Cathedral (1)

The symmetry of a cathedral is pleasing to gods and men. Amplified by the walls, the voice echoes around you and inside your heart. Prayer climbs mightily and higher. You visit the cathedral because there you are closer to heaven. The gods admire themselves in the architecture and draw closer to the people. The floor plan, diagrammed as a crucifix, expresses the meeting of two roads, a crossroads, or a bird in flight. The sky is below, and above. The threshold is where you kneel, half in darkness, half in daylight. Here is the sinner, the poet, kneeling at a pew. He offers a sacrifice, prayer, with his lips. His eyes are closed. But he knows that from the aisle the ceiling towers up and over the pews and down falls the other colonnade.

Up ahead, the lateral transept crosses the widening nave, beneath the cupola, and beyond the choir and the apse the multi-coloured panels of stained glass – tripartite, figurative, sumptuous – expose the interior to radiance and dominate the altar. Lit torches stand sentinel along the aisles. Smaller windows give light blessings to chapels hidden in the apse and in the alcoves off the transept. Generations of prelates lie sleeping in their tombs, in the catacombs under the floor. The cathedral has been a shelter against conquering armies, a bastion of the hierarchical order of faith and knowledge. The sinner is meant to feel awe. The ceiling ascends without whisper of support, in obedience to unseen powers. The cathedral suspends the law of falling bodies. The voice of the sinner rises and scatters amid the chiaroscuro. He prays to echoes, walls:

The echo sounds within the silence of the aisles and colonnades. But the echo has no source in perceptible things. It speaks from the shadows, like a miracle.

The things of this world return as from a long way off. The voyage obscures their path. The wilderness speaks. Whence comes this voice I hear? The cathedral replies: in the ribs of the ceiling, in the secluded artery of the aisles, in the tiered waist of the choir, in the stone faces of the altar, the cathedral abounds in symmetry.

Prey to illusion or faith, a humanist and poet, the sinner has no technical knowledge of the architecture, the physics of acoustic replication, the way the cathedral amplifies whisper, fragments and repeats it, and sends it back to him. Sound separates from sense, and finds another sense, like a migrant worker. Comforted and distressed by the echo, the sinner pleads with the echo to remove the mask. He addresses the unknown: show thy face.

The building interferes with the voice. The words of the sinner are not his words. The voice (‘undertakes this task’) belongs to someone else (‘Ask’). Words are missing. Symmetry breaks. The syntax has been changed. The cathedral mutilates the language.

Rhythm and Seriality

The Walls Series has 58 drawings (see Figure 1). The current arrangement has them in rows of five. As you can see, this layout is a grid, an array. By the title of the series, and the writing at the foot of the panels, the drawings are walls. The grid is a specific configuration of the series. In its original format, the series was displayed on 58 separate pages, arranged in a vertical stack. Effectively, the vertical stack functions as a codex, like an art book. Below the title page is the first drawing. To view the series, to read this book, the reader turns the pages, one at a time, placing the drawing face down, on the left, and uncovering the next drawing. Because the drawings are all on the recto, only one drawing is visible at a time. The verso is blank. In spite of their variety, the drawings have a common purpose: the representation of the object named by the caption.

Figure 1.

Walls: combined series.

Now, for the drawings to be able to illustrate the caption – the same word, the same object – there must be recurrence, community, and resemblance among them. The walls must somehow be equivalent, and the equivalence must have a basis in the behaviour of the lines. The viewer sees lines, in various dispositions, on every page: lines that are unequal in length, unpredictable in their number and position on the page, diverse in their angles relative to the frame, lines that bend and make a geometrical shape, that intersect and close, the fill up an area, that continue straight, that stop. To survey this behaviour, the viewer would like to be able to compare the drawings, to study the patterns, to name and classify the common elements. But all of this – the elements, the patterns – is indistinct. In the vertical stack, no sooner does a drawing appear than it vanishes. Only one drawing is visible at a time. As the series unfolds, the details are lost. They become invisible, and further away. The problem with trying to compare the drawings is that this can happen only in memory, and memory is not strong enough to allow the reader to hold up for his inspection, in mental space, an arbitrary sequence of panels at once.

