Can you live in a Fractional Dimension? – II

To quote Anthony Liccione, “A mind is not weighed by its magnitude, but by the dimensions of its thoughts”. Of course, it’s not the same dimension I am going to talk about here but the spirit of the statement is a desirable dimension (pun intended). Hope you have read the first half of this post. No? never mind just keep reading ūüôā . I will do my best to answer the question posed in the simplest form. To start of, think about a line segment like this

Straight_LineIf you double the length of this line then you will get 2 copies of the line segment. Okay, consider a square as shown in the figure. sqAgain, double the sides of the square. Now, you get 4 copies of the original square. I know, by now you would have extended this logic to a cube and figured out that 8 copies come out by doubling the length of each side of a cube. Okay, summary:

Line: 2 = 2^1 (2 to the power of 1)

Square: 4 = 2^2

Cube: 8 = 2^3

It is very easy to notice that the last numbers (exponents) are the number of dimensions in which these figures exist. In fact this is one of the definitions of dimension called Hausdorff dimension. Quite a cool definition actually. Of course, the definition I just gave is not totally correct in mathematical sense but for explanation, good enough. Have you heard about this amazing convoluted triangle called the Sierpinski gasket shown below?¬†As you can see there is a regular and conspicuous pattern exhibited by the¬†Sierpinski gasket, recurring all the way to infinity. Some of you would recognize this instantly as a fractal which, of course, is a bigger collection of objects having the same properties –¬†regular and conspicuous pattern¬†recurring all the way to infinity. There is a reason these structures are called fractals and I am going to explain you just that. Look back at the structure below.

Now, let’s double each side of this triangle.

If you observe closely you get 3 copies of the original triangle not 4. The implication of this is captured in this equation:

2^d = 3
Where d is the dimension in which the¬†Sierpinski gasket live. ¬†A quick evaluation puts d at 1.5849… which is a fraction……. fraction….. frac….. fractals?! It’s true the name fractals was coined because these closed figures exist in a fractional dimension. After ‘how’, some may ask ‘why’ fractional dimension? Because, Self-Similarity. Period. Remember Cantor Set from the previous post, yes even that falls under the category of fractals. There is an inherent redundancy (Self-Similarity) in fractal formation and even a rudimentary knowledge in set theory will tell you that sets don’t contain repeated elements which is exactly the reason that leads to non-integer dimensions. There are some beautiful fractals which, I think, I am obliged to share here.

 

Are there any fractals in nature? Oh, yes. The famous 1967 paper¬†“How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” puts¬†the dimension of Britain’s coastline at 1.25. Intriguing? What about this one: ¬†each branch of a cauliflower¬†carries around 13 branches, each 3 times smaller. This renders the dimension of a cauliflower to 2.33.

So, do you live in a fractional dimension? At least your brain surface does. The Hausdorff dimension of the surface of the brain is at 2.79 and that of the lung surface is 2.97. If such an essential organs can, then why not the whole body surface? Yes, it is possible but there is a key ingredient which is missing, remember, Self-Similarity. So one can safely claim that the dimension of our body is 3 (integer) in the conventional sense. Too bad, I know.

In summary, you can but you don’t live in a fractional dimension but if you desire to, ask this guy.


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Can you live in a Fractional Dimension? – I

To simply put, your expression after reading the question is puzzled, right? How can there be fractional dimensions, let alone living in one? Before heading to answer something that tricky, shall we get to ground zero and understand what a dimension is in the first place? Quoting Wikipedia: “In¬†physics¬†and¬†mathematics, the¬†dimension¬†of a¬†space¬†or¬†object¬†is informally defined as the minimum number of¬†coordinates¬†needed to specify any¬†point¬†within it”. Did you note the word “informally”? Obviously, there exist some rigid definition, what do you think mathematicians are for :). In fact there are multitudes of definitions in mathematics but will come to that later on. ¬†For now let’s just discuss some amazing thoughts based on the wiki answer which will really be a pleasurable experience. The most important thing in science is to ask the right questions so let’s start by a few interesting questions.

How many dimensions do our eyes see? The answer can be derived from this counter question: How many dimensions can a camera capture in an image? Well, that’s simple it is 2 and because our eye is an organic camera it can see only 2 dimensions. But why 2? Hmm, that’s a decent question. That’s because we live in three-dimensional space and one of the dimension(depth) in the image capture setup will be for light to bounce off the object and form image on the retina(photo sensors of our eye). Also, did you know because we have 2 eyes very close to each other we have stereoscopic vision and thanks to that we can visualize depth too. Okay that was more than asked, but how can you claim that the space we live in has just 3 dimensions? Because every point in space can be associated with three numbers in x, y and z direction.¬†But how can you be sure that there are only 3 dimensions? Because we perceive only 3 dimensions. Isn’t that answer more anthropological rather than scientific? Okay, I will end the soliloquy here but see I made my point. Which one, science has anthropology as its basis or soliloquies suck :). No idiot, the point is that there is no concrete proof of space being three-dimensional.¬†By the way, the picture below shows how a ¬†Tesseract, a 4-dimensional “cube” will look to us when it is rotating, so Yeah.

