The Discreteness of Infinity (a homage to the Swiss mathematician Euler 1707-1783)
© Wayne Anton Roberts 1999,
all rights reserved; watercolour on arches 640g, 74cm x 102cm, Collection B J Beven.
Commentary below image. This painting appears in the new book
The painting concerns a remarkable equation discovered by the Swiss mathematical genius Leonhard Euler (1707-1783). In it, Euler finds an unexpected connection between pi and the positive counting numbers (integers) from 1 to infinity.
It was unexpected because, as most people know (and knew in the 1700's), pi is defined not in terms of numbers at all, but in terms of circles: a circle's circumference divided by its diameter, and so what in the world should circles and their intrinsic proportions (pi) have to do with numbers (integers), in fact all the integers from one to infinity? (More about this in a moment)
An ancient precedence of connectedness
The Pythagoreans, two and a half thousand years ago, had also discovered an amazing connection between numbers and shapes (geometry). For example, not only did they discover the famous right-triangle relation of adjacent sides to their hypotenuse (known as Pythagoras' Theorem), but that there also existed a connection of these special triangles to the whole numbers. In fact they discovered a method of finding all the triples-of-whole-numbers which could form the sides of such right triangles! These became known as Pythagorean triads or triples. The simplest and most famous of these 'whole-number right-angle triangles' is the 3-4-5 triangle, since 32 + 42 = 52. (This may not at first seem remarkable until you realise that there are infinitely many more right triangles that do not have all sides in whole-number length or proportion.)
The Pythagoreans soon also discovered that whole numbers and their ratios seemed mysteriously connected to 'pleasing intervals of sound' from a vibrating string, in that if you divided the string in the ratio of ½, it sounded exactly like the original note except higher: today we simply call it an 'octave higher'. But our labelling of such mysteries of perception as 'the octave' does an injustice to the wonder of the perceptive fact. Certainly, we now understand more about what constitutes an 'octave' in music, in terms of the ratios of the frequencies of sound pressure waves, but why these particular ratios are recognised by the human mind as the 'same' note, except higher, and why other combinations of frequencies sound harmonious or discordant remains an enigma of human perception and perhaps more fundamentally, of the Universe itself.
There were other ratios, such as 3/4, 2/3, again ratios of small whole numbers, which also seemed to be of fundamental importance to sounds which could be combined variously to produce pleasing resonances, appealing to the human ear, and these 'chords' of related component notes or pitches of sound became enigmatically known as 'harmonious'. The kind of grammatical or logical interconnection of these unexpected discoveries led Pythagoras and his followers to infer that 'All is number'. It was a tremendous leap of insight to make such an inference on the basis of their few discoveries, but two and a half thousand years of history has tended to support rather than diminish the spirit of that vision.
1,2,3,... a,b,c,...a language emerges from counting
Counting and numeration began very humbly, from the simple need to keep account of your sheep. Piles of stones or pebbles were used as a one-to-one parity method of reckoning (any remaining pebbles left over from a pile of them which stood in one-to-one relation with the original flock would indicate a marauding wolf had most likely enjoyed lamb for dinner that night). These simple accounting practices of the shepherds of millennia long since passed are reflected to this day in words such as calculus (L. calculare, meaning pebble, stone).
From these unlikely beginnings, from the simple practical need to keep account of your sheep, sprang a most poetic understanding of our planet and universe which we now know as mathematics. So astonishing is this abstract world of number and shape, of 'measure and relation', that through its elucidation and application, we have been able to explore new horizons: to fly, to circumnavigate the Earth, to walk on the moon, to peer into deep space, to listen to the stars.
A curious question therefore arises from these simple facts. If the language
of number was merely an expedient (albeit clever) invention of ours to facilitate
human affairs ('count sheep'), why should this mathematical language be one
which the Universe apparently not only understands, but itself speaks? Why should
such connections exist between and among numbers and geometries, numbers and harmonies,
and why should such connections, once understood, enable us to formulate theories
and poems of our understanding in the form of equations which allow us
to in turn predict motions and events not only on Earth but at astronomical distances
in space and time?
Euler's equation: connecting circles (pi) and numbers
The particular equation of Euler's as depicted in the painting The Discreteness of Infinity (above) found a connection not between right triangles and numbers, but between circles and numbers.
On the left-hand side of the equation is a shorthand way of writing out an infinite addition: 1/(12) + 1/(22) + 1/(32) + 1/(42) +....1 divided-by-every-successive-whole-number squared ... all the way to 1-divided-by-(infinity squared) ...a very small fraction indeed!! [i.e. 1/1 +1/4 + 1/9 + 1/16 + 1/25 + ...] This infinite addition of successively smaller fractions has a finite sum which Euler discovered by ingenious means is equal to pi-squared divided by 6. Now, by simply rearranging that equation , pi may be re-defined, not in terms of circles, but of numbers. Pi is: the square root of six times the sum of the reciprocals of the squares.
Euler's equation is like a mathematical yin-yang sign. It finds a perfect balance between a 'digital' 'quantised' world on the left (represented by the integers) and an 'undivided continuum' on the right (represented by pi and the circle's seamless curve). If we consider the essence or spirit of the integers on the one hand, and of circles on the other, we find remarkable resonances. Resonances that hint at a higher principle that may underpin the diverse but interconnected edifice of science.
... on the one hand, the integers ...