Instead of relying on memory, one can lay out the drawings in physical space, on the floor, or a large table, and view them all at once. This is what I did, and it is what the array entitled ‘Walls: Combined Series’ (Figure 1) represents. In replacing the vertical stack with the array, I have externalized the plane of memory that the mind had wanted to sustain. In the new format, I don’t need to remember the past. All is materialized and on the surface. With the drawings arrayed in space, there is no hierarchy of presence and absence, visible and hidden, face-up and face-down, recto and verso, on top and under. Nor am I constrained to follow the page sequence of the codex. I can choose my own path. I can read the drawings from left to right, or high to low, or disregard contiguity and hop-scotch from panel to panel obliquely. I can double back, to take another look, can eliminate spatial separation at a glance.

Let us say that I respect the invitation of the array to proceed from left to right, as in a page of handwriting. The contiguity and lateral visibility of the drawings produce an experience of the series that is unavailable, or too thin to notice, in the other format: serial visual rhythm. In architecture and art, rhythm is defined as ‘the harmonious sequence or correlation of colours, elements, or masses’ (‘Rhythm’, definition 1.5, Oxford English Dictionary 2019). In the drawings, the rhythm of visual elements becomes perceptible in the lateral array. Line correlates with line, form with form. But rhythm in another sense also comes to the fore. Because the viewer peruses the drawings in a row, the correlation of elements appears as a sequence of consecutive changes: affirmation and departure. The rhythm of this structure – the row – bears comparison with the rhythm of poetic language in prosody: ‘the measured flow of words and phrases in verse’ (‘Rhythm’, definition 1.1.a, Oxford English Dictionary 2019). The array of panels gives rise to the perception of motion – of kinetic ‘flow.’ The layout affects the perception of form, and reveals the sway of an architecture that was hidden.

Imagine the drawings of the array projected onto the blocks of a city. Looking at a row of walls is like a pedestrian walking on a street and looking over at the buildings going by. Fall weather, Riverside Drive, in Manhattan. Sunlight in the maple trees. The cross-streets at intervals. He turns onto one of these. Broadway at the far end, distant traffic.

The pedestrian studies the facades up close. He notices that certain elements of the design do not change. For example, every townhouse has a door, a window to the side, a flight of steps from the sidewalk. The townhouses share a number of common elements; this he notices, but he also notices that there is variety. Some windows are larger, closed with shutters, others display interior curtains. The doors are different. But by the time he reaches the fourth townhouse he cannot recall which doors are closed with a knob, which a brass handle, the sequence of alternatives, the number of each type, the colours. How many flowerpots have irises? Like that. Any pattern formed by the number, sequence, and type of such elements is indistinct.

But suppose the pedestrian circles the block and starts again, this time from the opposite side of the street. He views the townhouses from a distance, walking as he looks, looking directly across, looking back, looking across, looking ahead. Although he had noticed that the facades change colour, there is a difference between seeing the colours individually, one at a time, in isolation, and viewing them arrayed in space: ochre, ochre, green, brown, ochre. Not only do the colours stand out because the whole painted façade is visible and there are five facades, like playing cards. It’s that colour – the sequence of colours, and how they repeat or vary – now appears worth paying attention to, as a visual element that recurs, persists and recurs, from building to building. Because all of the buildings are visible, he can review the itinerary (sequence), and in review the pattern made by the colours becomes increasingly distinct and pronounced. The buildings are part of a row.

Another way of getting at the difference between the experience of the vertical stack, or codex, and the array is to recall the operations of substitution and combination in the making of a verbal message. In the vertical stack, or codex, the axis of viewing available to the reader is that of selection and similarity. The appearance of a drawing exactly corresponds to the disappearance of a drawing. The one, as it comes to occupy the frame of vision, replaces the other, as it loses material presence and dissolves. Any two drawings are substitutes. In the array, the series progresses not by replacement but by combination. No drawing is hidden. No drawing replaces another. Instead, the drawings combine in a linear chain, a sequence of material displays tied to distinct spatial positions.

