74
Drametrics
What dramaturgs should learn from
mathematicians
Magda Romanska
The relationship between math and music has long been known and analyzed. We
know that a well-structured musical composition is like a well-structured mathematical
formula. The mathematician and violinist James Stewart argues that mathematics,
like music, is concerned with structure, “the way mathematical objects fit together and
relate to each other.”1 The same can be said of art: from classical realist paintings and
sculptures to the most abstract works of Picasso, Kandinsky, Malevich, Mondrian,
and Pollack, each composition follows a carefully arranged structural order with
colors, shapes, patterns, and empty space consciously complementing, supplementing,
and juxtaposing one another. Like music and art, Literature too has a long-standing
relationship with mathematics. Drawing on the relationship between math, art, and
music, in his 2012 New Yorker article Alexander Nazaryan attempts to trace implicit
similarities between mathematicians and fiction writers. Like mathematicians, fiction
writers create patterns that follow a well-defined structural order, from the meta levels
(the sequence of chapters and paragraphs) to the sublevels of sentences, words, and
syllables. Nazaryan quotes Ernest Hemingway, who is said to have written in 1945 to
his colleague Maxwell Perkins: “The laws of prose writing are as immutable as those
of flight, of mathematics, of physics.”2 From Hemingway to J. K. Rowling, fiction
writers are known to draw elaborate diagrams for their novels, structuring the
organization of their stories on the basis of mathematical formulas.
The close affinity between math and poetry has also long been known to poets
and literary theorists. As it is based on meter that defines the rhythmic structure of
each verse, poetry is nothing more than the linguistic equivalent of a complicated
mathematical pattern; it’s a highly intricate, word-based math game. Let’s take our
basic iambic pentameter, one of the most often used meters in English poetry, a
form favored by William Shakespeare, John Donne, and John Keats. Iambic pentameter has ten syllables per line that alternate unstressed and stressed beats. They are
arranged in five pairs following a clear mathematical pattern: “1 2 1 2 3 4 3 4 5 6 5 6
7 7.” In Four Riddles, Lewis Carroll uses the iambic pentameter quadratic x2 + 7x +
53 = 11/3 in one of his riddles. Other poetic forms, the octave, for example, follow a
different rhyming scheme that looks like “1 2 1 1 1 2 2 1.” In addition to the
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mathematical arrangement of their lines, the overall structures of poems follow
mathematical formulas. A villanelle, for example, has six stanzas, with three lines in
the first five stanzas and four in the last.
Most recently, mathematicians, humanists, social and computer scientists
have come together to devise algorithms that would explain such diverse phenomena
as the progression of history and the development of language. Kevin Slavin, an
MIT professor and a designer of large-scale, real-world games, argues in his popular
TED Talk that algorithms guide our world.3 Likewise, the Harvard mathematician
Jean-Baptiste Michel, for example, argues that there is a mathematical pattern to
the evolution of the English language and that math can explain certain historical
forces, including wars. Michel suggests that mathematics can be used to
predict future events and perhaps even to prevent calamities. For Michel, the
huge databases we’re currently amassing thanks to the digital revolution will allow
us to process unimaginable amounts of data, which will reveal mathematical patterns
in every sphere of life.4 This is where science and digital humanities are coming
together.
But what about drama and theatre? The idea that theatre is guided by the same
rules of mathematics appears both absurd and absolutely logical. If all other art
forms work according to the same rules, why shouldn’t theatre? Indeed, although
the relationship between math and dramatic structure has never been explicitly
explored, math has always been implicitly present in the development of Western
drama and dramatic theory. In 1863, the German novelist and playwright Gustav
Freytag published Die Technik des Dramas (Technique of the drama), in which he
outlined a geometric pattern of dramatic structure in classic Greek tragedy. With (a)
introduction, (b) rise, (c) climax, (d) return or fall, and (e) catastrophe, Freytag’s
triangle outlines the basic flow of dramatic action:
Figure 74.1 Freytag’s triangle
Although it doesn’t provide any actual numbers, Freytag’s triangle quite closely
resembles the golden triangle, in which the ratio a:b is equivalent to the golden ratio
Phi, φ = 1.61803398875.
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MAGDA ROMANSKA
Figure 74.2 Golden triangle
In the golden ratio formula, a + b is to a as a is to b, and is equal to φ:
Figure 74.3 Golden ratio formula
Figure 74.4 Golden ratio formula
Freytag’s triangle is based on the classic Aristotelian model of dramatic structure.
Aristotle, like many ancients, was as proficient in math as he was in dramatic theory.
