Elegantly Simple! — Quotidian — 416!

(Transcript of video originally posted on 4 Sep 2022)

So, in Mathematics, you would have approached a lot of problems algebraically. When you enter higher classes, the teachers would have said, “There is a geometric solution to this, which is a lot more elegant!” Multiple pathways, leading to the same answer… Don’t we have it in religion too? We think that there is but One God.. One Religion. But, when we encounter other religions, and the beauties that exist in them, that is where we realise the Hows can be many! This is what we are going to talk a bit more about. They call it “Elegant Solutions”. Beautiful. Beautiful in its Simplicity. Unexpected Simplicity, and Elegance. That’s what we are going to see! Let’s look at a couple of examples!

Namaste! Elegantly Simple! Should be beautiful. Should be simple. Should be elegant. There are many examples, in nature, in the work of mankind, smart, clever, things. We are going to review a few.

Pythagoras Theorem — I am sure every one of you remember this one! Two sides. They could be of any length. But, A Squared added to B Squared will always be equal to C Squared. Said Pythagoras, 2500 years ago! Till date, there have been more than 350 different kinds of proofs for this popular theorem. Among these unique and different approaches, there is one that I think is most elegant. This one comes from Mr. Pythagoras himself! Extremely simple. Just using four pictures. No higher-level maths! Look at this unfold!

So, A Squared added to B Squared.. Is it always equal to C Squared? For arbitrary values for A and B? That’s what we are going to see.

Take any random triangle, with random side lengths a, b, and c. (Oh, wait. Not any random c. But a particular c, that makes it a right triangle). Repeat it and make a set of four exactly look-alike triangles.

Now, move those four around to rearrange and form a square like this. As you set this up, you will notice an inner-tilted-square forming, with side length c. So, the area of the inner-square would be c squared. Everything is clear till now!

Now, remember the colors of these triangles.. Use the same triangles. But, let us take them on a further excursion. And, reconstitute them in the form of TWO RECTANGLES. I am not changing anything else! If you notice carefully, we now have two rectangles with side lengths a and b.

Okay! Now comes the clincher! As you can see on the left, the empty space inside the four triangles is c squared. And on the right, the empty space has now become two spaces and its areas are respectively b squared and a squared (as the empty spaces are squares of side lengths a and b respectively!)

From this, we realise that irrespective of whatever a or b values you begin with, c squared will always be equal to a squared plus b squared. Proved! Q E D! Whatever was to be proved, has been proved! Pythagoras himself has given this proof.

In Nature too, we can see such elegance presenting itself often! So, assume there are four cities arranged in a square form like this. And, the Municipality wants to connect these places up for travel. Or, so goes the story. And, they invite suggestions.. on how to lay the roads. People come up with suggestions. One of them comes up with … a criss-cross pattern. (Length 4 + 2√2)

But, .. the authorities complain that there are multiple paths between the same cities, and they are trying to minimise road-length so as to keep costs low. Another proposes a grand circle. From wherever to wherever, a circular path. Outer Ring Road! (Length π√2).

But the authorities feel it can be shortened further.

A straight line approach? (Length 4).

Or even a shorter-straight-line approach (Length 3).

This is good, but to go from A to B, is a circuitous hell through D and C… Is there a better way?

Another person came up with an X. (Length 2√2)

Is there a better way? Yes.. Another person came up with an I pattern. But, the road length is still 3..

Is there a superminimum? There is a field of mathematics called Calculus of Variations. A complex mathematical concept — and after deep analysis, one can prove that THIS IS the minimal distance possible. You may not believe the final answer when I show it to you.. You may not even be convinced that this is the shortest! But, take a look!

This is the minimal distance structure. The angles at the center are exactly 120 degrees. There are two convergence points. And these convergence points are called Steiner points. But, how do we even arrive at something like this?? We need a mathematician for this. For each city. For each highway department! Perhaps, only then! Or, .. do you have soap in your bathroom? See what this guy does! He is a mathematician. Look how he demonstrates the answer for that roadways problem.

