By Gabriella Edelstein and Claudette Palomares
The fractal beauty of Arcadia by Claudette Palomares
Art versus Science versus Life: Recursion in Arcadia by Gabriella Edelstein
by Gabriella Edelstein and Claudette Palomares
In a schoolroom on a large country estate in Derbyshire in 1809, Septimus Hodge, the family tutor, must deal with the furore that his erstwhile school friend—an unseen Lord Byron—has wrought upon the household. Meanwhile, his pupil—13 year old mathematical prodigy, Thomasina Coverly—forms precocious notions about rice pudding and bluebells that seem to lie on the precipice between madness and genius. Fast forward to 1993 and two warring historians, the Caroline Lamb-apologist, Hannah Jarvis, and Byron defender, Bernard Nightingale, try to unearth the mysteries of the historical events seen prior, including the possibility of a duel and murder by the hand of the poet so bad, mad, and dangerous to know. The play moves between the two time periods until merging together in an ineffably moving denouement that involves two very different pairs from two very different worlds dancing to the same waltz of offbeat time.
Thomasina (Georgia Flood) and Septimus (Ryan Corr), Arcadia, Sydney Theatre Company, Image: Heidrun Löhr
The fractal beauty of Arcadia
By Claudette Palomares
One of the chief thrills of a Tom Stoppard play is what seems to be an infinite repository of ideas, masterfully interwoven through dialogue and metaphor. One of the most striking ideas in Arcadia, and indeed, in Stoppard’s entire oeuvre, is his exploration of the “iterated algorithm”.
But what is an iterated algorithm, you ask? An algorithm is a formula, a recipe in mathematics. Input a number for x and you get a result for y. But what happens when you take y and input it into the formula again as the new x, repeating this action a thousand times over? Slowly, a pattern emerges, and in some cases, something that can only be described as transcendently beautiful:
The Mandelbrot set (Wikipedia)
What you see here is an algorithm, iterated; inputted hundreds of thousands of times over. It is also known as a fractal, a never-ending pattern.
Like a ball that breaks a frosted glass window, revealing sunlight on the other side, the calculator allowed mathematicians to explore the possibility of an iterated algorithm. A single fractal might take a lifetime to compute by pencil and paper alone, but can be created in a matter of minutes, thanks to a single fingertip pressing on a calculator button a couple of thousand times over. Yet a fractal is not only a picturesque pattern. As scientists came to discover in the latter half of the twentieth century, nature itself is often fractal-like.
Take for example, a snowflake. If you zoomed into a snowflake crystal, you will see that the vortices—the plumes—of the ice crystal are likely composed of smaller formations of wider shape, and as you zoom closer and closer the same pattern emerges again and again, in smaller detail. The famous Koch snowflake is a beautiful example of such a self-replicating iterated algorithm.
The first seven iterations of the Koch snowflake in animation (Wikipedia)
The fractal plays a central part in the play because it is what Thomasina Coverly—the thirteen year old genius living in a country estate in the year 1809—is reaching towards, albeit prematurely, approximately one hundred years before its time. Frustrated by the limitations of classical geometry, Thomasina complains to Septimus that “Armed thus, God could only make a cabinet.” She asks him:
God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose?
Thomasina tries to model an apple leaf, using what she titles as a “New Geometry of Irregular Forms”. In the part of the play set in the present, Valentine Coverly (a scientist and Thomasina’s descendant) explains that Thomasina has, in effect, invented the iterated algorithm. And as she predicted, the iterated algorithm goes a long way in describing and modelling much of nature—not just of snowflakes, but of fern leaves and the surface of the earth, among others.
In Arcadia, the iterated algorithm is not merely an idea expressed in the plot of the play, but a brilliant metaphor for the recursions and patterned elements that form its narrative. Dialogue lines are repeated in new forms and characters in the present mimic characters in the past until they are all combined in a sublime paradox of symmetry and chaos. The characters in the 1809 section of Arcadia make familiar choices: they seek fame, connection, love, sex and knowledge. We see these efforts repeated again in the modern section: Hannah and Valentine seek knowledge, Bernard seeks fame and all the characters in their own way are seeking connection and love. This is the fractal that emerges from Arcadia’s beguiling narrative.
The iterated algorithm is also a beautiful metaphor for the ways in which history appears self-reflexive and patterned. An overview of recorded human history seems to be a repetition of old lessons, relearned and unlearned. History is a fractal where the same narrative is inputted over and over. And yet, I think most of us would say that the narrative of human history—which is as much a narrative of the pursuit of knowledge—is on the whole, quite beautiful.
