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This article is taken from PN Review 234, Volume 43 Number 4, March - April 2017.

The Golden Ratio: Poetry & Mathematics Emily Grosholz
IN HIS BOOK The Poetics of Space, Gaston Bachelard talks about the house of childhood, the house we never leave because at first we live in it, and afterwards it lives on in us. The house of childhood organises our experience, first of all determining inside and outside, and then offering middle terms: the front porch and its steps are a middle term between the house and the town, while the back yard and garden are a middle term between the house and the wild. (In the proportion between two ratios expressed in A:B :: B:C (or  ‘A is to B as B is to C’, we call B the middle term, which brings A and C into clear relation.) It organises what is far away, because we measure ‘away’ by how far it is from home, how many hours or days of travel. Moreover, the windows of the house let in the distances, the dwindling train tracks, river or road, the fields and forest, even the cloudy-blue or starry heavens: they are set squarely on the walls within the window frames, as light comes through and we see what is outside. It also organises time.What lives in the basement or the attic? We ourselves do not eat or sleep or socialise or read there, though those rooms are part of the house: they are where we put the past, the discarded and the treasured. Finally, the house invites playing: the playroom with its gate and the fenced part of the backyard, enclosures where the toys are kept and children imitate the human activities of building and furnishing houses, admonishing and encouraging their dolls, rushing about on small basketball courts and soccer pitches, setting forth amidst the ceremonies of departure and return.

This brings me back to two songs. The first is, to my mind, the most beautiful of the poems in Robert Louis Stevenson’s A Child’s Garden of Verses (1885): ‘Where Go the Boats?’ As a child I owned a golden vinyl record with this poem on it, recorded as a song: thus I learned it by heart and I can still, and very often do, sing it. Many of the poems I know by heart I learned as songs, especially poems in other languages.

Dark brown is the river.
Golden is the sand.
It flows along forever,
With trees on either hand.

Green leaves a-floating,
Castles of the foam,
Boats of mine a-boating,
Where will all come home?

On goes the river
And out past the mill,
Away down the valley,
Away down the hill.

Away down the river,
A hundred miles or more,
Other little children
Shall bring my boats ashore.

The house where I grew up was not located by a river. However, earthbound as it was, my house was just a block south of the Main Line railroad tracks, in the suburbs of Philadelphia. We heard the train whistles, and the Doppler Effect lowering the pitch, all day long; it was my first, aural introduction to the sorrow of the Red Shift: why are all those galaxies leaving us? We were also a block north of the Lincoln Highway, one of the first transcontinental highways in the United States. Route 30, as it was designated under the auspices of the United States Numbered Highway system, established in 1926, ran all the way to San Francisco, and when my parents returned from California where my father had served in the Navy during the Korean War, that’s the route they came home on. We had a painting of the cypresses ‘the sailor wind / ties into deep sea knots’ (as Robinson Jeffers wrote) at Point Lobos over our fireplace, and I retained a few fugitive memories of California and the long trip back home. So for me that road always led to California, as well as Exton (where the best ice cream place was), Downingtown (summer camp lay on its outskirts), and Lancaster (where the Amish people at the Farmers’ Market came from), points west that seemed far, far away. On my first road trip in high school, I drove a friend past Lancaster, north to the Ephrata Cloister – which was like going to eighteenth-century southwest Germany, as I later discovered – and felt that I had achieved adulthood, navigating past the Pillars of Hercules into unknown waters. But I must return to early childhood.

As Tolkien wrote, in another of my beloved books The Hobbit (1937), ‘The road goes ever, ever on.’ So the Lincoln Highway set up a dialectic with my house, not least because Point Lobos was over the fireplace and my mother’s most romantic, and often repeated, memories were of California and Hawaii: she was never able to travel much during most of her short life, except to the New Jersey beaches and to New England where she went to college and still had friends. Her stories were the other side of my father’s silences, though he too had a trove of stories, set pieces with all the bitter absurdity of those in Joseph Heller’s Catch-22. Drafted in World War II, and then again in the Korean War, my father spent seven years of his life crossing the great Pacific again and again in destroyers and tankers, seeking refuge from his terror and displacement in alcohol, at sea. And though he made it back home, like Odysseus, he was often there but not there, sitting in his armchair reading through tome after tome of Naval history and smoking the cigarettes that eventually bore him away again.

