Civilisation as a child of mathematics

The following are six mathematical functions you can find speckled across the natural canvas, accurately simulating the mathematical behavior of everything from multiplying bunnies to the bending of light around black holes.

1. Sine

The sine function

The sine curve is a fabulous example that concretely solidifies the scientific basis of thousands of natural phenomena. The function that generates the curve is simple:

f(t) = A . sin(ωt + θ),

where A is the amplitude of the wave, ω the angular frequency and θ the phase angle. The behavior of the curve itself bears a close relationship with the circle: draw a circle on a sheet of paper and mark its center as the origin of a Cartesian system. Now, if you were to trace the locus of the figure and project the position of your hand on the x-axis, you will observe that for every rotation that you make a point on the projection will first move away from the center, travel to the axial extreme (a length which is equal to the radius), come back to the center, and then move on to the other. If you were to plot the distance of that point from the center of the circle, you will get the sine curve – of course, you’ve to keep in mind the sign changes.

Because of its convoluted relationship with the circle, the sine curve has basis in its applications as a periodic waveform – a typical wave that repeats itself in design over a specific time period. Remember the path traced by a pendulum in your grandfather clock? Same. Also goes with undamped oscillations of a block suspended from a rigid ceiling by a spring, the propagation of heat waves, the vibration of strings on a guitar, the intonations in human speech, the signal processing that is required in most electronic gadgets, Heisenberg’s inequality and proofs of quadratic reciprocity.

2. Cosine

The cosine function

The cosine function differs from the sine function only in that the phase angle θ is displaced by 90 degrees. In other words, the cosine waveform is the sine waveform that’s got itself a small headstart. Such a wave becomes useful when one studies the propagation and interaction of multiple sine waves that give birth to interesting interference patterns like the one shown.

An interference pattern

3. Exponential

The exponential function

The graph is generated by the following function:

x(t) = α . β^t/τ

Suppose that a gambler plays a slot machine with a one in n probability and plays it n times. Then, for large n (such as a million) the probability that the gambler will win nothing at all is (approximately) 1/e. And so Napier’s constant raises its fiery head.

At first, nobody saw nothing peculiar about the value of e – not until Jacob Bernoulli began to study the gambler’s issue and began to take note of a particular value that could be approximated to the value of another famous limit. Nestled between and 2 and 3 on the integer scale, the exponential growth factor manifests itself in hundreds and hundreds of mathematical problems, all of which have direct impact on business strategies and economic development. You will know the chessboard problem, wherein an Indian king was once gifted a beautifully crafted chessboard by a courtier. In return, the courtier asked for this: one grain of rice on the first square of the board, two grains on the second, four grains on the third, and so on. The king obliged, only to find that for 64 squares to be filled, 18,446,744,073,709,551,615 grains of rice would be needed – something that would weigh 461,168,602,000,000 kilograms!

Applications? Unsurprisingly many because of the way the function tends to grow with respect to time. If you observe the graph, you can see that for small increments of ‘x’ after some lower values, the value of the function skyrockets. This is similar to, if not the same as, the many growth factors manifested, for example, in the following:

• The Malthusian catastrophe
• Population of microorganisms in a culture
• Loudness of sound
• Nuclear chain reactions
• Ponzi schemes
• Moore’s law
• Compound interest
• The water lily problem

4. Logarithm

The logarithmic function

Draw a line passing through the origin and having a slope of 1, and using this line as a mirror, reflect the exponential curve. The image is the logarithmic curve, the inverse of the exponent. You must be reminded of having used the log tables, a small booklet with those long lists of extremely tiny numbers with which you could solve complex math problems. That’s the log function for you: it is the mathematical equivalent of scaling. The only difference is that, unlike in practical situations where it is constant, the scale in the universe of numbers is part of a sequence characterised by a growth ratio.

And as the exponential series represents growth, the log function represents a pattern on which to base that growth. The applications of both are the same since one is the inverse of the other, but it is worth detailing both because of the same reasons.

5. Asymptotes

Asymptotes

Asymptotes are like porn movies you find on the net for free: they tease and tease and tease, and then they end abruptly leaving everything else to your imagination. The one shown above is a rectangular hyperbola. The sections of the hyperbola that extend towards infinity along either axes are the asymptotes – and only because they exhibit a tendency towards touching the axes but never do. Put an other way, the first curve (+x-axis asymptote) gets closer to the second (+x-axis) as it gets farther away from the origin.

(x – h)(y – k) = m

m = x . y

The real application lies in the field of asymptotic analysis, which in turn is a key tool for exploring the ordinary and partial differential equations in the mathematical modeling of fluid flow through the Navier-Stokes equations.

6. Modulus

The modulus function

The modulus function is the noble gentleman amongst functions: you feed it with negative values but all it returns is their positive cunterpart. That is probably why it is also known as the absolute function.

Introduced by Jean-Robert Argand, and later conferred a denotation by Karl Weierstrass, the applications of this function pertain to the concepts of complex numbers, quaternions, ordered rings, fields and norms – which in turn are used in the modeling of real-world phenomena.

