Tag: mathematics

Posted in Science & Nature

Newton’s Apple

Posted on by Jinavie

Common belief is that Newton discovered gravity after an apple dropped on his head. Although there is no historical evidence to support this myth, it has become a popular story. There are two common responses to this story: the first is “Wow, Newton was a smart cookie” and the second is “Pfft, I could have discovered gravity without an apple, it is such an easy thing.”

The latter group of people are idiots. Newton did not “discover” gravity. Human beings have known that objects fall to the ground since the dawn of time and have utilised it in ways ranging from sports to killing other people by crushing them with giant rocks. Even animals know of the concept as seen by eagles dropping turtles on rocks to crack the shell. In fact, if you could not figure that out, then you would really be an idiot.

The reason why Newton is famous is not because he found that apples fall from trees, it is because he observed the phenomenon, noting that it was always perpendicular to the ground, which in combination with the knowledge that the Earth is round suggests that objects tend to fall towards the centre of the Earth. Again, Newton’s brilliance was not that he simply observed an apple falling, it was that he pondered it and spent years researching it until he discovered the way gravity behaves. He devised formulas to estimate how gravity functions, even applying it to predict how the moon orbits around the Earth. Thanks to Newton, we are able to model the world around us and send rockets to the moon without launching our astronauts in to the depth of space with no hope of recovery. 

Interestingly, physicists still do not know what causes gravity. There are many theories, such as particles called gravitrons attracting two objects to each other. Although the mathematics of two objects attracting each other has been accurately calculated, it is unknown what causes it. Only after you discover the truth behind how gravity functions can you say that “I could have discovered gravity in my sleep” (actually, even then you probably spent decades just trying to grasp the concept).

Before you criticise, know what you are criticising. 


Posted in Science & Nature

Golden Ratio

Posted on by Jinavie

The golden ratio is a magical number that divides a line into the most beautiful ratio. It bestows a mystical power in an object and allows for the creation of excellent architecture and art.
This magical ratio is (1 + √5)/2, or 1.618033988. If there is a line divided by the golden ratio called a + b, then b:a and a:(a + b) are both the same ratio.

We can find the golden ratio in countless values seen in animals and plants. A snail shell’s golden spiral allows for the snail to grow without changing shape, while the distribution of branches on a tree also follows the ratio. The golden ratio controls everything from the spiral pattern of galaxies to the pattern of our brain waves. The golden ratio is the law of the universe.

Using this magical ratio, we can find the most beautiful composition of a human being. The Venus of Milo, considered as one of the most beautiful figures in history, has a ratio of 1:1.618 between her upper and lower body (divided at the belly button) – the golden ratio. The same can be said for the ratio between the head and neck compared to the rest of the upper body, and the length from the belly button to the knee compared to the length below the knee. The exact same composition was used to construct the statue of Doryphoros, one of the most famous examples of ancient Greek sculptures. The diagram that illustrates these ratios is the Vitruvian Man by Leonardo da Vinci (Vitruvius was a Roman architect who utilised the ancient Greek knowledge of applying the proportions of a human being, i.e. the golden ratio, in constructing temples). 

The Great Pyramids of Giza, Solomon’s Temple and the Parthenon are all partially constructed according to the golden ratio. It is said that buildings constructed outside of the golden ratio will collapse over time. The same is seen in Eastern constructions, such as buildings and inventions from the Goryeo Dynasty of Korea. 

Interestingly, the golden ratio applies to intangible objects as well. For example, Chopin’s Nocturne pieces tend to climax at the point of the golden ratio (roughly two-thirds in). The ratio is still used in modern day design, with the standard credit card size being the best example.

The golden ratio is an eternal beauty that does not go out of fashion with time.

Posted in Science & Nature

Euclidean Geometry

Posted on by Jinavie

In 300BC, a Greek mathematician called Euclid wrote a series of texts called Elements. The Elements was a textbook that outlined many principles of mathematics (especially geometry) and it would become one of the most influential works in the history of mathematics. It is composed of a series of axioms (the axiomatic approach) from which many deductions and theorems can be made. Although many of these axioms sound extremely simple and like common sense, the implications are staggering.

The following is Euclid’s Five Postulates of Plane Geometry:

  1. Two points determine a line. 
  2. Any line segment can be extended in a straight line as far as desired, in either direction. 
  3. Given any length and any point, a circle can be drawn having the length as radius and that point as centre. 
  4. All right angles are congruent (can be superimposed). 
  5. Parallel postulate: If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side, if extended far enough. 

