Tuesday, October 11, 2016

3.14159… The Transcendental Periphery

3.14159…  The Transcendental Periphery

In our time, the term ‘awesome’ is over-used and trivialised.  It has become at its crudest a mere term of approval.  Much of the reality in which we generally agree we are living remains, however, truly awesome.  Sheer physical scale can overwhelm us.  On Earth we have the Grand Canyon, the Himalayas and the Aurora Borealis; we have earthquakes, tsunamis and hurricanes – sights or events whose magnitude reduces us to a sense of insignificance.  Look beyond our world, to the massive star which gives us energy and enables life to exist, yet which would incinerate any form of life which so much as approaches it and will eventually do the same to much of our solar system.  Look at the vastness of the distance between that star and its nearest neighbour, and then at the uncountable multiplicity of stars beyond.  That sense of insignificance blends with sheer incomprehension.

And yet ‘awesome’ can be found within us.  The human brain with its approximately one hundred trillion synapse connections.  The fact that our fragile bodies are constructed from atoms which, if split, could unleash the power of nuclear weapons.  The complexity of coding in our DNA which patterns us - and all living things – to be what we are.

In fact, ‘awesome’ lurks in every direction.  Including mathematics.  Back in the 1980s I was introduced to a book by Douglas R. Hofstadter titled ‘Godel, Escher, Bach’, in which the author looked at the paradoxes explored by those three and others in what he called ‘strange loops’.  These he found in the self-referential.  Think of the words: ‘this statement is false’ – which can neither be true nor untrue.  Hofstadter, in his 725 page ‘mental gymnasium’ (as the New Statesman reviewer described it) traces these loops through Godel’s ‘Incompleteness Theorem’, Escher’s impossible graphics and Bach’s extraordinary canons.  It was a book that enhanced my sense of the conceptual wonder to be found in the mathematical frameworks that appear (albeit incompletely?) to underlie all of existence.

Pi, it seems to me, stands as an epitome.  It is the ratio of the length of a circle’s circumference to that of its diameter.  That part is reasonably comprehensible, even to a non-mathematician such as myself.  It doesn’t matter what size the circle may be, the ratio is constant.  The name ‘Pi’, after the Greek letter, was coined by an English mathematician, William Jones, in 1706 and was probably meant to stand for the word ‘periphery’.  Pi is often inaccurately represented as 22 over 7, or 3 and one seventh, but when attempts are made to capture it with greater precision, the wonder begins…

The figure 3.14159, quoted in my heading, is more accurate, but in truth the decimal points continue, ever diminishing, and seemingly have no end.  Computing has vastly improved our ability to calculate it.  The record in 2015 ran to 13.3 trillion digits, and still no finality in sight.  This apparent infinitude results in Pi’s identification as an ‘irrational number’.  But the irrationality also extends to the nature of these digits, which never once repeat periodically.  Though they cannot be described as truly random, say the mathematicians, there are no appreciable patterns or sequences of repetition as the digits unfold.

Pi is also described as ‘transcendental’, which – in a mathematical sense – means that it exists but cannot be expressed in any finite series of either arithmetical or algebraic operations.  So, if you try to express Pi as the solution to an equation, the equation – like the decimal – goes on forever.  It transcends the power of algebra to display it in its totality.  That is awesome.

But there’s more.  There’s a mystery, in fact.  Pi doesn’t in any way actually need to have these apparently infinite extensions.  It has been calculated that an expansion of Pi to a mere 47 decimal places would be sufficiently precise to inscribe a circle around the visible universe that doesn’t deviate from perfect circularity by more than the width of a single proton (ie: less than an atom).  Now remember: it has been calculated to 13.3 trillion decimal places!  So why?  Why is this seemingly simple ratio of such an apparently infinite nature at all?  Perhaps if we ever understand that, we’ll understand a lot of things a lot better.

Because Pi is everywhere.  Any natural circle you see, for example.  The discs (to us) of the moon and sun.  The head of a sunflower.  The pupils of our eyes.  In ripples, waves and spectra.  The rainbow.  Anything with circularity involved: the double helix of DNA for example.  Weather patterns, hurricanes, whirlwinds…Mathematicians calculate formulae for all sorts of natural phenomena, and so very often Pi is part of the notation – but at this point I begin struggling to grasp, and have to take things on faith.

No wonder the circle is the core motif of mandalas.  If infinity is bound into its make-up, what better shape to contemplate in states of meditation?  The Sanskrit word ‘mandala’ itself can be interpreted as meaning ‘circle’.  So once again we find, although we must always be wary of jumping to conclusions, that our ancient philosophies seem to touch on concepts which we think we are only now discovering.  Yes: here we go again – re-inventing the wheel.

It is said that the universe itself is circular in its nature.  I cannot claim to understand this idea with my intellect, but it appeals to me greatly.  If you were somehow able to travel far enough so that you crossed the entire universe, you would end up back at the place where you began.  What a trip!

And perhaps still those decimal points would be continuing, trillions and trillions more of them.  And those formulae that never end, because they can never quite get to the bottom of Pi.

Think about it.



Sand mandala of Chenrezig, Buddha of Compassion, created by monks of  Tashi Lhunpo monastery on a visit to Shaftesbury, summer 2016.  Photo by RF.

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