The motion of the pedestrian along the block magnifies and dramatizes the correlation between the spatial location of the drawing and the drawing. To relocate his vision, to see the next building directly, the pedestrian moves along the block. The eye, affixed to his body, travels as he walks. Equivalently, the pedestrian could hold his position while the facades pass him in order, like panes on a strip of photographic cellophane. In the vertical stack, translocation, and thus the sense of motion, does not occur.

The difference between the two ways of experiencing the drawings comes down to the presence or absence of the visible row. And the visible row brings to light a characteristic of the series that remains indistinct or in abeyance in the codex: the ‘flow’ of elements. Because the row has a marked beginning and end, any pattern of visual elements – ochre, ochre, green, brown, ochre – arises in a well-defined and bounded space. This boundedness structures the sequence as, precisely, a row, which is to say, an interval of visual form. In being bounded, in allowing the viewer to retrace and repeat his itinerary, to see the same drawings again, to grasp the chain as a set of resemblances within limits, the row supplies the syntax and ground of rhythm. This is a rhythm that is proper to a subdivided interval in space. Inseparable from the lateral combinations, the perception of rhythm arises from the drawings when configured as an array.

In the vertical stack, or codex, as we have seen, the row is not materially present. The rhythm of the series is therefore not linear. The rhythm is that of a series of substitutions, and each replacement (substitution) interrupts and ends the display: the hand turns the page, uncovers a drawing for inspection, but then covers up the drawing. The lateral interval corresponds to one page and contains one drawing. In turning the page, there is no sensation of motion in space – no flow – from here to there.

The row discloses something that was hidden. In perceiving the rhythm, the reader becomes aware that a principle of construction – an architecture, a plan – runs through the whole series – all of the intervals – and binds together all of the individual pages. The underlying architecture favours symmetrical structures. ‘Rhythm is the architecture of being,’ Senghor says (Harney Reference Harney2004, 46). In the drawings, symmetry is the architect of rhythm.

The rhythm of the row derives from the mix of similarity and difference in the visual elements of the panels. But this mix is governed by a principle, as I have said. The behaviour and dialogue of the drawings cannot be arbitrary. The rows structure the series as an order of repeating intervals, and thus impose on the series the visual logic of sets – i.e. chains of similar objects. But prior to any such logic imposed through the format, the series exists and exhibits repetition. Visual elements do in fact reappear and recur, forming patterns – although buried in the vertical stack. The awareness of these elements, and the symmetries they exemplify and break, is shadowy, inchoate, ephemeral, inarticulate – victimized by oblivion. The reader cannot sustain knowledge of the elements. The row, the measured flow of its rhythm, brings us closer to that knowledge.

Dispositions of the Line

Moving from wall to wall in a row, and then down through other rows, the viewer notices the recurrence of elements. But there is a more basic recurrence: the panels. If you strip out the marks from the panels, you are left with the caption. If you delete the caption, you are left with empty space inside a frame. The series rests on this basic design. What recurs is the rectangle, projected onto the page and multiplied.

On the page, the drawings are arranged in a continuous row. The abstract rectangle functions as a frame, surrounding an interior surface. The frame positions the drawing somewhere, localizing it, disposing it relative to the borders. Contemplating the drawings in turn and consecutively, the viewer perceives rhythm. The rhythm is experienced as material flow, whose substance unfolds in terms of duration, speed, amplitude, momentum, inertia, acceleration, cadence, and force – the behaviour of physical things.

In general, rhythm is the perceptible disposition of finite elements organized in a series. And given any series, like the drawings, it appears that the disposition of elements might be catalogued, the rhythm explained. Where does the rhythm come from? Can we explain the rhythm as a function of discrete visual parts? Can we identify such parts, observe the manner of their combinations, and classify their patterns? As mentioned previously, the experience of viewing the drawings – the flow of rhythm – suggests an analogy with verse.

