Tag: mathematics

Posted in Simple Pleasures of Life

Simple Pleasures of Life #23

Posted on by Jinavie

Happy coincidences.

Life is full of chance and coincidences. Coincidences can range from something simple like bumping into someone you haven’t seen in a while at a supermarket, to what some people call “fate” or “miracle” or “destiny”.

Mathematics will tell you that coincidences are simply the product of the law of large numbers – that it is merely a statistical event. Psychology will tell you that we are just victims of the regression fallacy. Religious people will tell you that it is an act of god.

For me, happy coincidences are just little events in life that spice things up a bit. I don’t believe in fate or destiny or some omnipotent deity, but that doesn’t mean that I treat coincidences as unimportant random events. I find that rather than dismissing it, identifying it as something good makes my life a little bit happier. Who cares if the reason that all the traffic lights were green might be a sign? Who cares if the reason that the song that JUST happens to fit my mood comes on a shuffled playlist was statistical chance? Who cares if you call meeting your soul partner destiny?

What matters to me in the end is that it happened and it made me happy. I also find identifying the coincidences and the steps that led to it happening quite interesting (because I’m a huge nerd, which I say with pride). For example, if I had decided to stick with my plan of going to Asan Hospital instead of Severance Hospital for my selective this year, I wouldn’t have met the girl that invited me to WKMSO and I wouldn’t have met the awesome people I did, nor would I have had my unforgettable NYC/Vegas adventure.

Maybe the reason it makes me happy is that knowing that if even a tiny detail was changed, say if I was put on a different shift to that person, things would have turned out quite differently. For better or for worse. But the fact is that things happened to happen in the way it did and it led to me having an amazing time.

And that my friends, is a long-winding rant-y explanation to why I love the butterfly effect.

Butterfly

Posted in Science & Nature

Natural Design

Posted on by Jinavie

We look around the world we live in and marvel in all its complexity and grandeur. But Mother Nature focusses on one thing when it comes to design: efficiency. That is to say, that nature strives to design things that will do the job best. For example, stars and planets are always round because a sphere is the most effective way to get all the mass as close to the planet’s centre of gravity as possible (a process known as isostatic adjustment). The wings of a bird have evolved to maximise the thrust generated at the least energy cost, while the sleek, teardrop body shape of fish allow for them to slip through water with minimal resistance. One of the best examples of nature coming up with the best design solution is beehives.

If you look closely at a beehive, you will find that it is made up of tiny hexagons. Each hexagon is a room that a bee can fit in and the walls are made from wax. The interesting thing about hexagons is that it has many properties that make it the ideal shape in construction.

Firstly, hexagons can fit together perfectly to tile a plane, meaning that bees can tile thousands of columns without wasting any space. The little columns even end in a unique pyramidal shape that allows them to tile up nicely with each other at the centre.

Secondly, a hexagon has 6 rotational symmetries and 6 reflection symmetries, making it very easy to tile as every bee will know what orientation to build their cell in using the side of any cell as a reference.

Lastly, in a hexagonal grid each line is as short as it can possibly be when tiling an area with the smallest number of hexagons. Therefore, bees can use much less wax when constructing hives, while achieving remarkable strength as hexagons gain lots of strength under compression. This design also allows for the maximum amount of honey stored in each cell.

Bees have mastered this architectural feat not through physics and mathematics, but through evolution – the driving force of nature. Over millions and millions of years, various types of bees will have experimented with square-celled hives or triangular-celled hives, but they could not survive as long as the hexagonal-celled bees because their hives were less efficient. This is exactly why nature is so good at coming up with the best solution to a problem. Because in nature, the best solution to the problem an environment offers is rewarded with survival.

Posted in Science & Nature

Rain

Posted on by Jinavie

Let’s imagine that you are walking outside, when rain clouds catch you by surprise and suddenly pour down on you. Assuming that you have no umbrella or anything to cover yourself with, is it best to run back home or walk back? Or to elaborate, should you walk and spend more time in the rain, or should you run, which means you will run into rain sideways?