Aristotle’s Poetics, a book of dramatic theory, has been a guidebook for both playwrights and dramaturgs for centuries. Aristotle’s model of the dramatic arc, as
described in Poetics, has been traditionally illustrated by dramatic theorists like this:
Figure 74.5 Aristotle’s model of the dramatic arc
In music, many compositions are based on the golden ratio, including some of
Chopin’s most famous études and nocturnes. Likewise, in art, classical and modern
paintings are built around the golden ratio formula. In fact, in the visual arts and
architecture, the golden triangle and golden ratio have been a standard of design,
most famously perfected in Leonardo da Vinci’s works.5
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But what about theatre? Since (dramatic) theatre is a two-tiered art form based on text
and its performance, the space (number of pages) and time (the duration of the performance) continuum and the correlation between the two doesn’t lend itself to easy
geometrical calculations in the same way that visual art and music do. Theatre artists –
actors, directors, stage managers, designers and choreographers – have an innate
understanding of the importance of time (timing) and space (blocking). When we do
consciously calculate the time/space continuum of dramatic structure, not surprisingly,
we find that like in music and in visual art, in drama the best well-made plays – from
Sophocles’ Oedipus the King to Ibsen’s A Doll’s House – follow the same mathematical
formula, timing their climactic moments at or around the vicinity of the golden point.
Let’s see how it works in Ibsen’s classic, A Doll’s House, a play about Nora, a
housewife who decides to leave her husband, Torvald. Early in the opening of the
play, we learn that in the past Nora falsified her father’s signature to borrow some
money to save Torvald’s life. Krogstad, an employee in the bank that Torvald runs,
attempts to blackmail her with the document she signed; he asks her to intercede on
his behalf with Torvald, who intends to fire him. When the matter comes to light,
Torvald, rather than appreciating the fact that she saved his life, even if it meant she
had to break the law, disowns Nora, which prompts her to leave him. With a script
that’s 101 pages long, the moment of conflict – Nora trying to convince Torvald not
to fire Krogstad – falls on pages 38 and 39, with the golden point falling mathematically on page 38.57856. Likewise, the climactic moment – the conversation between
Nora and Krogstad and his dropping off the fateful envelope in her mailbox – falls
precisely on pages 62 and 63. The golden point falls mathematically on page
62.42143. The geometric breakdown of the play then looks something like this:
Figure 74.6 Script diagram of A Doll’s House
The golden ratio formula of A Doll’s House script of 101 pages would look like this:
62.42143 + 38.57856 / 62.42143 = 62.42143 / 38.57856 = 1.61803398875
Regardless of how we format the script, barring some major formatting disfigurations, even when the placement of the two golden points changes, the formula
remains the same.
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Can we then try to revise Aristotle’s model of dramatic structure to consider the
placement of conflict and climactic scenes to fall within the vicinity of the two
golden points on the spatial continuum of the script? If we do so, Aristotle’s dramatic
arc looks more like two golden triangles:
Figure 74.7 Aristotelian model of dramatic structure
Interestingly enough, the dramatic structure that constructs the best well-made
plays similarly constructs the best well-made movies. In fact, in cinema, the dramatic
arc of the narration follows the mathematical formula of the golden ratio almost to a
T. Drama theory scholars as well as film scriptwriters have long known that the
narrative arc of the best movies closely follows the Aristotelian model. In his book
Aristotle’s Poetics for Screenwriters (2002), Michael Tierno advises aspiring screenwriters that Aristotle’s Poetics “can’t tell you everything about writing an immortal
screenplay, but it’s a great place to start.”6 To test our thesis that mathematical rules
guide cinematic conventions in the same way that they guide dramatic conventions,
let’s use some basic observations about film structure made by Christopher Keane, a
legendary screenwriting teacher. Keane notes that the average well-made film script is
roughly 120 pages long and that it can be broken into three acts, with act I being 30
pages long, act II, 60 pages, and act III, 30 pages again. Plot Point I (interchangeably
referred to as an incident, crisis, or subclimax) falls approximately on pages 25 to 30,
and Plot Point II (also referred to as the second reversal) falls approximately on
pages 75 to 80. The so-called first reversal falls on page 45. Applying the golden ratio
formula to Keane’s observations, we see that Plot Point II falls exactly at the golden
point of 74.16407. Furthermore, applying the golden ratio model to the first
subsections of the plot, we see that Plot Point I typically falls at 28.647450, that is,
between pages 25 and 30 (act I ends on page 30). Other screenwriting teachers
have also noted that the first reversal generally takes place on page 45. Taking
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into consideration the symmetry of the two reversals and two plot points, our
mathematical movie diagram would look like this:
Figure 74.8 The diagram of Christopher Keane’s film plot structure model
The overarching structure follows the golden ratio model, as well as the sub-elements
of the dramatic structure resembling the golden ratio model of Fibonacci’s spiral.
The same mathematical calculations apply to the time dimension of well-made films.
With a typical 90-minute movie, Plot Point II falls roughly around the 55th minute
of the film, while Plot Point I falls roughly around the 21st minute of the film. The
first reversal should take place around the 34th minute of the film. Test it with some
old-time classic Hollywood movies or even the most recent Oscar winners.