The rhythm of verse arises from recurrence, the return of metrical units and values in the line: the metrical foot, and within the foot, the syllables, or syllabic values. Perhaps verse prosody can help analyse the drawings. There would have to be a corresponding taxonomy of parts: the row = verse; the panel = foot; and the visual line = syllable. By this taxonomy, the row of drawings is a pentameter of five visual feet. Like verse, the rhythm grows from the measure. But does the analogy hold all the way through? Prosody halts before the enigma of the line. Can the lines be counted – numbered – in the measure? Can the line function as a unit of display, in the way that the syllable is a unit of sound?

In looking at the drawings, when one tries to compare the metrical role of the line and the syllable, the analogy falters. Under scrutiny, the line is elusive. The line is an act of drawing. Drawn by hand, the limits of the line, specifically, are not replicas. This is because the line has no part in a visual object that exists at higher degree. There is no ‘lexicon’ of visual objects, outside of the array and held in common by all of us, from which the limits derive. Instead, the limits of the line are disclosed by the behaviour of the line itself: the line bends, or stops, or multiplies. Prior to bending, the line has no history.

A line that continues on a straight vector is akin to an arbitrary trace, a mere signans, a mark of presence. The line is a path. When the line bends, or stops, or multiplies, then it reveals itself, reveals its limits, reveals its disposition, the possibility of becoming a unit, the possibility of repeatable form.

The rhythm of the drawings arises from the recurrence of elements. But the line is elusive, a residue of the hand, a gesture. Let us make a distinction. In the pentameter of the row, the viewer sees a recurrence of parts: the frame, the surface in the frame. The viewer also sees lines, but the lines do not appear as recurring units. Instead, what recurs in the panels, from the perspective of prosody, is the mere line, the scoring of the page with visible traces – marks. The visual foot, the discrete unit of measure, is an amalgam of the mark, the surface, and the frame. The line is secondary, added to the mark, like a supplement. Devoted to mimesis, the line exceeds the mark and emerges from the mark as form. Needless to say, this phenomenology of the visual line maintains a separation and a distinction in purpose and function between the mark and its supplement, the line, a separation that is veiled by the drawing itself.

Rhythm involves the behaviour of the lines. As distinct from measure, the rhythm arises from changes to the disposition of lines in the drawings, in passage along the row. Subject to scansion, the row exhibits the lines in various perceptible dispositions: level or bent, rising or falling, high or low, right or left, open or closed, one or many, in parallel or oblique, abstract or mimetic, and so on. Encountered in the row, the changing dispositions of the line form the elementary parts – the elements – of the visual rhythm. Since a line’s disposition changes by degrees, the elements of rhythm are not dichotomies but contrasts: more open, steeper still, with fewer companions, etc. Like the linguistic sign, the elements have no inherent content. The elements are values, produced by negation, within the system of architecture.

The prosody of form shows that the rhythm follows from the progress of visual elements. The value of these elements is not established anywhere except in the space of the drawings. The values appear in the transition between drawings, in virtue of the contrast with other elements, themselves degrees or intervals of difference, and so on. There are contrasts beyond the perceivable horizon. The number of visual elements is probably infinite.

The elements are not only not fixed in advance, they multiply with progress through the series, both because new panels join the array, and because perception grows more sensitive to contrast. The longer the series the more space for elements to develop serially. But in reviewing the panels, and scanning the rows, the viewer’s awareness of elements also grows more precise. He perceives more elements, finer degrees and modes of replication. He experiences more subtle aspects of form. He is more sensitive to the exchange between recurrence and deviation: the repetition of non-identical parts.

Prosody provides only the framework of rhythm: the metronome. In the hesitation that gathers force, in the echo that calls forward, in the balance of forms and the tendency to interrupt that balance, in the flickering and clarity of pattern, in the detours of vanishing lines, the elements of the rhythm subdivide again and again.

The array is fixed, but the visitor scans it row by row. The arrow of time having been nullified, he may go back and start over. At some point in the itinerary, walking back and forth, the (sym)metrical form of the array seems to him to harken to echoes of long ago. Visual rhythm blends with the voice. No longer can he extract an individual wall from the row, or the row from the whole array. In the very act of repeating a scan, he multiplies the elements and their echoes.