—Claudette Palomares, March 2016
Bernard (Josh McConville) and Hannah (Andrea Demetriades), Arcadia, Sydney Theatre Company, Image: Heidrun Löhr
Science versus Art versus Nature: Recursion in Arcadia
By Gabriella Edelstein
At the climax of Arcadia, when Bernard attempts to raze all in his path in order to prove a theory about Lord Byron, he launches into a diatribe about the value of the arts over science. He believes that questions about mathematics and science are pointless in the face of art, “If knowledge isn’t self-knowledge it isn’t doing much, mate. Is the universe expanding? Is it contracting? … Leave me out. I can expand my universe without you.”
To him, what is more unfathomable and beautiful than the universe is Byron’s writing ‘She walks in beauty like the night’ “after coming home from a party”. Ultimately, Stoppard reveals that the art versus science debate constructs a false binary, seeing as art is just as much a part of nature as the science. His efforts to explore the ways that art and science work together trickle through Arcadia’s thematics as well as structure. The play is an exercise in how to be both, presenting the argument that art and science should be placed side by side in our attempts to understand the universe.
Thomasina’s fractals are Arcadia’s proof of the equality between the two disciplines. As Claudette adroitly explained, iterated algorithms provide a way to map out some of the shapes of the natural world. Iterated algorithms create an image of our world that is beguiling to the eye, and are just as present in nature as they are on geometric paper. Just as iterated algorithms predict, shapes in nature repeat themselves, creating the promontories of leaves, the curves of flowers, and the points of snowflakes. A fractal are a perfect example of the melding of art and science: it is an aesthetically pleasing image that is created by numbers which becomes the shape “for a curve like a bell… a bluebell… a rose” in application. It is not a matter of what came first, however, or what is more important – the numbers that predict the natural shapes or the development of the natural shapes themselves – but the fact that nature is both.
Stoppard uses the overlap between science and art as a means to structure Arcadia: the fractals of nature are reflected in his use of formal recursion, what is known as mise en abyme (literally, “placed in an abyss”). This is when an artwork is reproduced within itself in order to make obvious the illusionary nature of both art and life. The most famous example of this is Velázquez’s Las Meninas, which is a painting of Velázquez painting the Infanta Margaret Theresa whilst the audience, the King and Queen of Spain, are reflected back into the painting by a mirror.
Stoppard uses this sort of recursive manoeuvre in Arcadia’s double plot: two pairs of scholars trying to pin down evidence for two elusive theories. But as history moves forward, change is bound to occur, so the mirrors become somewhat distorted by the inevitable entropy. The talkative genius of Thomasina becomes the silent genius of Gus, and the gliding waltz between Thomasina and Septimus at the beginning of the nineteenth century becomes the rather stagnant one between Gus and Hannah at the end of the twentieth. Indeed, even the process of discovery between the two sets of characters mirror each other, which in turn are affected by chaotic interpersonal relationships that happen in the vein of history repeating.
By using iterated algorithms as a dramatic structure, Stoppard is able to reflect the universe in microcosm. Even on a metatheatrical level, Arcadia uses the artistic form of drama to hold within it the structures of science and mathematics. And by making a gesture towards the nature of his play as a play, Stoppard reminds the audience that what they’re witnessing is not history or fact or science, but art. This is a dramatic universe wherein the present is endlessly recurring in the pursuit of self-creation, which is, perhaps, the fundamental narrative of our lives.
—Gabriella Edelstein, March 2016
Nicolas Poussin – Et in Arcadia ego, 1637–1638, oil on canvas, 87 cm × 120 cm (34.25 in × 47.24 in), Musée du Louvre
“Oh, phooey to Death!”: Waltzing into the age of Chaos
By Gabriella Edelstein and Claudette Palomares
This is the way the world ends
This is the way the world ends
This is the way the world ends
When T. S. Eliot envisioned death as happening “Not with a bang but a whimper” , he could have been describing Guy Fawkes’ last breaths, the decline of Europe, or the eventual passing of the universe. In Arcadia, Thomasina realises that everything known will eventually peter out into nothingness, that the universe is fated to run out of heat and turn cold. Stoppard provides the audience with a pair of alternatives once they find that “we are all doomed”: we can either fall into nihilistic hopelessness, seeing ourselves “alone, on an empty shore”, or, as Thomasina insists, “we will dance” into the future instead.