The golden ratio calculations work all fine and dandy in classical art, architecture,
and perhaps in the classical dramatic structure, but when it comes to modern
abstract art, at first we feel like we cannot discern any patterns at all. Although it’s
hard to believe that Picasso, or Jackson Pollock, follows any structure, the modern art
works, just like their predecessors, are tightly guided by mathematical proportions.
Figure 74.9 Kazimir Malevich, Suprematist
Composition (Red Square and
Black Square) (1915)
Figure 74.10 Diagram showing the relation of
the golden triangle to the structure of painting (reprinted from
John Milner’s Kazimir Malevich
& the Art of Geometry)
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MAGDA ROMANSKA
One example is Malevich’s Suprematist Composition, which appears at first glance to
be nothing but a random arrangement of two squares. Only when we look more
carefully do we realize how painstakingly careful Malevich’s composition is, how
many hours of mathematical calculations it required for it to follow all of its golden
ratios and golden triangles.
But are the same mathematical patterns that are present in abstract art also present
in contemporary postmodern drama? Elinor Fuchs argues that modern works of
drama, even such seemingly plotless ones as Beckett’s Waiting for Godot, follow the
Aristotelian model of the dramatic arc.7 If Waiting for Godot follows the Aristotelian
model, it must also follow some kind of mathematical formula, right? Even if – as in
modern art – it is not readily discernible. Beckett’s existential story Waiting for Godot
is the tale of two tramps, Didi and Gogo, who wait for Godot, who (spoiler alert!)
never arrives. There are basically three kinds of scenes in the play: 1) the scenes in
which Didi and Gogo wait for Godot; 2) the scenes where the Boy comes to tell
them that Godot won’t come; and 3) the scenes where two wandering tramps, Pozzo
and Lucky, pass by Didi and Gogo’s waiting area. Nothing else happens – ever.
Like Malevich’s two squares in Suprematist Composition, the two acts of the play
remain in a mathematical relationship with each other through the two interconnected subplots. The entrances of Pozzo and Lucky are the two structural pillars
on which the two acts of the play are built. They also are paced according to strict
adherence to the rules of the golden ratio. If in the case of A Doll’s House we can
argue that our choice of climactic moments can be arbitrary, in the case of Waiting
for Godot the placement of entrances and exits is not. In the 98 single-spaced pages of
the script, Pozzo and Lucky enter on page 16 and exit on page 42 of act I. In act II,
they enter on page 75 (page 25 of act II) and exit on page 91 (page 41 of act II). In
both instances, their entrances fall at the golden point relative to their exits:
Figure 74.11 Script diagram of Beckett’s Waiting for Godot
Likewise, the entrances and exits of the Boy provide the connecting arc between
the two acts. In act I, the Boy enters on page 43; his entrance is a sign of hope for
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Didi and Gogo that Godot will come. The Boy’s second and final exit on page 94
(page 44 of act II) dashes their hopes indefinitely; there is no doubt that Godot will
never come. The golden point between these two markers falls precisely on Pozzo
and Lucky’s second entrance, thus connecting the two subplots.
Figure 74.12 Script diagram of Beckett’s Waiting for Godot
Perhaps it is a combination of conscious work and some innate sense of balance, symmetry, and composition that drives the great artists to create works that somehow fit into
the mysterious golden ratio design that seems to govern all other natural phenomena.
If the mathematical structure of Waiting for Godot, however, doesn’t seem too
explicit, Beckett’s other plays, in fact, have a deliberate and very explicit mathematical
formulas. One of them is the well-known Quad, a 1981 television play, in which four
actors dressed in distinctly colored robes (blue, red, white, and yellow) silently walk in
sync around the square stage in well-defined patterns: “Quad has a musical structure.
