{"id":1434,"date":"2021-01-04T17:12:31","date_gmt":"2021-01-04T17:12:31","guid":{"rendered":"https:\/\/silverpi.com\/blog\/?p=1434"},"modified":"2021-07-27T11:52:33","modified_gmt":"2021-07-27T11:52:33","slug":"the-chronicle-of-pi","status":"publish","type":"post","link":"https:\/\/silverpi.com\/blog\/index.php\/2021\/01\/04\/the-chronicle-of-pi\/","title":{"rendered":"The chronicle of Pi"},"content":{"rendered":"\n

This humbling aspect of Pi is also a perfect metaphor for Learning \u2013 as a process of smoothing the corners and \u00a0resulting in well-rounded individuals . It tells us that Learning is a continuous and never-ending journey, towards that unachievable perfection…. Thus Pi represents Learning \u2013 and in many ways, Pi represents, shall we say, Life itself!<\/span><\/em><\/p><\/blockquote>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Pi is arguably the most famous number in the world of Math. In Math, alphabets (English or Greek) are used to represent (yet) unknown numbers. There are typical letters used to represent specific types of unknowns (theta<\/em> for angles, delta<\/em> for small numbers, epsilon<\/em> for very small numbers etc.). But Pi is among the very few that almost always represent just one number (called Pi). <\/p>\n\n\n\n

We often assign Pi the values like 22\/7, 3.14, 3.14159 etc.  based on the exact application use case. But by definition, we do not know (and we cannot<\/em> know) the precise<\/em> value of Pi. <\/p>\n\n\n\n

The simplest way to look at Pi is as the ratio of the\ncircumference of a circle to its diameter. But that sounds simple \u2013 then why\nthis mystery? <\/p>\n\n\n\n

The mystery actually comes from the fact that we cannot<\/em>\nhave<\/em> (and hence measure) a perfect circle! A perfect circle is an Utopian\nidea<\/em> and what we create are just approximations of that idea. This now sounds\nmysterious and quite profound. Let us examine this further. <\/p>\n\n\n\n

In dealing with curves like circles, a technique usually employed by Mathematicians is to \u2018approximate\u2019 the curve with a number of small straight lines. Thus a circle can be approximated by a polygon (square, hexagon etc.). So let us start with a square (picture). Then the circumference of our circle would correspond to the perimeter of the square and the diameter would be its diagonal. Now if you take the ratio of the perimeter to the diagonal, you would get the value to be 2.828. This is our first approximation for Pi \u2013 which is far from the value we know.<\/p>\n\n\n\n

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Geometric estimation of the value of Pi<\/figcaption><\/figure><\/div>\n\n\n\n

Now consider a hexagon (6 sides). In this case, the value of the ratio of the longest diagonal to the perimeter is 3.0. OK, now we have got something closer to our value of Pi, but still not good enough for any practical purposes. <\/p>\n\n\n\n

Increasing the number of sides of the polygon to 8, 16, 24\netc., you can see that the ratio gets closer to our value of Pi. The table\nbelow summarizes this. You can also see that with 24 sides, the picture is visually<\/em>\npretty close to a circle. But to get a 5-digit accuracy, you have to go up to\nabout 1000 sides! <\/p>\n\n\n\n

But no matter how many sides you take for the polygon \u2013 you will never<\/em> reach the perfect circle. <\/p>\n\n\n\n

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Approximation of Pi with increasing number of sides of the Polygon<\/figcaption><\/figure><\/div>\n\n\n\n

So what does this say about Pi? It is a number with unending\ndecimal places, whose exact value can never be determined. The closer your\napproximation gets to the ideal circle, you have a more precise value. <\/p>\n\n\n\n

In addition to this classical (geometric) approach, there is also a modern (analytical) approach developed to estimate the value of Pi. This consists of evaluating the circumference of a circle for a given diameter by applying finite series approximations to a slowly converging infinite series. This method was first used by Madhava (1350 AD) and his disciples of the famous Kerala School of Mathematics <\/em>of the 14th<\/sup> to 16th<\/sup> centuries (Ref. below).  For the learners, looking at Pi as the sum of an infinite series  gives brilliant and direct insights to understanding the nature of this number \u2013 its infinite<\/em> and unknowable<\/em> nature. <\/p>\n\n\n\n

In terms of applications, Pi broadly shows up in scenarios where a circle <\/em>or a part of a circle<\/em> is involved. This includes circular and spherical shapes, circular motions, waves, oscillatory motions, spiral motions, spiral shapes, cyclical changes etc. This makes it useful in a variety of practical scenarios.  <\/p>\n\n\n\n

The wide variety of application areas include AC Electric Power, Electronic communication, Rocket Science, DNA structures, shapes of River Basins, understanding of earthquakes etc. <\/p>\n\n\n\n

Pi also appears in some scenarios where there is no obvious<\/em> circular pattern involved \u2013 for example, in the distribution of Prime Numbers in the number system <\/em>But some think that this indicates a deeper underlying aspect that we have not yet <\/em>understood!. (Note: this should not be confused with the pi-function,<\/em> which defines the number of prime numbers less than a given integer). <\/p>\n\n\n\n

Pi reminds us that as we improve our tools and models, we get closer to the perfection of an idea (perfect circle) – but we could never reach that. This humbling<\/em> aspect of Pi is also a perfect metaphor for learning<\/em> \u2013 as a process of smoothing<\/em> the corners and  resulting in well-rounded<\/em> individuals. It tells us that Learning is a continuous and never-ending journey towards that unachievable<\/em> perfection. <\/p>\n\n\n\n

Pi represents Learning \u2013 and in many ways, Pi represents, shall we say, Life itself!<\/p>\n\n\n\n

Bibliography
<\/strong>\u201cChapter 10.\u201d The Crest of the Peacock: Non-European Roots of Mathematics<\/em>, by George Gheverghese Joseph, Princeton University Press, 2011. <\/p>\n","protected":false},"excerpt":{"rendered":"

This humbling aspect of Pi is also a perfect metaphor for Learning \u2013 as a process of smoothing the corners and \u00a0resulting in well-rounded individuals . It tells us that Learning is a continuous and never-ending journey, towards that unachievable perfection…. Thus Pi represents Learning \u2013 and in many ways, Pi represents, shall we say,…<\/p>\n

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