How?! How did soap know? Did it hurriedly get a Ph.D in Higher Maths just before dissolving in the water? How does Nature know, to minimise it? When we need a maths Ph.D to derive those!? But, at least these, you can say it is soap, man-made solutions, etc, but here is an even more interesting demonstration of elegance for you!

Here, we have a river flowing — top of screen to bottom of screen. And, near the river, there are two little towns. One a little farther from the river, and the other a little closer. They want to set up a water distribution station somewhere in the bank of the river, so that these straight pipelines can serve water to these two towns, while keeping pipe-laying cost to a minimum. They think about it! Should they do it like.. This…

Or, should they do it like That…

Where should they keep the station so that the pipe-length total is a minimum.

Or, should they try this configuration…

You know what the answer is? The elegant, simple answer? Start by imagining a mirror-image of the closer-town on to the other side of the river. Lay a straight-pipe-line imaginarily from one town to the other, passing through the river. Look for the point where this imaginary pipeline crosses the river! THAT is your optimum point for locating the water distribution station. Start laying the pipeline from that point towards both towns! Hey! Light beam? Angle of incidence? Angle of reflection? Hey! How does light know to minimise pipe length?! Isn’t that simply amazing?!

VanMoof is a Dutch cycling company. VanMoof exported its cycles to America too. Their cycles were world-class, after all! But, the problem was, it started getting delivered with breakages. They tried larger boxes, better packaging, they even wondered if they should switch delivery agencies… They tried literally everything under the sun! But, nothing seemed to work. Great cycles, but when delivered from Europe to the Americas, the customers were unhappy and the company didn’t know what to do.

They brainstormed for ideas.. And, one of them quipped, “If only this were a smart-TV, do you know how carefully they would be handled?!” They pounced on that single thought, and said in unison, “That is what we are going to do!” From that day onwards, VanMoof cycles started being delivered in a box like this! It features a very large television inside which there is a cycle! So, the delivery agents looked at the “Fragile / Careful” notice, saw the image of the TV, and immediately, subconsciously, became a lot more caring and attentive.. They started delivering the cycle with as much respect as they used to deliver a Smart TV. VanMoof even included the word “Smart” on the packaging! Ride the Future! Smart! Electrified! How clever! Zero extra cost almost. For nothing changed in the design or the composition of the box!

And, finally, we can’t be talking about elegant, simple, out of the box, without referring to Steve Jobs! This was the first iPod. It wouldn’t even have fit into a pocket! And, very heavy too! When Steve Jobs was shown this first prototype, he was livid! “No way, make it smaller!” was his response. The engineers responded with, “Sorry, we can’t go back to the drawing board. It is nearly impossible. Everything has to be redesigned. I don’t think it can be made any smaller”.. While the engineers were walking with him in the corridor, Steve Jobs paused, turned back, went near an aquarium, and dropped the prototype into the water! Plop!

It settled in the bottom, and a moment later, there was a gurgling bubble rising to the top. “That’s it”, Steve said, “there’s space there. If bubbles are coming out, there must be empty space inside. So, make it smaller!” And, only because of this relentless pressure that Steve applied on his team, advice, tension, emergency, panic, pressure, … a couple of months later, Steve was able to do this. Ten thousand songs in your pocket! He was able to launch that magical device!

As a closing thought, we have been discussing elegant simplicity. We have read before — Simplicity is the Ultimate Sophistication. Albert Einstein himself (or at least some people say he did!) has said once:- “When the solution is simple, God is answering!” God herself is talking. It is not you. Not me. Man can’t do it by himself. There is something divine involved. Something far beyond human comprehension. What humility! Alright, we will meet next week! Thank you!

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Rajendran Dandapani

Rajendran Dandapani

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Business Solutions Evangelist at Zoho Corp. President at The Zoho Schools Of Learning.