Geometry starts with the house and field and town centre, as we find it in Euclid, for Euclidean geometry is the study of figures. Here is a sampling of his definitions, from Book I of the Elements.

3 The extremities of a line are points. 4 A straight line is a line which lies evenly with the points on itself.  5 A surface is that which has length and breadth only.  6 The extremities of a surface are lines. 13 A boundary is that which is an extremity of anything. 14 A figure is that which is contained by any boundary or boundaries.  
The Thirteen Books of Euclid’s Elements, tr. Thomas L. Heath (1956).

And then he gives us the circle, various triangles, the square, and the oblong, the rhombus, and various trapezia. This is the world of childhood: the yard is a rectangle; the house is a closed figure, a set of rectangles and triangles (the walls and roof) hemming in a cuboid; the lane is a bounded straight line; the center of town is a square. But the road is a line that goes on and on, like the river, like the train tracks that allow the train to glide so quickly, in a straight line at a constant speed, as the train whistle turns into a lament. The house is a finite figure, straight from the pages of Euclid, and thus a figure of finitude; but the road, or river, is not a figure, for ‘it flows along forever’. Or rather, it is a figure after all, thanks to the linguistic red shift given to the term ‘figure’ by the ambiguities of English: it is a figure of the infinite.

And we see a foreshadowing of the expansion of geometry into the infinite in the seventeenth and nineteenth centuries in the last of Euclid’s definitions, which he added to clarify the peculiar status of parallel lines.

23 Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

This definition doesn’t limit itself to line segments; it involves co-planar lines that are ‘produced indefinitely’ and yet never meet. So we are invited to think about what happens to a line as it goes on and on: in the seventeenth century, Desargues, inspired by optics and a novel theory of perspective, was one of the founders of projective geometry, and proposed that all co-planar lines intersect: parallel lines just intersect at infinity, ‘the point at infinity’. And Leibniz, following the work of Desargues and Pascal, took space itself as an object whose structure is revealed by studying the transformations of figures, and what remains invariant among the transformations: thus one could think of all the conic sections as variants of the circle. Indeed, in 1679, Leibniz briefly considered the possibility of spherical geometry (the easiest of the non-Euclidean geometries to understand, because navigation on earth more or less exemplifies it), based on the analogy between all lines in projective geometry intersecting (some in the point at infinity) and all geodesics on a spherical surface intersecting (a geodesic is the shortest distance between two points on a spherical surface and thus the analogue to a straight line in ‘flat’ Euclidean geometry). However, he veered off in another direction and left the explicit formulation of non-Euclidean geometry on a surface of constant positive curvature to the Hungarian mathematician Janos Bolyai in the nineteenth century, following upon the work of Euler and Gauss. Nikolai Lobachevsky worked out the non-Euclidean geometry on a surface of constant negative curvature around the same time, but entirely independently. And Bernhard Riemann, building on the work of his teacher Gauss, came up with the generalised notion of a two dimensional surface (generalisable to n-dimensional surfaces) which launches geometry into the realm of topology.

If only everyone had decided to stop fighting in 1940, or 1945, or 1950 – my father could have kept on sailing east, around the south coast of India and then of Africa, and just come back along a geodesic to my mother and me. My mother used to sing me this lullaby, by Alfred Lord Tennyson.

Sweet and low, sweet and low,
Wind of the Western sea.
Blow, blow, breathe and blow,
Wind of the Western sea.
Over the rolling waters go,
Come from the dying moon, and blow,
      Blow him again to me;
While my little one, while my pretty one, sleeps.

[…] Father will come to his babe in the nest,
Silver sails all out of the west
      Under the silver moon:
Sleep, my little one, sleep, my pretty one, sleep.

Like a baby on its mother’s breast, one can always dream. The house governs the poetics of space (inflected by time – and eventually Riemann’s geometry provides a model for Einstein’s space-time); the road and river govern a poetics of time (inflected by space – for we must all go home again, whether we can or can’t, in fact or in imagination, sooner or later). And what child does not thrill to the romance of departure, which is after all what eventually he or she prepares to do: depart from the house of childhood, aided and abetted by romance.

During my twenties, I was lucky enough to travel often to Europe, where I studied the art and architecture that links Minoan Civilisation (c. 3500–1500 BCE) to classical Greece and Rome, thence to medieval Europe and the Renaissance (c. 1500 CE). Here is the great irony: to escape the square of the house of my childhood, I launched myself on rivers of air from Philadelphia to London, on traintracks from London to Paris to Rome to Brindisi, on Mediterranean currents from Brindisi to Patras and from Athens to Heraklion, and soared above five millennia. What did I find? Again and again, at the heart of the matter, I re-discovered the square! And all the reasons why we can, and must, go home again.

