble’. So, even when the cow isn’t there in front of you, the mental image of it forms quite easily, thanks to the ensemble.
But here’s the surprising part. You might’ve seen a cow before. But we bet you’ve never seen a purple cow in a spacesuit. And yet, when you read that, it did pop in your head, didn’t it? (or so we hope.) This is achieved by a process called ‘mental synthesis’. New and weird images in your brain are created from those which are already stacked in there. The brain coordinates so that neuron ensembles of different things fire at the same time by the activation of posterior neurons due to electrical signals. Hence, a synchronized image forms in your head. Ta da!
+ + = TADA!
Now, we hope you’re still hooked and reading, unfazed in the ultimate truth of imagination. Dr. Scott Barry Kaufman from the Imagination Institute bursts a few other myths that we’ve been led to believe. Remember that whole ‘right brain- left brain’ story. Yeah, that’s a lie. Dr. Kaufman says that you can’t ever harness one part without actively engaging the other. He also defines three kinds of brain networks, two of which are:
- The Executive Attention Network:
This is responsible for holding onto a lot of information at the same time and is your best friend when you are preparing for and giving a test. It’s the lifesaver which ignores boring ideas in the process of creativity and tries to dig deeper to explore other possibilities.
- The Default Network:
Labeled as the ‘imagination network’ and your ticket out of that painfully insipid class, because it allows you to daydream to your heart’s content.
As the talk on imagination continues, we can’t help wondering how we start to stumble upon self-made conversations in our head or the plot for the craziest novel (or well, Facebook status) when we should be slaving with OHT preparation and are completely blank at other times?
It turns out, that when you’re paying full attention to something else, the default network springs into action! And that’s where all your Eureka moments are born! However, if they’re not triggered even when you want them to, it could be because you’ve become so familiar with your surroundings and the brain connections so efficient that your imagination becomes weak. In the words of Gregory Berns’, professor of neuroeconomics and director of the Center for Neuropolicy at Emory University, “a lazy piece of meat, which doesn’t want to waste energy.”
But what should you do about it?
“The surest way to provoke the imagination … is to seek out environments you have no experience with. … Novel experiences are so effective at unleashing the imagination because they force the perceptual system out of categorization, the tendency of the brain to take shortcuts.” Says Berns.
In other words, stop being a couch potato and go explore the world. Go take a walk, travel to a new place, read a new book while you wander through the woods and feel alive as your imagination engulfs you whole!
Life of π
We all know π is the ratio of the circumference of a circle to its diameter. But considering the importance of this constant, we know very little about its history and origins. We decided to do it justice and bring to light the amazing life of π.
Babylonians estimated the value of pi to be 3.125 while the Egyptians estimated it to be 3.16. However, it was in the 18th century that Mathematicians decided to give a name to the constant that was the ratio of the circumference of the circle to the diameter due to its ubiquity. They denoted it with the lowercase Greek letter pi from the Greek word for circumference perimetros.
Today we know the value of pi to the trillionth digit thanks to powerful supercomputers. The record for finding consecutive numbers, from 3.14 onward to the final digit, is held by Fabrice Bellard, who has calculated pi to 2.7 trillion digits!
The Babylonians and the Egyptians might have made their guesses but it was Archimedes who first calculated the value of pi to a very reasonable accuracy. But how did archenemies calculate it without a computer?
He drew a circle of diameter 1 units and made an inner triangle. Since the circumference is
Then he drew a triangle and measured the lengths of the 3 sides. Of course, this was smaller than the circle’s circumference. He then made triangles between the triangle and circle and measured the length of the 6 lengths. He did the same for 12 lengths, 24 lengths, 48 lengths and finally 96 lengths.
This gave him the lower bound for the value of pi as.
3 10/71 < π
Archimedes did the same with outer triangles
This gave him the upper bound value as
3 10/71 < π < 3 10/70
The proof for this is some very hairy mathematics so we will leave that out.
But such an accurate value of pi is more of an overkill. We don’t need so many decimals for pi. With just pi accurate to 39 decimal places, we can measure the circumference of the universe accurate to one half of a hydrogen atom. Now that is some accuracy.
So why do we bother calculating such accurate values of pi? The answer is that this has become a standard of measuring the processing speed of a computer. The value is now determined more as a competition and less for mathematical accuracy.
The digits cannot be predicted and hence have to be calculated by the computers and ergo it is among the most random things in the universe. Yet no one has found a statistical way of determining its randomness (we use it in normal distributions for randomness.)
There was however once a dark time in the history of pi. In the 1900s a legislation known as the Indiana pi bill was issued which tried to change the value of pi to 3.2. The legislation was however stopped due to the presence of a Professor C. A. Waldo of Purdue University, who happened to be present in the legislature on the day it went up for a vote.
Due to its ubiquity in trigonometry, calculus, and statistics, the number is in no way limited to geometry alone, even if it did originate from there. It has since rightly become the most important constant in mathematics.