Okay, you might be thinking where I am going with this. As I said earlier before moving on to a tricky answer let’s get comfortable with dimensions and related concepts. In the nineteenth century a lot of work was done in Mathematics by some of the most brilliant people such as Bernhard Riemann on higher dimensional geometry. These were later used in physics starting with the Special theory of relativity. ¬†In fact it was Hermann Minkowski (Yes, he was a Mathematician and¬†Einstein¬†was his student) not Einstein who gave the 4-dimensional¬†space-time model. ¬†Then came along Quantum Mechanics which was completely probabilistic and too random to make sense, although philosophers loved this and created all sorts of crappy theories using the¬†uncertainty¬†of the new science (as is always the case). The uncertainty of the new subject was obviously taunting to many physicists but in the recent¬†decades the dimension picture is attempting to explain why quantum mechanics may look probabilistic because of our 3-dimensional spatial perception. The amazing video below talks about this new approach, now very well-known as M-theory (apparently M stands for Minkowski) and also shows you how to visualize 11 dimensions.

Cantor Set

The above picture shows a Cantor set also known as Cantor comb. You get this by taking a line and opening up the middle third of the line. As you can see the second line has an open middle which is 1/3 the length of the line and the third line is got by applying the same rule on the second line. What is so special about this? Actually, everything about this set is special and the most amazing thing, before I stop this first part of my answer (I know, I was cheeky for not giving the answer in this part :)), is that the Cantor set which intuitively looks to have a dimension of 1 (as it is a line), actually has a dimension of 0.630929… Yes, I will explain this in the second part and also answer the question in the title. So, stay tuned

……………….to be continued………………….

Introspection by a young Indian

I’m sure most Indians would have heard two contradicting views about India. Firstly, the arguments claiming India to be a prosperous and well developed region (cluster of provinces) before colonization and claims to have proof for their argument. Secondly, the argument (especially foreigners) considering Indians as people who are boastful of their past arguable achievements, without having any present day significance. This contradiction made me think. Initially like everyone else I tried to find which one is the truth but soon realized that I’m missing something bigger and more important. ¬†I started so discover a certain degree of agreement in the contradictory views. So, the soul of this post is to find the implicit fact agreed by both the proponents.

In the former argument there is a call for the Indians to not forget their roots which were apparently glorious and says that latter argument is developed by westerners (in particular by the British) to destroy our self-esteem. The latter argument just points out at our present day reality and asks for our recent contribution to the world development. If we get into verification of authenticity of either of the arguments we will be lost. We can’t deny the second argument but at the same time we can’t fall prey to it since we were indeed a slave nation under the British and we do see lack of self-esteem among Indians, who are willing to accept anything from the west and don’t like to appreciate the east. So let’s move on and check if there is any common ground in these two arguments.

Being a person who likes to define rules of the game before playing one, I want to define what is progress? Here is the way I look at it: Anything which makes us understand our universe better is progress. The only way to understand the universe better is through science. Thus my yardstick of progress is scientific progress. When we contribute to scientific progress it gives a sense of right to enjoy the modern day technological facilities or else metaphorically speaking, if we don‚Äôt contribute to science and enjoy its benefits then it is like living on other’s income by theft or robbery. Being a man with moral conscience scientific progress is the epitome of all kinds of progress which will make me proud in front of foreigners. It is the last frontier of a nation’s developmental challenge.

Now coming back to the two opposing views, one fact is accepted in both perspectives i.e., Indian contribution to scientific development (synonymous with development) in modern times is very less if not nil (thanks to CV Raman, Ramanujan, JC Bose). Major chunk of scientific work happens in America after WW2 (though CERN and some European universities continue to hold on) and before WW2 it was Europe which was the knowledge bowl of the world. So in past 400-500 years we are constantly lagging in the quest for knowledge. Whoever claims for Indian contribution inevitably goes before this period and in case they come up with some examples they will be exceptional people with a foreign institute affiliation. So where did we fail? I am forced to accept that we failed, but where?

Scientific progress asks for economic prosperity since the quest for satisfying the knowledge hunger comes only after satisfying the real hunger. India certainly had all the resources for economic development. Any simple study will reveal the reason behind huge population in India as abundance of natural resources here, apart from petroleum. Hence economic progress with men and material power must not have been a problem. But, both scientific and economic progress asks for one key factor which was completely missing in India and it is often neglected around the world. That key factor is political stability. The political instability and lack of governance not just derailed our scientific progress but also our economic and every other forms of progress. This is where our ancestors failed us. They couldn’t give political stability and protection because of lack of foresight, will, interest and presence of selfishness.