There are two ways you can get wet in the rain: it will either fall on top of your head, or you will run into it from the side. The amount of rain that falls on your head is constant whether you are walking or raining, as the entire field you are travelling through is full of raindrops. Therefore, one would naturally think that running would not add much benefit as you run into more rain by moving faster, as you essentially hit a wall of raindrops.

But this is not true. No matter how fast you travel, the amount of rain you hit sideways is constant. The only variable that affects the amount of rain you hit sideways is the distance you travel. This is because the amount of raindrops in the space between you and your destination is constant.

Summarising this, the wetness from rain you receive is:

(wetness falling on your head per second x time spent in rain) + (wetness you run into per meter x distance travelled).

Since you cannot really change how far you are from your destination, the best way to minimise getting wet is to run as fast as you can to minimise the time you spend in the rain.

Then again, this is only the most practical option to keep you dry. If you are feeling particularly romantic or blue, then feel free to stroll through the rain, savouring the cold drops on your face (or wallow in the sadness that is your life).

(Here’s a very good video explaining the maths/science of it all: http://www.youtube.com/watch?v=3MqYE2UuN24)

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Posted in History & Literature

Hitler

Posted on by Jinavie

There are millions of units for various things. Some are simple and standard like the metre and gram, while others are quirky and humorous like the Helen unit and the Banana Equivalent Dose. As long as you can justify it with logic and objective quantification, you can virtually create any unit. Using this logic, some historical knowledge and a cruel sense of humour, one can come up with an extremely disturbing unit called the Hitler unit.

As most people know, Adolf Hitler is one of the most notorious criminals against humanity in the history of mankind. He was responsible for the death of at least 17 million people, including the 6 million Jews and 5 million other ethnicities killed during the Holocaust, and victims of World War II. For simplicity’s sake, let us only consider the victims of the Holocaust as direct victims of Hitler’s ambitions.

Using this statistic, we can now create a new unit called the hitler – equivalent to 11 million human deaths. Ergo, killing a single human amounts to a crime of 91 nanohitlers. The “hitler level” of some of history’s worst notorious dictators are as follows:

  • Kim Il Sung: 1.6 million deaths = 145 millihitler
  • Pol Pot: 1.7 million deaths = 154 millihitler
  • Hideki Tojo: 5 million deaths = 455 millihitler
  • Adolf Hitler: 11 million deaths = 1 hitler
  • Jozef Stalin: 23 million deaths = 2.09 hitler
  • Mao Zedong: 78 million deaths = 7.09 hitler

The hitler unit gives us a clear picture of “how much worse” someone’s crimes are compared to those committed by Adolf Hitler. For example, Jozef Stalin could be considered twice more evil than Hitler.

The true utility of the hitler unit is that like other units, it allows for useful conversions to other units. For example, the EPA currently values a human life at $6.9 million (USD). A simple unit conversion thus tells us that 1 hitler is equivalent to the loss of $75,900,000,000,000 (-$75.9 teradollars). In 2008 when the US Congress failed to pass a stimulus bill following the subprime mortgage crisis, the market lost $1.2 trillion over one day – the equivalence to 15.8 millihitlers. Conversely, if a mugger took $200 from you, they have technically committed a crime of 2.64 picohitlers.

Although the crimes of Adolf Hitler were beyond tragic, this imaginary unit teaches us that almost anything can be quantified in the field of science and mathematics.

Posted in Science & Nature

Pi

Posted on by Jinavie

Pi (π) a mathematical constant that is defined as the ratio of a circle’s circumference to its diameter. It is approximately equal to 3.14159, but since it is an irrational number (cannot be expressed as a ratio), the decimal places go on and on with no repeating segments. The history of pi extends back to almost 5000 years ago, as it plays such a crucial role in geometry, such as finding the area of a circle (A = π ²). It is not an understatement to say that pi is among the top five most important numbers discovered in history (0, 1, i and e being the others).