Waleed Bin Khalid
The Circadian Rhythm
Have you ever wondered if your sleep cycle was normal? Sleeping at 3 in the morning and waking up at 12 pm, or maybe having only 3 hours of sleep during your exams and wondering every day how you haven’t died by now? It isn’t NORMAL, but it is normal… Does that make sense? It shouldn’t, to be quite honest.
Our body has its own internal cycle or the “circadian rhythm”. It is biologically defined as, roughly, a 24-hour cycle in the physiological processes of living beings, including plants, animals, fungi and cyanobacteria. Translating that to English, it includes all the changes our body experiences during 24 hours.
But is it what we call the BIOLOGICAL CLOCK? No, not really. The circadian rhythm is controlled by the biological clocks in various groups of cells. These are, in turn, controlled by MASTER CLOCKS that consist of a group of nerve cells in the brain called SN or the suprachiasmatic nucleus (say it out loud 3 times to get it right). It controls the production of melatonin, a hormone that makes you sleepy and as it is located just above the optic nerves (they relay information from the eyes to the brain), the SN receives information about incoming light. So, when there is less light—like at night, the SN tells the brain to make more melatonin making a person drowsy.
So, basically, Master Clock->Biological Clock->Circadian Rhythm. Too many clocks, too little time, eh? I’ll see myself out…
However, it’s not just the biological clock telling us to get into bed. Sleep is regulated by another body system “sleep/wake homeostasis”.
When we have been awake for a long period of time, sleep/wake homeostasis tells us that a need for sleep is accumulating and that it is time to sleep. It also helps us maintain enough sleep throughout the night to make up for the hours of being awake “morning person” or “evening person.”
Now, is it our fault or can we blame our parents for our weird sleep cycles? In a strict sense, circadian rhythms are endogenously generated, meaning researchers have identified genes that direct circadian rhythm cycles in people. By now you must be thinking “Yes, I can tell my parents it’s their fault! I’m not the ‘awaragard insan’ they call me!” Wrong. Although they are dependent upon and produced by natural factors within the body, they can be modulated by external cues such as sunlight and temperature. Matlab, it’s your fault, you ‘awaragard insan’.
Don’t fret, child. According to the Sleep Foundation (yes, it exists. Google it), research has shown that blood melatonin levels in teenagers naturally rise later at night than in most children and adults. Thus, they experience a sleep phase delay making the body feel unnaturally alert at night. So if not your parents, you can blame science till the age of 19 years and 364 days.
Circadian rhythms are also linked to various sleep disorders, such as insomnia, and abnormal circadian rhythms have also been associated with obesity, diabetes, depression, bipolar disorder and seasonal affective disorder. But no, that doesn’t mean you’re going crazy. You must have heard about jetlag, one of the more known circadian rhythm disorders. It occurs when travellers suffer from disrupted circadian rhythms as they pass through different time zones and the body’s clock doesn’t match their wristwatch, making them feel groggy and disoriented. Nothing to worry about, though. The body’s clock will take a few days and reset itself. So, your Mami Basheera is lying about having jetlag one month after coming back from the States. She’s basically trying to remind you every moment of your waking life, that she saw some goras.
Seriously, Mami Basheera?
However, these symptoms can also occur in everyday life when the circadian rhythm is disrupted by keeping long and irregular hours [read: exam season]. Because of this, it is important to keep a regular sleep schedule and allow plenty of time for quality sleep, allowing our vital biological components to help us perform at our best… which is pretty much never, eh?
Maham Javed Sultan
Could a Jurassic Park Ever Come into Existence?
Do sudden ripples in a glass of water throw you in wild terror? Does your imagination go on to supply a deafening thud? Do you turn your head slowly and tell yourself you’ve lived a good life, because you’re going to get your head ripped off by a…dinosaur? Still aren’t over Jurassic Park, are you? Here’s a little secret: neither are we!
We stumbled upon one of our ‘Eureka Moments’, with a piece of news which surfaced last year. It all started with a Chinese scientist browsing through a market in Myanmar. He found a 99-million-year old piece of amber, from the mid-Cretaceous period (127- 89 million years ago, when dinosaurs were in the full glory of their reign), suspended in which were fragments of a coelurosaur; a feathered, sparrow sized…DINOSAUR!
The coelurosaur is thought to be a pocket sized member from the family of Tyrannosaur Rex. Suddenly, our hearts skipped a beat as we expected a Mr Hammond to come forth and hire a team of scientists to revive Jurassic Park! Then, our juvenile excited selves let the science side kick back in and we realised just how flawed Jurassic Park was. No, we’re not saying anything to the perfection that is Spielberg. But we are going to dive into why Jurassic Park is biologically impossible.
The whole essence of Jurassic Park lies in the gut of
a blood sucking mosquito, entombed in amber. The DNA was then extracted from the blood, amplified and injected into an egg and voila! Dinosaurs take over the world. Right? Wrong. Here’s why:
Firstly, DNA is inherently a very unstable molecule. A group of researchers recently found out that the half-life of DNA is 521 years. Scientists who have tried to extract DNA from fossilised bones estimate that after 1.5 million years, DNA is too short to be recognised and at 6.8 million years, every bond broken. The actual age of dinosaur bones? 65 million years. The DNA would have simply decomposed into nucleotides, rendering it useless.
