Eureka Moments

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

C= πD

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 next stage in calculating a more accurate value of pi came in the enlightenment era when calculus was in development. This was found via the Madhava-Leibniz Series stating:

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

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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.  1

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 produ2ction 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.”3

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.

4Circadian 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.

5

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?

Sources:

https://sleepfoundation.org/sleep-topics/sleep-drive-and-your-body-clock

https://www.nigms.nih.gov/Education/Pages/Factsheet_CircadianRhythms.aspx

https://www.sciencedaily.com/terms/circadian_rhythm.htm

Maham Javed Sultan

 

 

Could a Jurassic Park Ever Come into Existence?

jp

 

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!

feathers
A magnified form of the discoverd piece of amber. Courtesy: Science Mag

 

coelo
Courtesy: Royal Saskatchewan Museum (RSM/R.C McKellar) 

 

 

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.

mr-hammond

 

mos

 

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 whodna-nucleotides 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?

dnaa

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*

 

frog
Not quite!

kid

 

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.

embryo.png

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.

barney

 

Sources:

http://www.forbes.com/sites/alexknapp/2013/09/14/scientists-demonstrate-that-jurassic-park-couldnt-happen/#7940282f1a4d

http://www.foxnews.com/science/2016/12/09/feathered-dinosaur-tail-fragment-trapped-in-amber-amazes-scientists.html

http://www.nature.com/news/dna-has-a-521-year-half-life-1.11555

http://www.telegraph.co.uk/news/science/dinosaurs/10303795/Jurassic-Park-ruled-out-dinosaur-DNA-could-not-survive-in-amber.html

http://www.iflscience.com/technology/could-jurassic-park-ever-come-true/

http://nypost.com/2017/01/07/how-scientists-actually-could-bring-dinosaurs-back-to-life/

 

 

Mahnoor Fatima

 

 

 

Where did e come from?

2.71828182846…

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.

1

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:

2

What if they gave you a 25% interest every 3 months (0.25 years)?

2

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.

(http://math.stackexchange.com/questions/420018/proving-lim-limits-n-to-infty-left1-fracxn-rightn-textex complete proof here)

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.

2

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.

 

2.png

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.

Muhammad Waleed Bin Khalid