Using these postulates, mathematicians are able to deduce more advanced theories. For example, the Elements also describes the famous Pythagorean theorem, which states that “in any right triangle, the area of the square of the hypotenuse (the diagonal) is equal to the sum of the areas of the squares of the other two sides” (a² + b² = c²).

Thanks to Euclid’s works, we are now able to accurately model and measure the three-dimensional space around us. Not only did Euclid set the foundations for mathematics, his works were also instrumental in the development of logic and modern science.

Posted in Science & Nature

Dimensions: To Infinity And Beyond

Posted on by Jinavie

So far we have covered 6 dimensions: point, length, width, depth, time, alternate universe and travelling between possible futures being the key feature of each. Hopefully, you as the reader have caught on to the pattern of dimensions so far: point (0D) -> line (1D) -> branch (2D) -> fold/point (3D) -> line (4D) -> branch (5D) -> fold (6D)… This pattern of point/line/branch/fold continues past the 6th dimension. Therefore, the 6th dimension acts quite similarly to our 3D world in that we can assume it to be a point.
The 3rd dimension was a point in time. So what could the 6th dimension be a point of? The answer is infinity.

It was mentioned that the 5th dimension carries all of the possible alternate timelines that are created from a certain point. In the 6th dimension, these branches fold up to meet so that we can travel freely between every point. Therefore, the 6th dimension is a point that contains every possible timeline – where anything that can happen in the universe exists. This is infinity.

But by definition, infinity encompasses everything as there is no “end”. Then has our journey ended? In a fascinating turn of events, it turns out that we can not only ascend to the 7th dimension, but there are still 3 more dimensions to travel through.
By now we know that as the 6th dimension was a “fold/point” dimension, the 7th dimension must be a line connecting different 6D points (infinity). How can there be more than one infinity? Actually, infinity is only as great as the initial conditions from whence it was born – the Big Bang. In terms of universes, these initial conditions are the laws of physics such as gravity, the speed of light and hundreds of other constants. For example, gravity is 9.81ms-² in our universe. But if this value was off by even 0.0001, our entire universe would be completely different. Ergo, our universe and all the timelines that have and will form depend on the Big Bang. This also means that there can be many other “infinities” with different laws of physics. The line that these infinities lie on is the 7th dimension.

A good analogy for this is genetics. People’s lives have different outcomes depending on their choice, actions and random chance, but they cannot escape their pre-programmed genes. For example, it is not expected that a boy will (naturally) grow into a woman or sprout wings and fly. But if they were born with two X chromosomes or born with the DNA of a bird, this life would be possible.

Now let us follow the basic pattern to move to the 8th dimension. Here, the 7D line branches to meet yet another point of infinity (6D). And yet again, we can bend these branches through the 9th dimension to jump from one universe to another.

Lastly, we can take all of these branches and folds that encompass all possible timelines and all possible universes and draw it as a single point in the 10th dimension. This one point is the relative and absolute “everything”.

But what now? It is impossible to reach past the point of “everything possible”. This means that we cannot jump up another dimension as no other 10D point exists to be connected to. Ergo, the highest possible dimension is 10D and this is the basis of string theory. The 10th dimension is where the so-called superstring vibrates to form the subatomic particles that are building blocks of every matter in our universe.

As mind-boggling a journey it was, if you were able to follow through from the start, we have travelled from a single point that occupies no space to another point that encompasses all things possible in our universe in all possible timelines. We have zoomed out to the point that there is no longer a box to “think outside of”.

Can one ever reach that point where one knows everything that was, is and will be? To know every piece of knowledge that is the absolute yet relative truth? Although we cannot physically jump through dimensions, our minds can keep rising up to raise our level of understanding and enlightenment higher and higher. As we only live in the 3rd dimension, we have no less than 7 more dimensions to explore and understand. Only when we have reached the 10th dimension can we say that everything possible has been discovered.

Until then, anything is possible.