—

Have you ever wondered how 10 digits that were introduced a few millenia ago gave birth to so many different and varied functions and behaviorisms? The need that gave rise to them in the first place was that of quantification: simple and abstracted notations that each stood for a particular value that was a multiple of one. The second step lay in the classifiability of these numbers into similar-seeming sets, each of which was deigned to behave the same way. After this categorization arose the applications, where real world objects were compared to the numbers in terms of their respictive classifiabilities. Next in line was dimensions: the number of objects in a particular direction. Ultimately, there came modeling which represented the interface between understanding what was already there and what we could do in order to mimic it. The computer and the engine, two machines that completely changed the way the world understood and functioned, are both conclusively based on the functions shown above. The computer uses the functions to generate higher values via (complex variations of) Boolean and set logic, while the engine uses them to magnify inputs to result in larger outputs. However, their true importance lay in the fact that each one of them represented hundreds of us. One computer or one engine did what a thousand humans could have together in a single day. They saved time; rather, they brought in extra time, time that was devoted to other purposes, time that quickened the process of civilisation.

The art of licking ass

This is a very specialised skill if you will, and requires a lot of conviction on the doer’s part in order to get the right message across. In today’s world of money and everything being as close as possible to materialistic, licking asses can get you a long way if you lack merit. It’s the single most efficient method that has been proven, each and every single time, to succeed. The amount of vanity that the people around us bear forth with pride is staggering, and years of evolution and civilisation have done nothing to castigate it. And the silver lining ultimately comes out to be the permission of that vanity to continue to stay and spawn from mind to mind. It is present in every individual, within me, within you, within Osama, within Obama, and speaks forth through one form of the arts or other. Money can do nothing to quell it: a poor man will have it, and it will show forth in his perseverance; a rich man will have it, and it will show forth in his philanthrophy. Wellbeing can boast of nothing it has done to defeat it: a sick man will have it, and it will show forth in his steadfastness in the belief of the value of life; a healthy man will have it, and it will show forth in his early morning jogs. And it’s not as if vanity and education are parallels – the illiterate is capable of being as vain as the educated are, and the fundamentalist will be as vain as the liberalist. It has its mark everywhere.

And that is good, nay, fabulous! What health or wealth can’t do, vanity does. It fires up the dying spirits of a man, it boosts his surging morale, and it gives an instantaneous solution to every conceivable problem. Suppose a society bereft of vanity, and everyone becomes Jesus. That is not good: we need to be human to feel human, and we have our needs and desires. These desires are what propels us forward, don’t you think? We need to feel good about being who we are, and we can get nowhere if not for vanity. It is an element that requires no faith nor belief. You don’t have to pray that you or someone else has it: it comes with the package. The highs and lows that speck our days or nothing but vanity and the castigation of that vanity, respectively. All it needs is acceptance, and once it sinks in that we are vain and can not help being so, then vanity is a boon. It is the fruit of knowledge in disguise.

There is no single governmental or private institution that is free from it. It begins with begging and ends with holy matrimony, at least the ones don’t work out that well. If you expect yourself to be sufficed by the happiness that emanates from the “admiration of nature’s beauty and her bountiful fruits”, then you and I are Aryan sages from 4,000 years past. Changes abound, and we need to keep up with them to stay around as ourselves! We, gentlemen, need to lick ass to be able to do that. We have to go to little children and tell them how cute and pink and chubby they are, and only then can we hope to sell them an ice cream even if they don’t want one, and only then can we hope for a commission on that. We have to go to young ladies and tell them how bright their skin is, and only then can we either get their phone number or sell them a skin cream. We have to go to sardonic men in coats and polished boots and tell them how we admire them and how we think they’re so inspirational in their business achievements, and only then can we hope to bag a contract to build a gate for their factories. We have to go to the local money lender, tell him how poor we are (this is a special one: others can feel good about themselves only if they feel bad for you), get some money for an exorbitant rate of interest from him, go to the police academy, fall on people’s feet (special one again), convince them that we won’t let them down, and sign up as a constable whose only job is to fetch tea for his bosses, while all the time hoping the government’s pension funds are still burgeoning. As for the man who is convinced that he is not vain and does not carry a single ounce of vanity, praise him on that and he will give you that loan you want to tint the windows of your ill-purchased car.

Vanity is everywhere. It is in the air, it is in the waters, it is in the sky. It is in the non-believers, it is in the zealous, it is in the dying, it is in the just-born. Whole governments stand on it, and the very same governments go to war if that vanity seems to have been mocked at. The law may have narrowly slipped through its cluthces, but lawyers and judges have not. I was hoping that the jury pools were still pure, but that thought went to hell if Boston Legal’s anything to go by. The only thing that is not vain is vanity itself. Moments after it has taken a firm grip over a young man’s mind, it will be disappointed. Humans don’t deserve to be vain, but since vanity is everywhere, we know not definitely of a world otherwise. And if my opinion counts for anything, I don’t want to be in a world otherwise!