Tuesday, 16 October 2018

Squaring the Circle: The Sufi Way

A friend who was a mathematician once remarked, in relation to my Sufi studies, that I was "trying to square the circle". What this is in his field of work is to construct a square equal in area to a given circle, a problem that you can't solve using geometry alone, though what he meant, of course, in layman's terms was that I was trying to do something that is considered to be impossible, or even insane. The Sufis' use of the octagonal symbol may, in one sense, represent an approximation, or half-way house, for Squarelanders who need to understand circles – the Sufic materials, according to the writer, thinker and Sufi teacher Idries Shah, being half way between "mere literature", shall we say, and active Teaching.

In the case of the constant, pi – which is the numerical value of the ratio of the circumference of a circle to its diameter – we simply cannot calculate the exact value of this irrational number, because even if we calculated pi to a million digits (and this has been done, indeed some can even recite the first hundred or so from memory), the answer will still, and always, be no more than an approximation of pi. "It's turtles all the way down," as someone once remarked.

There are ways around such difficulties, however, and we need not despair. pi expressed as the fraction 22/7 will be sufficiently accurate for many everyday uses, while those with more precise needs might use 3.142 or 3.14159. These values will be perfectly adequate "for most practical intents and purposes".

In a similar way, even something as simple at first glance as calculating the square root of a number like 2 is not a trivial task, since it is also an irrational number, 1.4142135623730950488016887242097... (ad infinitum), but again we can satisfactorily make use of an approximation like 1.4142.

In a sense we could say that the task of attaining Sufihood, or of coming to comprehend the answer to "life, the universe and everything", is a similarly boundless and irrational task.

Thankfully, there's another way in which we can tackle such difficulties and it is known as "successive approximation". Without going into the details, suffice it to say that, given an equation which we can treat as a "black box" – not necessarily knowing how it works but content in the knowledge that it does work – we can quickly arrive at an approximation of our desired answer.

In the case of calculating a square root, even that of a complex number like 4.56789, we initially make an educated guess (here, for example, that the number is going to be greater than 2 and less than 4). So we plug that guestimate into the equation and out pops an answer. Now, this answer will not be the precise answer, but it will be a little more accurate, most likely, than the initial guess with which we seeded the process.

What we do now is plug this new value into the equation, and we repeat the recursive process, and again out will pop an answer that is still more accurate. All we have to do is repeat or reiterate this process perhaps five times (or more, depending on the exact nature of our needs), and we will quickly arrive at a good-enough approximation of the square root (let's say 2.137), and we don't even need to resort to a calculator or a computer to arrive at this answer.

One thing to take into consideration (for example in calculating the square root of a complex number), is that the process of successive approximation will not just generate "real numbers" (like 1.234), but also "imaginary numbers" (such as -2.345), and the process needs to be able to set aside the imaginary, and utilize the real.

What we're talking about here is the mathematical use of successive approximation, but this process of gradual honing is basically a negative feedback process, and it applies to a great many other things, such as steering a boat down the centre of a winding river, avoiding the banks to either side, by making mid-course corrections, robots that follow white lines, and automatic gain (volume) controls on amplification systems. In the same way, it also applies to many things that "home in" on a changing, moving target, such as a bird of prey or a ballistic missile.

And, most of all for our purposes here – though there are other more rigorous approaches possible (utilizing powerful "flux linkage" and magnetic attraction, for example) – this gradual, often heuristic, intuitive orientation, induction, assimilation, and polishing process applies just as much to our progress along the Way, with more than a little help from our friends, who have successfully walked this Path before and returned from a higher sphere, bringing with them a precious secret formula of knowledge and love.

Meanwhile, as the physicists and mathematicians pore over the complex calculations, through experience, "touch" and serendipity more than anything, the batsman or woman simply hits the ball for six, right over the boundary, and lovers will simply share a kiss. What was once thought impossible has become second nature for such people.

As for me, I'm still at 22/7, and this is still very much a work-in-progress.

By the way, did I mention that pi is what is known in the trade as a "transcendental number"?

~ Image 1: "Mandel zoom 11 satellite double spiral".
~ Image author: Wolfgang Beyer.
~ Licence: Creative Commons Attribution-ShareAlike 3.0 Unported (and alternatives).
~ Source: Wikimedia Commons.

~ Image 2: "Projection stéréographique d'une sphère sur le plan équatorial" ("Stereographic projection of a sphere on the equatorial plane").
~ Image author: Sylvie Martin.
~ Licence: Creative Commons Attribution-ShareAlike 3.0 Unported (and alternatives).
~ Source: Wikimedia Commons.

Text: Copyright (c) 2018, Eric Twose. Licence for re-use: Creative Commons Attribution 4.0 International (CC BY 4.0).


  1. interesting analogy ....Mike Lease.

  2. Thanks, Mike. It's a start, at least. The main thing (eg across at the Caravanserai) is to encourage people to open up.