Et in Arcadia Ego

  Et in Arcadia Ego , by Nicolas Poussin

Et in Arcadia Ego, by Nicolas Poussin

Tom Stoppard's Arcadia: a play that alternates between 1809 and the present (well, 1993 present), begins with a mention of Fermat's Last Theorem (which had not yet been proved--Wiles finally got it a year later) and ends as a metaphor for the Second Law of Thermodynamics, and whose structure itself can be modeled (loosely) as a discrete dynamical system.  It skewers academia.  It is a postmodernist work that jabs at postmodernism.  There's sex, Romantic poetry, tortoises, waltzing.  So, yeah, lots to talk about.

Eric and I really geeked out on this one. The more you read it, the more you find, and the more interesting it becomes.  The story is actually not that complicated, but the structure of the play can make it seem that way.  Arcadia opens in the English countryside in 1809 at the home of the Earl and Lady Croom (we never meet the Earl).  The garden is being completely redesigned in the new Romantic style by a Mr. Noakes, who is using the only Improved Newcomen Steam engine in England to drain the pond. All the action in the play takes place in the drawing room of the home; the table in the center contains an assortment of objects that gets more cluttered as the play progresses.  The Croom daughter Thomasina is being tutored by one Septimus Hodge, a friend (acquaintance) of Lord Byron who is quite the Lothario, having seduced one of the house guests, Mrs. Chater.  Mr. Chater is a poet (we are led to believe) whose first major poem was skewered in the Picadilly Review by an anonymous reviewer (but guess who it is) and whose recent work, Couch of Eros, is being read by Septimus in the opening scene.  Thomasina is quite gifted at mathematics and Septimus has given her an assignment for the morning:  prove Fermat's Last Theorem.  Of course she cannot, but she begins doodling in her notebook by iterating a certain function (we don't know which).  This is an explicit reference to discrete dynamical systems, which were not at all understood (or even much thought about) then, and even if they had been there was not enough (any?) computing power available to run thousands of iterates.  Note that when Stoppard was writing the play, "chaos" and all the pretty pictures had seized the popular imagination thanks to the caffeine and nicotine-fueled work of Benoit B. Mandelbrot (math joke:  what does the B. in Benoit B. Mandelbrot stand for?  answer: Benoit B. Mandelbrot.)

Scene 2 takes place in the modern era.  We meet Hannah, who is writing a book about the transformation of the garden at the Croom estate.  I forgot to mention that part of Mr. Noakes's plan included a hermitage.  Lady Croom wants to know who the hermit will be; after all, Mr. Noakes should supply one.  Hannah has a theory about it, which proves to be correct in the final page of the play.  We also meet Bernard Nightingale, an English scholar always on the lookout for fame and academic bragging rights.  In conversation with Hannah, he deduces, via some of the materials in the library, that (a) Lord Byron had been at the estate; (b) had seduced Mrs. Chater; and (c) had killed Mr. Chater in a duel, prompting him to flee England for the continent.  We also meet Valentine, who is trying to understand the grouse population on the estate.  The records of how many grouse were shot is extensive, stretching back more than 200 years, but he can't find the pattern ("There's too much noise in the system.  The noise!").  Of course there isn't much of a pattern, as we know from studying the logistic equation--populations can exhibit chaotic behavior, even when the inputs are known completely.  Upon stumbling on Thomasina's notebook on the shelf, though, he is astounded to find that she was experimenting with just such an equation; at first he dismisses it--"She couldn't have discovered it."  Academic snobbery at its finest.

Scenes 3 and 4 are in the past and present, respectively.  Act Two, whose first scene is Scene 5, begins in the modern period, then moves to the past in Scene 6.  Scene 7 is where all hell breaks loose; more on that below.  So, here's how the play is modeled like a discrete dynamical system:  the end of each scene provides the foundation for the beginning of the next.  That is, we learn something at the end of the scene and this gives the impetus for how the next scene begins.  Back and forth in time, this iteration proceeds as we move along.  Bernard makes a lot of assumptions, which may or may not be reasonable, and writes a paper claiming that Byron engaged in a duel, killing Chater.  When we go back in time, we find out the truth: that Chater was really a botanist who died after being bitten by a monkey on an expedition in Martinique; his wife then marries Lady Croom's brother, the captain of the expedition, who had brought the Chaters along only because he was in love with Mrs. Chater.  Hannah discovers the truth and tells Bernard that she will expose him as a fraud, humiliating him.  Back in the past, the final scene shows that Thomasina is in love with Septimus (and he tries to pretend he does not feel the same way).  We know that she dies in a fire that very night as the play ends with them dancing a waltz on stage at the same time Hannah and Gus (who I haven't mentioned before now, but he never speaks; he does find all the relevant documents which disprove Bernard's theory and prove Hannah's theory about the hermit correct) are clumsily waltzing as well. 

The table in the center becomes cluttered with objects--increasing entropy.  In fact, since there are two systems contributing to the disorder, the total entropy is greater than the sum of the two individual entropies (this is one of the fundamental properties of entropy).  The Second Law of Thermodynamics is sometimes referred to as "heat death"--the universe will eventually be a completely disordered mass at room temperature.  The action of the play behaves this way a bit, but there are also obvious references to heat death; Thomasina literally dies from heat. 

Because I couldn't help myself, here's a plot of the play as it bounces back and forth in time.  There's not really a time scale to measure, but generally, the scenes have varying length, tending to get shorter as the play goes on (as if the function being iterated were converging on some fixed point).  Scene 7 is chaotic in nature, fluctuating wildly between the past and present, with some dialogue lasting only one or two lines in each time period before bouncing back to the other.  It's difficult to visualize this, but the graph below is one attempt.

 A rough graph of the action.

A rough graph of the action.

There is really too much mathematics and satire to summarize, so I'll stop here.  Up next, Virginia Woolf's To the Lighthouse.