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)

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Posted in Science & Nature

Dimensions: Time Warp

Posted on by Jinavie

The first three dimensions covered the three variables that determine space: length, width and depth. Then what could the fourth dimension possibly add? The answer is duration. The third dimension effectively becomes a point and the 4th dimension connects different 3D points to form a line that we refer to as time. For example, the “you” at this exact time is different to the “you” in five years time. These two “you”s are different (3D) points that lie on a 4D line (more specifically, your life). This is exactly the same principle as the 0th dimension being points in a 1D line, except three dimensions higher.

As a “moment” in 3D space is just a point on a 4D line, travelling from one point in time to another (i.e. time-travelling) would be as easy as walking along a straight line for a four-dimensional being. This concept is mind-blowing for us as we cannot fathom the concept of manipulating time. We are unable to see time as a dimension as we exist in a lower dimension. Every single moment in our lives is a brick that paves the road called time, meaning that we can only see each individual brick and not the overall picture. This is exactly the same as how a Flatlander could not understand the concept of depth and how we can be above them.

Although we see time as linear and straight, we are under the same illusion as the ant walking down the Mobius strip. Time is actually twisting and turning in the fifth dimension, creating multiple timelines that branch out like a tree of possibilities. These branches are influenced by our own choice, chance and the actions of others.
In other words, if a man proposes to girl A then he will go down the branch where he marries girl A. However, if he chose not to propose, he would end up marrying a different girl (or not at all). Therefore, he has entered a different branch than the girl A branch.
Now, if the man wanted to go from a timeline where he married girl A to another timeline where he married girl B, what could he do?

One method would be if he bent the 4th dimension (time) on itself through the 5th dimension to travel back in time to when he met the girl and not ask her number. This is exactly like folding a 1D line into a 2D circle to make the end point meet the starting point. However, to marry girl B he would have to make the right sequence of choices that lead to him marrying girl B (much like a role-playing game). This is the long-way round that would be too time-consuming and complex.

A simpler approach would be if he folded the 5th dimension through the sixth dimension and jump from the ending of timeline A to timeline B. This would be like the finger-lifting analogy we have been using time after time. If we pretend that the 5th dimension was a piece of paper, then we could fold it into a 6D cylinder so that the two edges meet. Now we are able to jump from one ending to another effortlessly, just as we did in the 2nd dimension.

It is easy to confuse the 5th and 6th dimension as they both deal with “alternate realitites”. Here is one way to differentiate the two: 5D space is like a 2D space for time – a flat plane where different timelines cross each other. Therefore, it can contain all of the possible outcomes from an initial condition – that depends on an action, choice or chance – such as your conception (visualise a dot on a piece of paper with many lines radiating out from it).
If we were to put a dot on a piece of paper above the first piece of paper (in the 3D space for time), we have entered a universe where there is a completely different starting point (before your conception), such as dinosaurs not existing. This means that the line called “you” may not even exist, and the two pieces of paper would never meet. The only way to jump from one piece of paper (5D) to the other would be through the 6th dimension.
Therefore, by jumping up a dimension, we gain a degree of freedom where we can move in yet another direction. This is seen between every dimension, such as 2D versus 3D. The 6th dimension merely lets us travel between different sheets of 5D paper.

So somehow we have reached the 6th dimension where one can not only time-travel, but jump from one alternate reality to another. Shall we venture further into the seventh dimension – infinity – and beyond?

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)

(Stick figures from xkcd)

Posted in Science & Nature

Dimensions: Flatland

Posted on by Jinavie

As we live in a three-dimensional world, it is difficult to imagine that there are higher dimensions. To illustrate this, the thought experiment of the hypothetical “Flatland” can be considered. Let us assume that there is a two-dimensional world called Flatland. Here, the concept of depth does not exist. Only forwards, backwards, left and right exist; there is no up and down. Everything that happens here would look like it was drawn on paper.

Now let us interact with Flatworld. If we were to touch Flatworld with our finger, it would be like poking your finger through a newspaper. The inhabitants of Flatworld would see a circle suddenly appear out of nowhere that grows larger and larger. A person would appear as if they were being seen through a CT scanner – in sections. The concept that things can be above or below would sound crazy to a Flatlander, even though to us it appears as a simple concept.

Let us take an ant walking along a piece of paper as an example of a “2D object”. If the ant wishes to go from one edge of the paper to the opposite edge, it must walk along the 2D plane. However, with our 3D powers, we can fold the paper into a cylinder; now the ant can walk to the other point in an instant (across the fold). To another ant on the other side, the ant would look as if it teleported and suddenly appeared out of nowhere.