The interesting thing about pi is that it is an irrational number. As mentioned above, this means that pi has an infinite number of non-repeating decimal places, with numbers appearing in random sequence. For example, pi to a 30 decimal places is 3.141592653589793238462643383279… Because of this feature, pi contains all possible sequences and combinations of numbers at a certain point. The corollary to this fact is, if pi is converted into binary code (a number system of only 0 and 1, used by computers to encode information), somewhere in that infinite string of digits is every combination of digits, letters and symbols imaginable. The name of every person you will ever love. The date, time and manner of your death. Answers to all the great questions of the universe. All of this is encoded in one letter: π.

That, is the power of infinity.

Posted in Science & Nature

Units

Posted on by Jinavie

In September of 1999, NASA ambitiously launched a Mars weather satellite. But the satellite did not even reach its destination, instead exploding in the atmosphere soon after launch. Why was this? The reason was so stupidly simple. The failure was because of units.

The satellite that was designed by Lockheed Martin was designed using the imperial system (pounds, feet and yards), whereas NASA’s systems used the internationally-used metric system. Because of this simple error, the pride of the USA space program fell to the ground and an astronomical amount of money was burnt to ashes in the air.

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Posted in Science & Nature

Mathematical Beauty

Posted on by Jinavie

What is the most “beautiful” mathematical equation? For millenia, many mathematical formulas and concepts have been described as beautiful (and some defining beauty, as the golden ratio does). In the mathematical world, the adjective “beautiful” is used in the sense that certain mathematical concepts, despite the fact they are rational and objective, are so pure, simple and elegant that they can only be described as art.

One such formula is Euler’s identity:

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Renowned physicist Richard Feynman described it as “the most remarkable formula in mathematics”. What makes this array of symbols and numbers so beautiful? Firstly, it contains the three basic arithmetic operations exactly once each: addition, multiplication and exponentiation. It also connects five fundamental mathematical constants with nothing other than themselves and the arithmetic operations.

0 is the additive identity, as adding it to another number results in the original number. 1 is the multiplicative identity for the same reason as 0. Pi(π) is one of the most important mathematical constants in the history of mathematics that is ubiquitous in Euclidean geometry and trigonometry. Euler’s number(e) is the base of natural logarithms and is used widely in mathematical and scientific analysis. i(√-1) is the imaginary unit of complex numbers, a field of imaginary numbers that are not “real”, allowing for the calculation of all roots of polynomials. Euler’s identity neatly sums up the relation between these five numbers that are so crucial in the field of mathematics. It is also interesting to note that these five numbers were discovered at different points in history spanning over 3000 years.

Some people describe mathematics as a distinct language in itself. Not only that, but mathematics is considered the universal language as it is both universal and ubiquitous. If that is the case, than Euler’s identity can be considered an extremely pithy literary masterpiece.

Posted in Science & Nature

Folding Paper

Posted on by Jinavie

Take any piece of paper and fold it in half. Then fold it in half again. Chances are, you will not be able to fold the paper more than seven times. Try it. No matter how thin the piece of paper is, it is extremely difficult to fold a piece of paper in half more than seven times. The reason? Mathematics.

A standard sheet of office paper is less than 0.1mm thick. By folding it in half, the thickness doubles and becomes 0.2mm. Another fold increases it to 0.4mm. Already, the problem can be seen. Folding a paper in half doubles the thickness, meaning every fold increases the thickness exponentially (2ⁿ). By seven folds, the thickness is 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 times the original thickness. This makes the piece of paper so thick that it is “unfoldable”.

Another limitation is that folding the paper using the traditional method means the area also halves, decreasing exponentially. With a standard piece of paper, the area of the paper is so small after seven folds that it is mechanically impossible to fold it. Furthermore, the distortion caused by the folds is too great for you to apply enough leverage for folding the paper.

Could these limitations be overcome by using a larger piece of paper? Sadly, no matter how large the piece of paper, it is impossible (or at least extremely difficult) to fold a piece of paper over seven times. This has been a mathematical conundrum for ages, until it was solved in 2002 by a high school student named Britney Gallivan. Gallivan demonstrated that using maths, she could fold a piece of paper 12 times. The solution was not simple though. To fold the paper 12 times, she had to use a special, single piece of toilet paper 1200m in length. She calculated that instead of folding in half every other direction (the traditional way), the least volume of paper to get 12 folds would be to fold in the same direction using a very long sheet of paper.