Secondly, any extracted DNA (if you manage to do that in the first place) would be too short to even make sense. It needs to be increased, by a process called PCR, which can start with a single molecule and keep replicating until it reaches the desired length. Now, let’s say you piece out the whole genetic map. Then comes the problem of putting it together. Imagine a book through a shredder, after which you have to put it back together, sentence by sentence, page by page. You’ve never read this book, and you can’t find a copy of it to act as a guide (unless of course we ask Universal Studios to send in some of the remains, since that played out so well the first time round) Replace book with dinosaur DNA. Impossible, right?
Let’s suppose that, miraculously, you do manage to put all the pieces of the genome puzzle together. Then we would be up against the fact that the amplified DNA could be contaminated by insect DNA, from the gut of which it was retrieved, along with the insect’s intestinal bacteria, or even the DNA of another dinosaur if it fed on two. Firstly, this would never result in a successful organism. And if it did, it wouldn’t be the dinosaur of your dreams, but a creepy hybrid which would have mosquito eyes, a dinosaur body, or something similar which never walked the earth to begin with. *Shudders*
Many scientists complain that, as the eternally-irked-with kids, Dr Alan points out in the movie, birds are more direct relatives of dinosaurs. So, it would have been suitable to substitute the missing gaps with bird DNA sequences, rather than frogs. Also, on a side note, we think it’s kind of insulting for the mighty dinosaurs to have their DNA substituted by slimy amphibians. (But, bird DNA would never let the dinosaurs switch genders. So, well played, Spielberg)
Lastly, you can’t just replace dinosaur DNA into an ostrich and expect it to be viable. A dinosaur that size would probably have a developing foetus too big for an ostrich, hence leaving the dinosaur embryo alone and homeless, to die a painful death.
A dinosaur clone is very possible in theory, but many practical problems stand in the way of it. However, science is making new gains by the day. Just last week scientists successfully found the oldest proteins ever, from a 195-million-year-old dinosaur. Maybe, in the next century, the greatest theme park known to man might become a reality. But you’ll probably be long dead by that time. As far as we know, your life will be dinosaur free, (unless it’s Barney) and we are SO sorry for bursting your bubble.
Where did e come from?
I am certain you have used this number a number of times in your college education. From its use in integration to growth and decay formulae.
But what is so special about the number e? Let’s check out its Eureka Moment.
But where did it come from?
In 1683, Jacob Bernoulli was studying compound interest. Here is what he was thinking.
Suppose you have 1 Rs (yep you’re broke) and you deposit it in a generous bank that is willing to give you a 100% interest rate per annum. Basically, they will double your money after 1 year. So after 1 year, you have 2 Rs.
But what if the bank were to give you a 50% interest rate per 6 months (0.5 Years)? Then after 1 year:
What if they gave you a 25% interest every 3 months (0.25 years)?
In simple words, what Bernoulli is trying to say is what would be the amount of money you would get if the interest rate of the bank was equal to the number of segments the year was divided in.
50% for 2 segments (payment every ½ of a year)
25% for four (payment every ¼ of a year)
1 % for 100 segments (payment every 1/100 of a year).
And so on…
So what if we have n years? We would have the following equation:
Money you have after 1 year =
Where 1/n is the interest rate. Simplifying this gives us
Now if you remember your college math the above equation is a binary expansion and in a condensed form this gives us:
Now what Bernoulli wanted to know was if there was a limiting value to what the bank could pay him. Or would he reach an infinite amount of money as n approached infinity?
Don’t be too demoralized if you don’t get this sum. Bernoulli himself was never able to solve the sum. He knew the answer had to lie somewhere between 2 and 3 but 50 years later Leonard Euler found the answer to this equation.
After some hairy mathematics which we will ignore here, Euler proved that the series did converge to a limiting value and that limiting value was, you guessed it, e.
e is Everywhere!
Euler’s number is not just in your calculus books. It is everywhere. Why? Because it is the defining number for growth and decay, something all natural and even many man-made processes follow. It is one of those magical numbers that reveals to us how things around us are all somehow linked. Don’t believe me? Let me show you how it is everywhere.
The number e is used in biology to determine the rate of population growth of different species, from bacteria to human populations. In nuclear physics, it is used in determining the rate of decay of different radioactive substances.
This is because the gradient (slope) of a y=aex graph (a is any constant) is proportional to the height of the curve.
Meaning the rate of growth/decay is proportional to the current amount of whatever we have whether it is a bacteria population or the amount of radioactive substance.
Moreover, Euler’s Number is crucial in geometry.
From the shape of galaxies to the movement of tornados, all follow a spiral shape. And how do we define a spiral? The simplest form is r=eθ
Moths fly around sources of light in a spiral since they have to maintain the light source at a constant angle from their eyes.
This is barely the surface of the amazing properties that Euler’s Number holds. It has been used in many other fields such as normal distribution tables in statistics to its use in economics in determining financial models.
But I hope that every time from now on you use this constant either in class or experience it around you, you appreciate how intricate and interlinked everything is.