The state of synaesthesia endured by the mind attests to Symmetry. We might even say that the array is a symbolist device for recalling the unity of the mirror and the echo. The rhythms are both seen and heard, and felt – like sensuous mathematics. The Greek word graphé means both writing and drawing.

Cathedral (2)

In the cathedral, the echo is formed by symmetry. In the array of walls, symmetry generates echoes. The visible marks on the page speak through context – like the walls of a cathedral. Seeing a drawing, the previous drawings are context. The word Wall under the drawings is context. The wall out there – Belfast, Berlin – is context. All the contexts collaborate in the drawing, its echoes, how it seems, how the rhythm flows (‘the feel of the thing’). Behind the echoes is symmetry. There is the breaking of symmetry in the imperfect reappearance of the line. The face of the drawing is masked.

As for the pedestrian, standing in the doorway, beneath the arch of saints, he senses that the harmony of the block – the row – was designed. The form of the series is not the result of chance. Nothing like the exact proportions and spacing of the cathedral, the multiple identical columns, the equidistant ranks of the pews, the framed faces of dead prelates, in perfectly aligned rectangles, height matching height, width width, along the aisles, and all the rest, confirming the rule of symmetry over the city, but still the sequence of facades unveiled to the pedestrian, after the formal interior, has the look of an ordered set, assembled by symmetry.

There is more than one set, he thinks. There is, it is likely, an array of sets, visual elements divulged through the city in echoes, unplanned by the architect.

From the principle of symmetry derives the possibility of the labyrinth. The cathedral is how he comes to the idea.

Andreas’ Device

In a seminar room in Freiburg, not too long ago, were gathered a company of European academics. They assembled to debate the meaning of symmetry to the disciplines of the liberal and practical arts: music, mathematics, philology, history of art, architecture, philosophy and science. The discussion continued for three days. Each morning stirred the appetite for more. The master of physics spoke of narrative models; the neuroscientist of culture as a transitional object, placed between the womb of the mother and adult melancholy. Someone supplied the diagram of the Labyrinth as seen by Theseus, dwarfed by the walls, and by Daedalus, who soars above them. We were in a building of luxurious discourse – a cathedral ornate with ideas.

Meanwhile, across the Mediterranean, in the Levant, the world was breaking. The refugee boats had set out from demolished entrepôts, on their way to intact nations. Some capsized and plunged under the sea. Others reached the beaches and borders of Europe to confront the hostility of crowds. The broken world had no symmetry. The balance of power was tilted. This was too obvious to matter. What mattered was the setting up of camps. Whether to admit a million, or a thousand, or none. Whether to alter national policy in response. Whether to build a wall.

In the seminar room, what mattered was not what mattered outside of the room. In the seminar room, the scholar recalled us to symmetry. In a previous epoch, he said, symmetry had to do with proportion and measure. The etymology, in metron, points to this usage. Symmetry means observing the calculus, the measure, the balance conducive to harmony in a physical structure – a basilica, let us say. Absolute quantities – length, width, height, circumference, angle – are less important than ratios: the width of the transept to the length; the height of a barrel (column) to its radius; the slope of a ceiling relative to the density of prayer; the spacing of buttresses.

Picture a naked man, with arms and legs extended inside a circle, how the feet and palms rest on the circumference, and you will have seen the human form of symmetry and the Golden Ratio.

Expressed numerically, these ratios of symmetry can be ridiculously precise: 23/29 of 125/131. The surveyor cannot register such fine discriminations. Make the ratio of (A + B) to (A) the same as (A) to (B). This means: 1 to 1.6180339… Such symmetries are hidden.