Mathematics, along with science, is what makes something that seems so simple, impossible.

Posted in Life & Happiness

True Love

Posted on by Jinavie

Let us assume that everyone has a person they are destined to end up with. Can we calculate the probability of such a fateful meeting between a man and a woman?

Suppose that the woman is X and the man is Y. Firstly, X and Y need to be born as human beings. They cannot be born as a worm or an onion or something. Here, we will say that the total number of species is M and the population number of each species as P (technically this part is forcing it slightly, so we can skip it).

Although the two have to beat ridiculous odds just to start, just being born as human beings is not enough. One must be born with XX chromosomes to be a woman, and the other must be born with XY chromosomes to be a man.

Let us assume that the two were lucky enough to be born as a man and a woman. Next, they must live in the same space. If one lives in some Korean city and the other lives in some American rural village, it is unlikely the two will ever meet.

Even if they did live in the same place, X and Y must have subjective qualities that the other person finds attractive. If they are not interested in each other, nothing will happen even if they did meet. By this stage, we have clearly gone past the scopes of mathematics.

Then let us assume that a man and a woman, who fit each other perfectly and born as people, are living in the same space. We are still missing one variable: time. Even if we took only the 5000 years that civilisations have existed, the odds of the two being born in the same era as similar ages is less than 0.001%.

Species, sex, space, time… Statistically speaking, the chances of a man and a woman beating all of these odds to establish a perfect couple seem nearly impossible. But we can clearly see that “true love” exists all around us. Numbers are just numbers. If you find a person that makes your heart skip a beat when your eyes meet, that makes you feel that the more you get to know them, the more you think you cannot live without them; in essence a person that makes you think “this person is The One”, do not let the person slip away. The scenario of you and that person existing on the same space-time and loving each other is something that verges on the impossible.

There is no treasure as rare as true love. If you have found true love, or believe that you have found it, fight to seize it and do everything in your power to protect it. That is the greatest accomplishment you can make in life.

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Posted in Science & Nature

Seven Bridges Of Konigsberg

Posted on by Jinavie

The city of Königsberg (capital of Prussia, now Kaliningrad, Russia) has the Pregel River running through the middle, with islands at the centre of the river connected by seven bridges. Is it possible to cross all of these bridges while only crossing them only once each?

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If you try to solve this problem, you soon discover that it is incredibly difficult not to cross the same bridge twice. But it is difficult to tackle this problem in a brute force manner. To calculate all of the permutations in the order of bridges, you use 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040, meaning that there are 5040 possible arrangements of bridges. Then how can you prove if the problem is solvable or not?

The great mathematician Leonhard Euler, upon being asked to solve the problem, is reported to have said that the problem is impossible to solve on the spot. In 1735, he proved his answer by modelling the seven bridges of Königsberg in a diagram of four dots connected by lines (representing the bridges).

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By using this model, the problem is converted into a “draw in one stroke” problem, which is also called a Euler walk to honour Euler’s contributions. Euler discovered many properties and laws regarding such problems. If a certain point is the starting point, then the line must first leave the point, then even if it comes back to the point, it must leave again. Ergo, the starting point must have an odd number of lines connected to it. The opposite applies to the ending point, where a line must enter the point, and if it leaves the point it must come back to it. Ergo, the ending point must also have an odd number of lines connected to it. In the case of a Euler walk, the starting and ending points are identical, so the number of lines is the sum of two odd numbers, making it an even number. Thus, to find out whether a picture can be drawn using one line, use the following laws:

  1. If there are no points of odd degree (odd number of lines), the starting and ending points are identical.
  2. If there are two points of odd degree, the starting and ending points are different.
  3. If there are one of more than two points of odd degree, it is impossible to draw using one stroke.

Thus, a Euler walk is only possible if there are 0 or 2 points of odd degree. Looking at the seven bridges of Königsberg problem, we can see that A is connected to 5 lines and B, C and D are connected to 3 lines each. As there are four points of odd degree, we have thus proved that it is impossible to draw a path that crosses all the bridges while not crossing any bridge more than once.

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