In another experiment, we make a Mobius strip (a ribbon is twisted once then its two sides are joined) and make an ant walk along it. Although the ant would think that it was walking in a straight line along a two-dimensional surface, it would have walked on both sides of the strip – a three-dimensional concept. If the Mobius strip concept is confusing, think of a garden hose instead: an ant walking along a straight garden hose is walking in: 1D (straight line), 2D (hose is actually a flat surface) and 3D (the ant can walk in a corkscrew pattern along the hose).

If we were to tell that ant that it had just travelled in a higher dimension, that ant would either scoff at us or be genuinely terrified of the experience. To it, we (or the giant pink circle that it sees our finger as) would look like some omnipotent being that can see everything going on in its world and teleport from one place to another. And although the concept of depth would initially intimidate the ant, it would bring the level of the ant’s understanding of the world up one dimension. For if we see what we only know, then how can anyone see anything new? The only way to truly learn and understand new things would be to jump out of the box and see everything from the outside – just like an ant seeing the piece of paper it was on from a higher ground.

Although we may laugh at the foolishness of the Flatlanders (and the ant), to a being of the 4th dimension, we would appear just as stupid and naive. By applying what we learned from the world of Flatland to our three-dimensional world, we can expand our horizon of knowledge and understand what the fourth-dimension is.

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)

Posted in Science & Nature

Dimensions: Exploring The Dimensions

Posted on by Jinavie

In the 21st century, films and television have evolved to show 3D images. However, most people only have a crude understanding of what dimensions actually mean. This is a guide that will explore the incredible journey from a zero-dimensional point to a tenth-dimensional point and all the wonderful lines and folds that lie in between.

A point in space has no area – this is the zeroth dimension.
When two points are connected, it forms a line – the first dimension. This line allows one to travel from one 0D point to another, introducing the concept of length.
Another line is drawn branching off this line in a different direction – we have entered the second dimension. Now we have the concept of width.
By adding another concept – depth – we ascend to the third dimension. Now it is possible to go from one point on a 2D surface to another as we have “folded” a branch in the second dimension to meet the other branch. A simple explanation would be lifting your finger off one point and placing it on another point.

To simplify our journey to the third dimension we have:

  • Assumed a “point” in space as a dot (.)(0D)
  • “Joined” two dots to form a line (|)(1D)
  • “Branched” the line to create two ends (Y)(2D)
  • “Folded” the branches together to make the two ends meet (P)(3D)

These four concepts of point, joining, branching and folding are crucial in understanding how the different dimensions interact.

An interesting thought regarding the concept of dimensions is perception. How would inhabitants of each dimension view different dimensions? This is easy for lower dimensions (2D and below) because we can see them as a dot, line, square and cube (our dimension). Ergo, we can easily understand all the concepts of the lower dimensions (e.g. width). However, the opposite would not be possible (e.g. a 2D being trying to understand depth) as the concept does not exist in their dimension.

To further explore this thought, we must explore the world of Flatland.

(This post is part of a series exploring the concepts of dimensions. Read all of them here: https://jineralknowledge.com/tag/dimensions/?order=asc)

Posted in Philosophy

Prisoner’s Dilemma

Posted on by Jinavie

The prisoner’s dilemma is a famous example of how game theory functions. It predicts the behaviour of two people when forced to cooperate. The story goes as follows:

Two accomplices in crime are arrested by the police. They are interrogated in separate rooms. As the police have insufficient information, they offer a deal to each prisoner to confess that the two committed a crime (or deny). The deal is:

  • If you confess and your partner denies taking part in the crime, you go free and your partner will serve ten years (and vice versa).
  • If you both confess you will go to prison for four years each.
  • If you both deny taking part in the crime, you both go to prison for two years.

Assuming the prisoners act rationally (i.e. for their best interest and minimising their jail time), the prisoner will obviously choose the “confess” option as this is hypothetically the best choice (minimum time = 0 years, compared to only 2 years minimum for denying). However, because both prisoners are thinking this, the result is almost always that both confess and end up with four years each. Therefore, because human beings are unable to trust another human being enough, people always end up acting irrationally (benefit not maximised).
If the two had been trusting (assuming the other would deny too) and cooperated, both would have served half the time. But people always assume (correctly) that the other person will betray them for their selfish gain and this win-win result is unattainable.