Symmetry is related to rhythm through the concept of measure. In verse, rhythm is defined as ‘the measured flow of words and phrases’ (‘Rhythm’, definition 1.1.a, Oxford English Dictionary 2019). These words and phrases are combined and ordered by symmetry to evince or educe patterns of fluidity and harmony. Like a cathedral, verse has an architectural design, with a bias toward proportion, equivalence, and measure in the parts (syllable, foot, line, and stanza) – the work of symmetry. However, to turn things in the other direction, an architectural structure is said to possess rhythm. In the case of a basilica, for example, there is a rhythm to the combined spatial elements (distances, volumes, masses, colours, shapes, textures) surveyed by the American tourist. Properly sequenced, the forms are perceived to blend harmoniously together and to flow.

Like an architectural interior, like metrical verse, the walls drawings have flow.

Later on, as the seventeenth century became the eighteenth, and time passed, the pursuit of symmetry alighted on other characteristics of physical structure. Instead of an obsession with proportion, the concept of symmetry came to mean replication. This idea of symmetry is exemplified by the mirror, in visual aesthetics, and by the inverted pendulum, or metronome, in poetics. In the design of a modern city on the ruins of the old, the architect searches for balance, but like a face bisected on the axis of symmetry. A mirror replicates the body. One tick of the metronome is equal to every other tick of the metronome. Alexander Pope translates Homer in strict iambic pentameter.

On the final morning in Freiburg, a mathematician offered a definition of symmetry to satisfy the scientists and philologists, the students of music, art, architecture, poetry, the experts in narrative and logic: ‘invariance under substitution’, he said. In the idea of invariance (I thought) begins the symmetry of mirrors, and more, the symmetry of clocks, the symmetry of orbits, the symmetry of pendulums, the symmetry of unified forces, strong and weak, the symmetry of walls, the symmetry of drawing, the symmetry of repetitive series, the symmetry of string, the symmetry of the heart. Verse is an array of replicas: vowel for vowel, foot for foot, line for line, stanza for stanza, echo for echo. The repetitions that are fundamental to the array of walls harken to proportion and invariance. Symmetry pervades the visual field.

Symmetry of the Walls

The array has fixed and repeating ratios, which frame the drawings. The viewer sees a wall through the ratio of one drawing per page (or 1:1), or five drawings in a row (5:1), or five drawings in three rows each (5:3). How the series frames the wall – the ratios – is intrinsic to what you see. Different ratios produce different rhythms.

The array has the property of equivalent measure: one panel equals one foot; one row equals a pentameter line. The foot, repeated, displays degrees of contrast by length, bending, angle of slope, direction, interval, and number in the line. Proportions matter. Degrees are infinite. The elements of form constitute the rhythm.

The array has the property of ‘invariance under substitution.’ And so we come to the general definition. Every panel shows a wall, every row shows five walls. The same wall. The caption is an ally of symmetry.

The symmetry of lateral displacement (translation) is a type of invariance by position on the axis of combination: this wall beside this wall beside this wall beside this wall. This symmetry underlies the monotony of thresholds. How much longer.

The symmetry of mirrors and mimesis is a type of invariance on the axis of resemblance: this wall looks like a wall; this wall looks like every wall. The row is a materialized sequence of resemblances.

The look of a wall is influenced by the symmetry of all the walls. As a series, the walls imitate, combine with, and signify other walls – walls of the array, and walls in Belfast, walls around cities, walls between rooms, walls in Berlin, in the labyrinth, in between nations, between temporary camp and city, walls in imaginary places, like the habitat of distant primes, or the landscape to plot zeroes in the Riemann zeta function. A mathematical series exists as a projection from the mind onto lamina. The representation of an imaginary and infinite array can only be partial. The walls of the array exist on paper, as a fond projection of symmetry.

I looked up. There was Andreas with a device (Figure 2). The device had a long arm, divided unequally about a pivot; to this arm had been affixed a second arm, shorter than the first, also unequally divided. Both arms were able to rotate. The second arm pivoted on the first, but it influenced, in return, the spinning of the first arm. Andreas nodded, but when he spun the pendulum and explained its motion, the word (symmetry) seemed to mean the order of cause and effect, whatever allows the mind to forecast the future across an interval crowded with discrete influences: predictability. In my mind I saw that symmetry is a property of objects handled by mathematics, in the tongue of symbols. To ask Andreas – a physicist – to explain symmetry is to require a translation, not between two languages, but between exclusive paradigms of the real.