What is the character or nature of the whole-numbers? (represented on the left-hand side of the equation in the denominators of the fractions which are added together). What do they remind us of? What is their essential nature? Their essential nature is very much tied up with the idea of the unit. That's because we get from one number to the next by jumping ahead one more unit at a time. But what exactly is a unit? It's a very abstract concept and one that is 'fundamental' within the Universe. That probably sounds a bit circular, but then that's because circles are very much like units anyway (as Euler's equation reminds us!). This idea of the unit runs very deep in history, science, linguistics, maths, music, politics, society, and in virtually every discipline. The world is a 'unit-ary' kind of place, and it's made up of billions of smaller units, and units of innumerable in-between sizes. This idea resonates for example in the Classical Greek 'atomic' idea, and later, in our own time, in the 'quantum' idea, and in the discovery of 'photons' (particles of light): all are units of various kinds. The fractal idea of Mandelbrot also echoes these concerns in affirming that in Nature, big units are like smaller units: shape-units recapitulate across many scales. So the Universe is definitely an itty-bitty-gritty kind of universe. We can't squish stuff all the way down to nothingness, because we come up against this grit, these little 'units'. Even spacetime itself has a kind of quantised grid-like nature. But this is only half the story, because the 'opposite' is also true: the Universe is smooth.
...and on the other hand, circles, and the undivided whole ...
The Universe is smooth and seamless like the circle. Now we begin to contemplate the right-hand side of Euler's equation, and circles come to mind straight away, because pi is sitting up there in pride of place on the top of that fraction. From an intuitive-feeling point of view, circles seem to have more to do with 'waves', 'seamlessness', the 'continuum', than with discrete chunks or jumps from one number to the next. It seems to be more 'analog', than 'digital'. Again, there are parallels, resonances in the history of science, in the nature of our Universe, which have been enigmatically discovered by Euler and encapsulated in this little equation: for instance, light behaves also like smoothly changing waves and not just particles! Yet light is both, revealing itself as one or the other depending on how you look at it (like the alternate and equal sides of our equation). Music too can be recorded in 'smooth' ways, and in 'itty-bitty-gritty-or-dotty' ways. Vinyl records contain a continuous groove with small smooth undulations (waves) that make the needle of the turntable vibrate and which is 'translated' into sound. On the other hand CDs record the music as millions of bits of information (ones and zeros) which are later translated back into sound (digital recording). Circles very much give you a feeling of not just the hole, but of 'the whole'. Units too are a kind of 'whole', but when you contemplate the circle, it's of a very special kind: this particular whole emphasises 'inclusiveness', 'connection', 'completeness', 'all-at-once-ness', 'all things returning again'. Gazing at the circle draws you to the idea of 'centredness' and origins. If you think these sorts of connections I'm drawing sound a bit bizarre, that's nothing compared to the 'quantum-weirdness' scientists have been discovering within the subatomic world (of the extremely small) that undergirds our entire Universe including how we think, what and who we are! Here particles flash into and out of existence from nothingness, but even more bizarrely, there seems to be a 'circularity' operating at this level in that there are what is called, 'non-local' effects. These are basically 'connections' that 'link' things like photons which are 'supposed to be separated' and separated by arbitrarily large distances (and, it's of the instantaneous-kind-of-connection, I mean connected-connected; faster than light-speed) in ways we don't yet fully understand, or even half understand. The circle, the loop, closes back on itself. The discreteness of the Universe yields again to utter connectedness. Separation is an illusion of the lower dimensions. It's a sobering thought - this one-ness thing - this whole of which we speak, and of which we are part.
Euler's equation is thus like a resonance or 'harmonic' in music in that it reverberates beyond itself, and the 'note it sings' echoes in different parts, places and times. It is 'connected' beyond itself. It encompasses the outer reaches of the Universe. It's like a lead-violinist of an orchestra who before the performance of a symphony, plays 'A', and is soon joined by a buzzing and droning of other instruments all adjusting and approaching through their tuning and resonance-matching that same 'A'. Euler's equation is a kind of 'sweet-spot'. It hits upon an important 'something' that the rest of the Universe seems to know or 'recognise' and resound to.
Interpreting the equation through the genre of landscape
In the painting, I wanted to reflect or echo these thoughts that Euler's small but powerful equation seemed to conjure up. For instance, the opposite yet complementary relationship between numbers and circles or geometries I tried to echo in a landscape sense. From a landscape point of view, sea and mountains are in one way 'opposites': one is static, the other dynamic and ever changing. Yet they have similarities too (connections). To look at from afar, they are very similar rhythmically. So too, the fractal pattern of foam on the sea is quite like the pattern of snow-drifts in the clefts of mountains. Thus things which appear to be opposites from one point of view, can be understood to be also connected ('equivalent', or one) from another equally valid point of view. The mountains (intuitively painted from the imagination) are obvious to any viewer of the painting. Perhaps less obvious is the sea (hinted at far right-middle of painting, and also lower middle of painting) I wanted the feel to be somewhat symphonic, of sea becoming mountains, and of the other way around, of sea and mountains 'precipitating out' of a togetherness into separate entities.
There is a lot of diversity on planet Earth. A lot of things that seem to be unrelated to other things, a lot of separation, a lot of division, millions of parts, trillions of units. Yet, when, in the twentieth century, humankind was able to (vicariously) leave their celestial place of abode for the very first time, the vision of wholeness and unity, of circular connection mirrored back from space was a stunning fulfilment of the spirit of Euler's little equation. Here was seen not a conglomerate knobbly asteroid-like monstrosity, but a sublime incarnation of the language of geometry and number that laces the human quest to understand ourselves and our Universe from the earliest of times: the soaring arc of Earth's circle gloved in vaporous blue.
Text & images © copyright 1999-2002 Wayne Anton Roberts. All rights reserved. Collection B J Beven