But what if the other prisoner was yourself? Let us assume that the prisoner’s dilemma game was played by you and an exact copy of you. A copy that thinks like you, acts like you and identical to you in every single way. Can you trust yourself? Do you trust yourself enough to deny the crime, when it is entirely possible that he or she rats you out to walk free while you suffer for 10 years? How do you know that he loves you more than himself? 

Your greatest enemy is you.

Posted in Life & Happiness

Formula Of Happiness

Posted on by Jinavie

Ask yourself the following questions using a scale from 1 to 10, where 1 is “not at all” and 10 is “to a large extent”:

  1. Are you outgoing, energetic, flexible and open to change?
  2. Do you have a positive outlook, bounce back quickly from setbacks and feel that you are in control of your life?
  3. Are your basic life needs met, in relation to personal health, finance, safety, freedom of choice and sense of community?
  4. Can you call on the support of people close to you, immerse yourself in what you are doing, meet your expectations and engage in activities that give you a sense of purpose?

Add the scores for 1 and 2. This is P for personal characteristics.
The score for 3 is E for existence (health, financial stability and friendships).
The score for 4 is H for higher order and covers self-esteem, confidence, ambitions and sense of humor.
Now input each score into the following equation:

Happiness = P + (5 x E) + (3 x H)

This gives a score out of 100. The greater your score, the happier you are (over 80 is considered a “happy life”). This formula was devised based on a psychology study of 1000 people. It takes into account the various aspects of life that contribute to your overall happiness and emotional well-being. The study showed statistics such as 40% of men reporting that sex made them happy, while 25% of women reported that losing weight made them happy. Men found more happiness in romance, hobbies and a pay rise compared to women.

Although everyone has a different definition of happiness, there is no doubt that there are common “happiness” factors to all of us and this equation tries to objectively quantify how happy you are in life.

Are you happy?

Posted in Science & Nature

Titanic

Posted on by Jinavie

Titanic is a film telling the story of the sinking of the eponymous ship, the RMS Titanic, directed by James Cameron in 1997. Most people are entranced by Leonardo DiCaprio and Kate Winslet’s excellent acting, the cutting-edge special effects and the waves of emotions that it projects to the audience, but there is another component that is just as amazing.
Most films and television shows tend to sacrifice science in the name of drama. Thus, science fiction movies are ironically quite inaccurate in even the most basic scientific facts. However, Titanic is strangely true to science despite being a drama film.

To start with, we can take the scene where Rose, embraced by Jack from behind, spreads her arms wide open like wings while on top of the stern of the Titanic. Here, Rose is seen standing so high that she is above the rails from the thighs up. In this position, even a slight push would cause her to lose balance and make her fall, causing the movie to end prematurely. But on closer inspection, it can be seen how Jack has his arms wrapping under the cables. To be so attentive to detail even in the moment of heated passion – Jack is surely a calm, cool-headed man.
In the scene where the Titanic is sailing, it takes 25 seconds for the ship to completely pass a point. Considering that the ship was 269m in length, this comes to a cruising speed of 38km per hour. This is 21 knots when converted – almost identical to the actual cruising speed of the Titanic which was 22 knots.

The movie is accurate in even finer details. Let us study the climactic scene of the sinking. When the ship is tilting at its highest point, a person took 4.3 seconds to fall and hit the water. This equates to a height of 91m, which can be achieved by a 269m ship tilting at about 40 degrees.
When Jack is bound by handcuffs, Rose bravely cuts the chain with an axe. But can a fair 18-year old girl summon such strength? If the chain is the thickness of two 5mm diameter metal rings, then the blade requires 49 Joules of energy to cut the chain. To achieve this, a 3kg axe must be swung at the speed of 20km/h, which is the same as dropping the axe from a height of 1.6m. Ergo, Rose can create enough energy simply by adding a little more strength to the axe as she swings it down from above her head.
Lastly, in the tragic scene where Jack sinks away, he disappears in 6.4 seconds. If by a rough estimate he sank about 2m, then it suggests that he descended at about 1/100 strength of free falling. This means Jack’s body density is about 1% greater than sea water. As the density of sea water is 1.04g per 1cm3, this is perfectly reasonable assuming that Jack is big-boned.

A film focussing on such fine scientific detail can certainly be called a masterpiece of the century. If only Rose’s voice did not echo in the final scene…