Figure 2.

Hand-drawn illustration of Andreas’ device.

Take, as an illustration, elementary particles. The word ‘top’ in ‘top quark’ has no meaning in ordinary speech. It does not mean, in French, or Hindi, or Latin, or Xhosa, the quark on the top, the highest or tallest quark, the best quark, or the boss quark. If you extract the phrase ‘top quark’ from mathematics, it explains nothing about quarks. The metalingual inquiry produces no content.

Sinner: But what is a strange quark?

Echo: Strange quark.

(Scholion: Dixit quark nec summum sed vero non solum.)

Sinner: That’s what I asked.

Echo: Ask.

Sinner: What is a bottom quark?

Echo: Bottom quark.

(Scholion: Nec aliemum nec non summum est.)

Like the quarks, symmetry is not what it is not.

Andreas saw the problem of translating symmetry into our speech, and tackled it by visual experiment – by spinning the arms of the pendulum and pointing to the pendulum. We watched the spinning arms hypnotically, like strange propellers, affixed to a column, arms with opposite spins, clockwise and anti-clockwise, accelerating unevenly and slowing, both arms in balance, slowing and falling, moving together in rhythm and conflict: the symmetry of unequal levers.

However well-defined the initial position of the two arms and the force of the push, the rhythm of the pendulum enters a labyrinth of unstable vectors, a chain of minute deviations from predictable behaviour, so that the future of the pendulum ever more widens into a probability fan of incalculable width. The symmetries are ‘hidden’, Andreas said. There is no perfect symmetry in a pendulum. There is no unbroken symmetry in a wall drawn by hand. One day I will ask Andreas to explain.

Labyrinth

By now you will have noticed that the walls of the matrix or array, the medieval city, the cosmos, the cathedral, the refugee camp, and the double-pendulum have symmetry in their design. Because of their foundation in symmetry, all these forms are alike. They mimic the Labyrinth that Daedalus built.

In my Walls series, the way echo calls to echo calls to echo calls to echo assembles the counter-format, the one in the mind, held in place by memory. The wider and deeper the storerooms of the mind the more it holds of turns that confound the hero, like a city in film noir. The pedestrian comes down the steps.

The Labyrinth develops in the journey – paths scrawling in the night like barbed wire, a tangle of angled paths, crossing the boulevard obliquely, repeating elements that differ by a molecule, paths that seem bent in the distance, paths on top of walls, paths made of mirrors and paths made of amphoras, and paths falling like the atoms in the pre-history of the sun-warmed cosmos, as told by Lucretius.

Symmetry fabricates the Labyrinth. But the Labyrinth is cunning. Born in symmetry, its symmetries are hidden.

This is the dilemma of Theseus, inside the Labyrinth. The hero acquires knowledge of the symmetries (so as to elude them) by walking the streets and doubling back and checking his impressions. Each row is an interval of rhythm. Symmetries escape. The rhythm soothes his fear, temporarily. He catalogues what he can. The visual echoes multiply. One wall equals a wall. Inside the cathedral, he prays to the cathedral. One wall equals a wall. The series inspires terror without limit. Theseus has travelled a great distance – immeasurable. His mind holds so many echoes. And yet, the journey ahead of him is equally long – infinite? At first enjoying the liberty to go where he likes, even to revisit previous rows, he now believes that the walls are endless. In the farthest regions of the array, the hero experiences the monotony of thresholds – twilight, moats, doorways, altars – leading to thresholds. Beguiled by nostalgia and synaesthesia, he remembers the sound of a drawing, the cadence of falling light. There is no symmetry between departure and return.

Whether in virtue of his laborious manoeuvres, obedient to the walls left and right, the hero extracts the plan of the array from his first-hand encounters and so climbs the air in his mind to the bird-path of Daedalus – the engineer as visionary – and depends on his mastery of the impossible series. Such mastery is not guaranteed. Absent the ordeal of symmetry, what brands him a hero? Theseus is just an average pedestrian until he conquers the device.