It is a kind of canon or catch – a mysterious square dance. Four hooded figures
move along the sides of the square. Each has his own particular itinerary. A pattern
emerges and collisions are just avoided.”8 The geometric diagram for the play drawn
by Beckett accompanied the first publication of the play in 1984.9
Figure 74.13 Diagram of Beckett’s Quad (1981). Reprinted from Samuel Beckett, Collected
Shorter Plays of Samuel Beckett (London: Faber and Faber, 1984)
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Beckett’s Quad is perhaps the extreme example of geometry as applied to theatre,
but in our increasingly fragmented and abstract postdramatic theatrical universe,
where the traditional Aristotelian model no longer always fits, how are we to think
about dramatic structure? What new rules and principles can guide the dramaturgy of
postdramatic theatrical works? Robert Wilson is known to use mathematical formulas
for his pieces, including Einstein on the Beach, which is structured according to predetermined sequences. More and more contemporary theatre artists are turning to
computer science and mathematics to design their performances. One of the most
famous is Anne Dorsen, an Obie-winning director and writer, who uses algorithms
to create theatre works, a practice she calls “algorithmic theatre.” Dorsen’s first
piece of algorithmic theatre, Hello, Hi There (2010), was a “conversation” between
two computers, which were programmed by “inputting a huge dataset of possible
language/responses and then creating a natural language processing algorithm that
allowed the two computers to respond to one another.”10 The computers replicated
Michel Foucault and Noam Chomsky’s 1971 debate about the eternal issue of
nature/nurture. In A Piece of Work, Dorsen creates a literal “hamletmachine” by
“using a far more complicated algorithmic modality for fragmenting and assembling
the text of Hamlet. In this piece, light, sound, and text [are] all controlled by
probabilistic algorithms called hidden Markov models. The hidden Markov model
we are all probably most familiar with is T9 texting; your phone will guess which
word you might be spelling based on what letters you have already typed and the
frequency.”11 Thanks to Markov’s model, the audience of Dorsen’s take on Hamlet
experiences a new version of the play each time it is performed.
Likewise, Ruth Little, a dramaturg and the winner of the 2012 Kenneth Tynan Award,
uses chaos theory in the creation of newly devised works. Little writes: “I’m interested in
dramaturgical dialogue that goes beyond linear determinism – the orderly predictable
world of classical physics and Aristotelian dramaturgical models – to an understanding
of non-linear dynamics and living systems. In fact, the majority of natural phenomena
are non-linear, and energy is replacing matter as a fundamental feature of reality. We
need to shift the register of our thinking to gain new perspectives on our own
experience.”12 Employing chaos theory, Little is able to channel the serendipitous
energy of the rehearsal room into non-linear theatrical narratives.
In my own play, Opheliamachine, which premiered at the City Garage in LA in 2013,
I use simple mathematical formulas to arrange seemingly random scenes. Their order
and sequencing creates dramatic structure which is not Aristotelian, but which nonetheless creates its own meanings.13 Such Drametrics, a combination of mathematics and
dramaturgy, will become more and more prominent, as both theatre artists and scientists try to make sense of the world, and as our globalized, fragmented reception of
reality renders the traditional Aristotelian storytelling obsolete. The new meanings will
emerge out of surrealist yet premeditated arrangements of concepts, images, and ideas.
Notes
1 Katharine Merow, “Mathematics and Music,” Mathematical Association of America, n.d.,
available online at www.maa.org/meetings/calendar-events/mathematics-and-music, accessed
October 12, 2012.
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2 Alexander Nazaryan, “Why Writers Should Learn Math,” The New Yorker, November 2,
2012, available online at www.newyorker.com/online/blogs/books/2012/11/writers-shouldlearn-math.html, accessed November 6, 2012.
3 Kevin Slavin, “How Algorithms Shape Our World,” YouTube video, 15:22, from a July
2011 TED Talk, posted by “TED,” July 21, 2011, available online at www.youtube.com/
watch?v=TDaFwnOiKVE, accessed October 6, 2012.
4 Jean-Baptiste Michel, “The Mathematics of History,” TED video, 4:26, from a February
2012 TED Talk, posted in May 2012, available online at www.ted.com/talks/jean_baptiste_
michel_the_mathematics_of_history.html, accessed October 12, 2012.
5 For more connections between da Vinci’s art and math, see Bülent Atalay’s Math and the
Mona Lisa: The Art and Science of Leonardo da Vinci (New York: Harper Perennial, 2006).
6 Michael Tierno, Aristotle’s Poetics for Screenwriters: Storytelling Secrets from the Greatest Mind in
Western Civilization (New York: Hyperion, 2002).
7 Elinor Fuchs, “Waiting for Recognition: An Aristotle for ‘Non-Aristotelian’ Drama,”
Modern Drama 50.4 (2007): 532–44.
8 Synopsis from Radio Times, quoted in database entry for “Quad,” British Film Institute,
available online at http://ftvdb.bfi.org.uk/sift/title/326964, accessed October 30, 2012.
9 Samuel Beckett, Collected Shorter Plays of Samuel Beckett (London: Faber and Faber, 1984),
293.
10 Annie Dorsen, A Piece of Work (Formerly False Peach), digital program for February 21–4
performance, On the Boards, 4, available online at www.ontheboards.org/sites/default/files/
dorsen_digital_NEW_title.pdf, accessed November 12, 2013.
11 Ibid.
12 “Ruth Little’s Thoughts on Dramaturgy,” available online at http://ee.dramaturgy.co.uk/
index.php/site/comments/ruth_littles_thoughts_on_dramaturgy, accessed October 12, 2012.
13 Magda Romanska, Opheliamachine, City Garage Theatre, available online at www.citygarage.
org/opheliamachine, accessed November 20, 2013.
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Magda Romanska