Arcadia ends with the death of a character—as it turns out, its most incandescent— who, in a single line spoken offhandedly, is cruelly snuffed out. Yet the effect is not half as bleak as you would suppose. In fact—and this is one of the reasons why I seem to return to this play again and again—instead of the cynicism and nihilism that you might expect from such a worldly mind, Stoppard gives us hope in the face of a seemingly incomprehensible, existential future. We know that the world will end, but to use Valentine’s words, at this very moment, and moments a million-fold, it is largely trivial. It is only Thomasina who instinctively understands this, as she says: “Oh, phooey to death”.
I’m of the impression that the hopeful message at the heart of Arcadia – that life is meaningful despite certain death – is made manifest by Stoppard’s reliance on the second law of thermodynamics. Although Thomasina does not have the calculator necessary to work out the maths behind the theory, she realises that heat – and thereby time – can only go forwards. The second law has two components: firstly, that heat cannot pass from a cooler to a hotter body without additional assistance; and secondly, entropy increases in a closed system over time. Thomasina explains this by way of a bowl of rice pudding:
When you stir your rice pudding, Septimus, the spoonful of jam spreads itself round making red trails like the picture of a meteor in my astronomical atlas. But if you stir backward, the jam will not come together again. Indeed, the pudding does not notice and continues to turn pink just as before… You cannot stir things apart.
How does the law affect mechanical bodies? Firstly, energy is wasted when heat is transferred between bodies and furthermore, once energy is expended, the process is irreversible. This theory has enormous implications for the fate of our universe: if temperatures can only move from hot to cold and cannot naturally go from cold to hot, the stars will simply run out of energy and the great expanse of the heavens will die. Or as Valentine puts it, like a cup of tea, “we’re all going to end up at room temperature”. How can we cope with this knowledge?
Thomasina and Septimus literally dance in the face of an entropic future, while the pursuit of knowledge is yet another form of dance encouraged by Stoppard. In the middle of the play, Valentine has a crisis in confidence, rooted in the belief that his life’s work, his true inheritance (patterns in grouse population found in his family’s estate records) are inconsequential. He is comforted by Hannah, who tells him that everything is, after all,
…trivial—your grouse, my hermit, Bernard’s Byron. Comparing what we’re looking for misses the point. It’s wanting to know that makes us matter. Otherwise we’re going out the way we came in.
It is the “wanting to know” that sustains our lives in the face of finality. The pursuit of knowledge then is like a waltz: a ball breaks the frosted window, we are brought out the cave into the sunlight, we take a step forward in the progression of ideas. But chaos happens, the unpredictable predictably appears, we step back and we are drawn back to the darkened cave. As a result, rather than linearly, we move in circular, multivalent patterns, in indeterminate ways. The acquisition of knowledge moves in ¾ time.
Indeed, what can we do but pursue what is personally meaningful? Bernard needs his Byron, Valentine his grouse, and Hannah her hermit in order to shape the purpose of their existences. As a production of the late twentieth century, Arcadia is a reflection of existentialist concerns of free will and personal impetus. Now more than ever we are seeing our lives in terms of a light that will be switched off. The audience is perhaps meant to look at their own lives and ask how they stand up against the death of the universe. Have we come to terms with it? At the end of our day, can we proudly say that we followed a trail of individual meaning? Although much of the action of Arcadia performs like a comic interlude out of a play by Oscar Wilde, underlying it is an accusatory questioning of whether we are succeeding in our obligation to ourselves. The second law may dictate the destiny of the universe, but it doesn’t have to dictate the breadth of our lives.
Valentine speaks of the ways in which chaos theory broke through the ennui of the age, when relativity and the theory of everything seemed to indeed, encapsulate everything. As Valentine tells Hannah:
It makes me so happy. To be at the beginning again, knowing almost nothing… It’s the best possible time to be alive, when almost everything you thought you knew is wrong.
Through Stoppard we recognise that incomprehensibility is not a state to be avoided, but rather, it is the prevailing mode of our lives, as well as the necessary beginning to the pursuit of knowledge. Only when we begin to challenge our assumptions, as scientists do every day, do we move towards greater understanding.
Arcadia asks us to participate in the waltz of ideas, rather than stand by the sidelines. Some of us will be naturally proficient, like Thomasina and Septimus. Some of us will be stilted, as are Hannah and Gus, the mute and awkward boy she dances with. Nevertheless, in the glow of slowly dimming candlelight, each dance looks equally beautiful, and necessary.
What other option do we have knowing that life is futile? When we come to terms with our individual candles being snuffed out, as well as those enormous things beyond the boundaries of our understanding going cold, what then? We can only accept the chaos alongside the determined whimper of the universe, but we must ourselves be the bang that gives life meaning.
—Gabriella Edelstein & Claudette Palomares, March 2016