The perspective he now finds needful is that of the architect. Unlike the pedestrian, trapped in the knot of detours, the architect soars above the Labyrinth. Look down on the array: this is what the architect sees.

To this patriarch, the cunning passageways reveal their itineraries; the structure of beginning, blind alley, detour, and return, and the exit – all of this is visible. Echoes are immaterial. The observer may choose where to start: at the centre or edge, or in between. No peril surrounds the choice. The penalty for error is suspended, and any error can be redeemed. The eye leaps over walls, which, from his point of view, have no height: one passes from passage to passage without effort.

At the eye-level of Theseus, this disregard appears as the ability to pass through the walls like air, as if the walls of the Labyrinth are merely mathematical, or metaphors – echoes at best.

The architect has built symmetry into the Labyrinth, crafted a figure that expresses rhythm, like the Walls. But that is not all. Since the Labyrinth was built by hand, imperfections mar the lines and break Symmetry. In this breaking is the beginning of symmetry, the founding order of the architecture, that which had been imagined – overturned. The cathedral was built by stonemasons.

The future of Theseus is restricted to an itinerary dictated by the form he enters. Presumably he retains free choice, to turn this way or that, but within a maze where whatever he chooses has already been planned, precisely defined and fully articulated within the structure, and the set of choices – all but the proper chain of choices – designed to frustrate. Theseus cannot outwit the labyrinth: it already foresees his departure, generations hence. The limit of his resourcefulness corresponds to a path that the labyrinth already supplies. Accordingly, Theseus can be equal to the Labyrinth, but no more. There can be no victory over the structure.

Monotony of Thresholds

The Berlin Wall stood for 29 years, from 1961 to 1989. Each series of the Walls drawings has 29 panels. There are walls in Belfast, another city of troubles for which the built barrier promises diminished strife. But does not deliver.

In 2016, I visited Berlin and toured the Stasi prison compound, inside the wall of the prison, the exercise yard and offices with telephones: the labyrinth of cells beneath. In previous years I had toured the walls in Belfast and seen the beautiful graffiti murals and mythology: the art of walls. I know of walls, having grown up in Kingston with the unwalled garrison communities face to face. A wall divides, all right, the most intimate tearing.

Like an infinite series, the array of walls runs on indefinitely. The tale of years unscrolls from left to right, or top to bottom, or obliquely, exhibiting new patterns. The array is a machine for generating infinity – a labyrinth of echoes – from finite representations.

In a Medieval codex, if you look at a page, then turn it over, then look at the recto, you see a story organized in one series. Each episode has beginning, middle, and ending. To turn a page closes the face, and begins a new episode. The recto dominates the horizon. Closing the recto reveals the contiguous drawing that had been hidden on the back, as if it were bones beneath the surface.

The array (labyrinth) multiples echoes, but promises a way out of them. The torment of the labyrinth is precisely this ruse.

The story is that the people came to the temporary camp, in the time of their ancestors, leaving their belongings at home. Soon they would leave to return, or to settle in a new land. There will one day be an exit. The symmetry of the unknown is their future.

Behind a wall, whole lives pass in obscurity, as if they had not been, like bones rhythmically ironing a shirt, monotonously, like a pendulum, back and forth in a dream.

As a type of labyrinth, the pendulum obeys the law of angular momentum. One pendulum affixed to the arm of another – Andreas’ Device – defeats calculation: too much to consider. The symmetries are ‘hidden’ (Andreas’ word), like a dream within a dream.

As for verse, the metronome imposes fidelity where none is desirable. The tick of mark echoes the tock of no-mark, duration and betrayal. And held prisoner by the symmetries, the rough tongue succumbs to artifice.

The poet might observe that lyric verse began in the green shade of a mountain, on the rainless abdomen of the desert, terrain of overloaded beasts: camels, horses, donkeys, human captives, slave-girls and boys fetching water, dancing in the halo of a fire.

In time, the line becomes infinitesimal, just as it is in physics, no more than a representation of thought. Walls are internal to the body. The stopping-place is told by political history.