The square is often nestled in the golden ratio (symbolised by the Greek letter phi, φ). Two magnitudes stand in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the magnitudes. So, assuming A > B > >0, we can write the proportion of the two ratios this way:

A : B :: A + B : A 
(or ‘A is to B as A plus B is to A’)

Note that A is the middle term between B and A+B in this case. We can also define φ (which is, like √2, a quadratic irrational number) by using the late medieval trick of reconceptualising ratios as fractions and proportions as equations:

φ = A/B = (A+B)/A = (1 + √5) /2

And finally, we retrieve the square tucked into a golden rectangle, which illustrates the golden ratio: the yellow side of the square is A, the purple top of the adjacent rectangle is B, so the longer side of the golden rectangle is A+B, and its shorter side is A. In this particular diagram, one stipulates that A=1, so that you can see especially clearly why φ = (1 + √5) /2 = 1.6180339887…

illustration of the golden triangle

Two places where I arrived on my travels made an especially deep impression on me, so that I feel as if I could relive the very moment when I first beheld them. One was the Parthenon, on its hilltop overlooking Athens, where in the summer of 1970 I followed the shade of Socrates around, listening to him use reductio ad absurdum (that paradoxically valid deductive argument form) to make the citizens of his city puzzled, reflective and ultimately philosophical. Recall that his student Plato was convinced that mathematics was the middle term between Becoming and Being.

The Parthenon the golden triangle

I kept a journal, and later I turned those journal entries into a narrative poem, Cypress and Bitter Laurel, which was eventually published in The Reaper (Vol. 17) by Mark Jarman and Robert McDowell. Here is that moment, recorded in the iambic pentameter I employed almost unconsciously:

Such an odd life we lead, the life of tourists
(And of spies, anthropologists and poets),
Moving slowly through the world’s locations,
Transparent, unremarkable, all eyes.
Yesterday, we got up at five, to see
The Acropolis in the very earliest light.
Large enough to count as monumental,
Small enough to be visible all at once,
The Parthenon stood white against the sky
And taught us both a lesson in proportion
Even in ruins, its roof blasted away.
For what remains is so insistently formal,
The series of marble steps and columns require
Completeness in the act of the mind’s eye.

The second place was Notre Dame in Paris: there it was again, the golden rectangle, and another embodiment in stone of the golden ratio and its iterations.

Notre Dame the golden triangle

Three years later, in 1973, I bicycled around Normandy with Henry Adams’s Mont-Saint-Michel and Chartres (1904) in my backpack, and then around Burgundy, with various writings by Viollet-le-Duc and the memoirs of my beloved Colette. A few months after that, starting graduate school at Yale University, I discovered Otto von Simson’s The Gothic Cathedral (1956) under the tutelage of Karsten Harries. Adams and Simson both refer to the Sketchbook of   Villard de Honnecourt, a thirteenth-century mason and architect from Picardy. On one page, he includes the following geometrical diagrams; the second (in red) shows how to halve the square (or, going in the opposite direction, how to double the square), as he analyses the ground plan of a cloister.

This diagram is especially significant because it recalls the construction in Plato’s Meno where Socrates leads an unlettered, unschooled slave boy to this very construction just by asking him questions, and then suggests that any human soul has access to mathematics: it is the middle term that orients us toward the eternal. (This of course means that everybody should be able to go to university, women are good at mathematics, and nobody should be a slave.)

Von Simson asserts, ‘the Gothic builders […] are unanimous in paying tribute to geometry as the basis of their art. This is revealed even by a glance at Gothic architectural drawings […] they appear like beautiful patterns of lines ordered according to geometrical principles. The architectural members are represented without any indication of volume, and, until the end of the fourteenth century, there is no indication of space or perspective. The exclusive emphasis on surface and line confirms our impressions of actual Gothic buildings […] With but a single basic dimension given, the Gothic architect developed all other magnitudes of his ground plan and elevation by strictly geometrical means, using as modules certain regular polygons, above all the square.’ And a bit later he adds, ‘The church is, mystically and liturgically, an image of heaven.’ One expression of that order is mathematical structure, manifest in both architecture and music. St Augustine, c. 400 CE, invoked the science and art of music, because it was based on mathematics. (Pythagoras influenced Plato, and St Augustine was in turn influenced by Neoplatonism, despite his conversion to Christianity.) The most admirable ratio, he claimed, was 1:1, equality, symmetry, and unison; next in rank were the ratios 1:2 (the octave), 2:3 (the fifth) and finally 3:4 (the fourth). Just as the stone squares in their golden rectangles are beautiful and stable, so the notes in their admirable ratios are consonant. Augustine, von Simson tells us in The Gothic Cathedral, loved both architecture and music ‘since he experienced the same transcendental elements in both’.

