Mountain Geometry

by Eric Miller


On occasion, a long epoch of concord with a favouring breeze may seem to grace us: inspiration in the sense that birds must relish it. What a divine—almost avian—thing it was for us, the hotel kitchen staff, to pack cheesecake, kiwifruit and champagne into our rucksacks, to tighten the straps that secured these dainties on our shoulders, and to climb right from the back exit with its tubs of lard to the stark summit of a Rocky Mountain in Alberta.

We felt we flew or, to speak more precisely, scudded upward. I wore shorts and a clean undershirt and sneakers with no socks, my colleagues wore nothing more substantial, and we scaled the steep flank of the darkling peak as though, like magpies, we half hopped, half sailed, never shaking off an appearance of indolence in spite of the winged celerity of our ascent. We might have kicked altitude away beneath our flexing feet, bubbling giddily like divers whom ebullience, in shimmering snorts of submerged laughter, expedites to the water’s surface.

After night fell, a thunderstorm broke out well below us in the valley. We were dry. We watched the lightning flash and fret; it resembled the dome and tassels of a jellyfish aglow in a cove. The spectacle of electrical unrest promoted our repose. When one couple began to thrash in a single sleeping bag, they rejoiced us, intimately clustered on that narrow summit, with the audible excess of their droll yet solemn ecstasy. The sounds they made as they scaled in duet the scarp of their pleasure amounted to musical improvisation, disclosure rather than presumption of form: a gasping flag or panting plume to mark—to augment—the height of the mountain and our happiness. Making instruments of each other, musicians of each other, they performed for and they warmed us. Sonorous fire!

The next morning we woke to find ourselves above a mountain lake and its crisp ripples turned like prisms, rose, golden, kiwifruit green, sky-blue. We dashed into it naked through frigidity fizzier on the nerves than champagne on the tongue. We might have crashed past pane after pane of stained glass, without cutting ourselves. The commotion we made among these aqueous fractals simplified instantly after we subtracted ourselves. A Varied Thrush (the V of its name visible on its breast) improvised in the mist, a series of whistles each on a different pitch, expressive (as it seemed) of the varying altitudes and perspectives from which it sang.


I brought Euclid’s Elements with me up that mountain, hoping for elucidation of angles acute and obtuse I suppose. The geometrician flourished in Alexandria, around 300 BC. I hardly had the opportunity to consult him. I did glance at the first three definitions Euclid offers. They seemed easy to apply.

A point is that which has no parts. Often when happy we feel as though we have no parts.

A line is length without breadth. When we scamper up or down a slope, we seem ourselves an entity without breadth—breathlessly so.

The extremes of a line are points. At the point when we left the hotel kitchen, we were full of extreme misgiving. On arrival at the top of the mountain (with its paired climaxes, electrical and sexual), we were pleased with our achievement. Still, the prospect of resuming what we had left behind (dishes to wash and dishes to make) already troubled us.


One day—it was many centuries ago—two Athenians happened to discuss the theme of virtue. Socrates? He was an ugly old man. Meno? Young and attractive. Can virtue be taught? This was the topic of their conversation.

Socrates warned Meno that a sincere inquirer into this subject can’t just break the concept of virtue into pieces, and say that there is one virtue proper to a politician and another to the tailor who mends your jacket. To split up virtue in this way (said Socrates) is like shattering a plate. It’s of no use then. (His image amuses me, since I have shattered my share of dishes. I have broken crockery on the job, and off the job. Too easily it smashes.)

Socrates and Meno did not debate the nature of virtue as such. Instead, they kept arguing about whether someone could really ever teach it to anyone else.

At this point, the old philosopher called over a child—a slave belonging to Meno. Socrates took a stick. Drawing a diagram in the dust with the sharp end, he asked the child, who could speak Greek, to solve a problem. How would the child go about determining the length of the sides of a square of a certain stated area? In order to arrive at the answer, it turned out the child needed to use diagonal lines as well as right angles.

Afterward, Socrates told Meno the child had not learned a thing. He argued instead that the child had recalled every bit of the geometric knowledge that allowed him to solve the problem. Not from any lessons received in this life! No. Like us (so Socrates claimed), the child relied on cosmic reminiscence. He remembered—and could remember—everything. Socrates explained that this was because our soul is immortal. At one time it dies. It is born again at another time. All seeking and all learning are in truth recollection from past lives, even from the intervals between past lives.

The English poet William Wordsworth may have had Socrates’s theory in mind when he sat down to write his “Ode: Recollections of Immortality from Early Childhood.” After all, in this poem, which he began in 1802, he says: “Our birth is but a sleep and a forgetting.”


If you have heard of Thomas De Quincey, it is probably because he confessed in 1821 to being an opium eater. He also knew the poet William Wordsworth well. According to him, Wordsworth “was a profound admirer of the sublime mathematics; at least of the higher geometry.” De Quincey confides, “The secret of this admiration for geometry lay in the antagonism between this world of bodiless abstraction and the world of passion.”