Now that I have arrived at some sort of conclusion what is more worrying for me is that we continue to commit the same mistake as that of our ancestors’. We continue our¬†apathy and indifference towards politics. Majority of those who indeed show interest does it for personal gains or to pessimistically ridicule the developments. Politics is something which everyone must know. Irrespective of a person studying science or arts or economics, he must know politics and its implications. He must have a political ideology of his own. He must be able see through the political developments and actively participate in it. Hopefully we will realize this before it is too late.

Chaos before the Chaos theory

Engineering is not an exact science and approximation theory lies at the very heart of it. We assume and approximate a lot, and a whole branch of mathematics called numerical analysis caters to our need. For instance, in analog electronics many parameters are inter-dependent non-linearly but we approximate and bring all that down to a neat Linear model (Remember this ‚ÄúLinearity …..Obvious?‚Äú). Avoiding the jargon, the output current depends on various powers of the input current in a transistor but the complexity is reduced by assuming the output current to be linearly proportional to the input (Approximating, like a Boss ūüôā ). Yeah, you can do that because the error is insignificant but, what about a system where even a small deviation can result in a chaotic ending. An analogy would be better, imagine a guy standing on the hilltop as shown. If he starts at a slightly different point, he would end up below but if he is exactly at the tip, then he stays there. So even a small variation can give rise to drastically different results. This is the premise of the big subject of 20th century, Chaos theory. Understanding Chaos theory explains how butterfly’s wings might cause tiny changes in the atmosphere which can ultimately cause a tornado on the other side of the globe. An interesting story is behind the accidental discovery of this new branch of mathematics and I am going to continue with that.

un1The year was 1885, King Oscar II of Sweden and Norway declared a prize of 2500 crowns ¬†to anyone who can once and for all establish, mathematically, if the solar system will keep working as it does today or will it suddenly shatter apart. The problem is as simple as it sounds? ¬†No Way. ¬†Even the great Newton, who had earlier showed that 2 bodies in space will have stable orbits, had tried and failed. The problem can be reduced to 3-body (sun, moon and the earth) but even that is a tough one; one has to deal with eighteen variables i.e. the position and velocity of each body in each of the three dimensions. The hero, also the poster boy of french mathematics then, Henri Poincar√© took this challenge upon himself and started working towards this mammoth task. He invented new techniques to simplify the problem, making successive approximations to the orbits which he felt won’t affect the final outcome significantly. Eventually he couldn’t solve the problem in its entirety but he was still awarded the prize for his techniques (To all teachers and graders, do you see this? The final answer doesn’t matter, the path traversed does)

When the paper was almost ready, one of the editors realized that there was a problem. Contrary to Poincar√© ¬†assumption, even a small deviation in the initial conditions can end up in completely different and unstable orbits. This observation came as a nightmare to¬†Poincar√© who had already sent a few copies of this paper out. The silver lining though was the new subject of Chaos theory which was very important and everybody soon realized its implications. ¬†In the end¬†Poincar√© became the key proponent of Chaos theory and even the short turmoil was eclipsed by the magnitude of this discovery. Maybe,¬†Poincar√© started with some good initial conditions ūüėČ

Fun way to find your nerdiness quotient

Are you ready for the truth? Ok this is how it works. There are 10 nerd jokes below. If you enjoy a joke then +1 for you, if you understand but don’t enjoy it then +0.5 for you, if you just don’t get it then you get zero. After going through jokes you can find your position on the following scale:

nerd scale

so lets start:

  • 1. Two variables discussing about religion:

x^2: Do you believe in god?

x^3: Well, I do believe in higher powers!

  • 2. Meanwhile in another part of universe of the polynomials

Spider-polynomial: “with great powers come great difficulty in factorizing”

  • 3. There are two kinds of people: Those who can extrapolate an incomplete data…
  • 4. There are only 10 types of people in the world: those who understand binary and those who don‚Äôt.
  • 5. Geeky pick up line: I wish, I was your derivative.. I could lay tangent to your curves!
  • 6. Another pick up line: I’m attracted to you so strongly, scientists will have to develop a fifth fundamental force.
  • 7. And one last pick up line: I wish, I was Adenine so I could get paired with ‘U’
  • 8. You may think glass is half empty or half full but engineers know that glass is twice large as it should be!
  • 9. A majority of people have an above average number of legs.. think about it.. its true!
  • 10. I’m not lazy, I’m overflowing with potential energy

Hope you got the score that you wanted to get (either more than 5 or less than 5). Let me tell you that my heart goes out for those who wanted to score less than 5 but scored more. Do write a comment below about your position on the nerd scale.