the construction of Plato's Meno

Later, he tells us that Villard’s book contains ‘not only the geometrical canons of Gothic architecture, but also the Augustinian aesthetics of ‘musical’ proportions. In one of his drawings, the ground plan of a Cistercian church, ‘the square bay of the side aisles is the basic unit or module from which all proportions of the plan are derived […] Thus the length of the church is related to the transept in the ratio of the fifth (2:3). The octave ratio (1:2) determines the relations between side aisle and nave, length and width of the transept, and […] of the interior elevation as well. The 3:4 ratio of the choir evokes the musical fourth; the 4:5 ratio of nave and side aisles taken as a unit corresponds to the third; while the crossing, liturgically and aesthetically the center of the church, is based on the 1 : 1 ratio of unison, most perfect of consonances.’ Von Simson goes on to explain in some detail how the architect of Notre Dame de Chartres elaborated on this structure in surprisingly innovative ways (‘he turned to advantage the restrictions that tradition imposed upon his design’) that allowed the vault of Chartres to be sprung at a much greater height than that of any of its predecessors. It was the first cathedral where the flying buttresses were aesthetically as well as structurally part of the overall design. Moreover, the golden ratio occurs in the figures of the west façade, as well as in the elevation: ‘The height of the piers […] is 8.61 m. The height of the shafts above (excluding their capitals) is 13.85 m. The distance between the base of the shafts and the lower string-course is 5.35 m. The three ratios  5.33 : 8.61 : 13.85   are very close approximations indeed to the ratios of the golden section.’ The dimensions of the elevation are also closely related to those of the ground plan. And so von Simson sums up: ‘Medieval metaphysics conceived beauty as the splendor veritas, as the radiant manifestation of objectively valid laws.’

This is the poem I wrote, inspired by the churches of Normandy, in particular one especially lovely set of ruins in Jumièges:

[…] An ancient abbey stands in Jumièges.
The western towers, twin battlements
Against the tides of darkness, still remain,
But every wall is gutted, overgrown,
And the high roof, the paradigm of heaven,
Is long since stormed away.
Between the nave and choir there is no stair,
No screen to keep the crowds from their desire.
Only a copper beech, the prince of trees,
Whose monumental bole could bear
The manifold thin nervures of the air,
Divides the floor. The leaves divide the sky
In panes of bronze which fan
Around a spectral cross of red and green,
So the lost vault becomes a sheer
Translucent window, and the blue between
The blue of distance, as it ought to be […]

Sometimes I think that the closest I have come to heaven were moments when I found myself standing in the nave of Chartres, looking up; but that was not just because of the radiant geometry and arithmetic manifest in the cathedral’s stones. The cathedral is also astonishingly luminous: the great architect of Chartres also replaced solid walls, to an unprecedented extent, with the transparent walls of stained-glass windows. The clerestory is the upper part of the nave and choir and transepts: because it rises above the lower roofs its windows admit a flood of light, which animate the colours of the stained glass (as James tells us in his rhapsodic Chapter viii of Mont-Saint-Michel and Chartres, ‘The Twelfth-Century Glass’). And James reminds us that Villard sketched the western Rose Window at Chartres. When we look through the windows of the house of childhood, we see the town we live in and the woods and fields beyond, and the sky above, which sometimes (especially when it fills with wheeling stars) becomes a figure for heaven. But when we look ‘through’ the windows of Chartres, we see heaven: there it is. Thus as I look back on other poems I’ve written that include cathedrals, I notice that images of their windows evoke not only the sky, but also the edge of a forest backlit by the rising or setting sun as it scores the horizon with its slanting, golden light. The Clerestory: a middle term between earth and heaven.

This article is taken from PN Review 234, Volume 43 Number 4, March - April 2017.

Readers are asked to send a note of any misprints or mistakes that they spot in this article to
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