De Quincey divides passion from abstraction. I will not follow him: not just now. To explain the grounds for my refusal, let me ask, “What does the word ‘geometry’ even mean?” The answer is, “measurement of the earth” (γῆ + μετρία). The origin of the word lays bare the basis of the ideal art of geometry in the act of surveying real terrain. Wordsworth himself argues in the preface to his 1800 Lyrical Ballads for the entanglement—even unity—of geometry with feelings. “The passions of men,” he says, “are incorporated with the beautiful and permanent forms of nature.” Repeated exposure to these shapes deepens emotion, he says, rather than blunts it. Wordsworth has in mind his home territory—the Lake District of England, imprinted with mountains and valleys. An elevated prospect hoists stirringly into view the latency of the “higher geometry,” of “sublime mathematics.”

As it happens, recent authorities concur with this claim. Alexandru Kristàly has worked out a “mountain pass theorem” in a treatise on variational principles. J.R. Sack and J. Urrutia in their handbook discuss the angular geometry of peaks, pits and line features—ridges and ravines. P.M. Petroff shares in his topographical work the insight that “Whenever the processes shaping a landscape favour the growth of sharply protruding features, channels develop amphitheatre-shaped heads with an aspect ratio of π.” What about James Harris? Well, Harris models for his amazed reader how fractal geometry can form mountains.


Thomas De Quincey was lucky. He heard Wordsworth read from The Prelude, decades before this poem appeared. De Quincey says he loved a passage that described a dream. It could not be more sublime. It illustrates “the eternity and the independence of all social modes or fashions of existence” granted to “mathematics on the one hand, poetry on the other.”

What De Quincey remembers is an episode from Book 5. Wordsworth’s narrator pages through Don Quixote beside the ocean. Weighing the perishable book, he begins to brood. Where else is knowledge to be found? Only in “shrines so frail.” The vulnerability of material culture! It comes on him like a wave. Turning his gaze toward the tide, he thinks of poetry and geometric truth. He would grant those things the “privilege of lasting life.” He falls asleep. He dreams he sees a figure hurrying over desert sands with a stone and a shell. The spectacle relieves him, for the dreamer believes himself to be in the presence of a guide.

The stone, without relinquishing its stoniness, somehow embodies Euclid’s Elements. As for the shell, Wordsworth says it has voices more than all the winds, with power to exhilarate the spirit and to soothe the heart. It is literature. Yet at the same time, it is just a shell. From its empty interior issues a prophetic blast: an ode announcing destruction to the children of the earth. Poetry’s oracular vessel—the shell, the sounding organic calcareous accretion of sea and seascape—makes time manifest: tempo, pacing, the past, the present and the future.

Geometry, the science mysteriously condensed in the stone, governs space. Still, we continue to grasp clearly enough that the stone also remains an unregenerate chip of our globe—a representative fragment, it may be, of the Lake District: a synecdoche for the rocky mountains Wordsworth loves.


If Wordsworth’s Prelude remains known for anything today, however, it is for a different claim in behalf of time and space:


There are in our existence spots of time

Which with distinct pre-eminence retain

A fructifying virtue, whence, depressed

By trivial occupations and the round

Of ordinary intercourse, our minds

(Especially the imaginative power)

Are nourished, and invisibly repaired.


Wordsworth’s phrase “spots of time” assumes the inseparability of the spatial from the temporal. Space over time—over a lifetime, the life we impart to time—becomes (so our poet claims) wholesome, reparative. All this, even though Wordsworth’s own spots of time amount to a sinister group. Items in the array centre or spiral, for example, around a visit to the site of a mouldered gibbet or to a lake from which a dredge retrieves a swimmer’s rigid body. Whatever else may be true of them, these spots happen—keep happening—in a landscape specified by geometric law. James Harris may help us here. When a chaotic process has generated a form (he says), a fractal character is the basis of that form. Copying itself, nature comes up with something else.


In Book 6 of The Prelude, Wordsworth remembers his first acquaintance with Euclid. He calls the relations and proportions proposed by geometry “nature,” and he calls nature a “leader” of the mind. The callow mathematician reports how, in the course of his inquiries, he felt each increment of his ignorance exquisitely pass away. To gain experience—to be experienced—can be as pure as people say innocence is. And doesn’t innocence keep growing proportionally to the growth of knowledge? Drag and lift are both necessary to the flying bird.

Wordsworth recalls his university days. Not the best scholar, not the happiest. But he learned to heighten still farther “the pleasure gathered from the elements/Of geometric science.” Transcendent peace accompanied this study. He says it was a comfort to him in those years. The narrator digresses into an anecdote, apparently about a castaway. “I have read,” he says,


of one by shipwreck thrown

With fellow sufferers whom the waves had spared

Upon a region uninhabited,

An island of the deep, who having brought

To land a single volume and no more—

A treatise of geometry—was used,

Although of food and clothing destitute,

And beyond common wretchedness depressed,

To part from company and take this book,

Then first a self-taught pupil in those truths,

To spots remote and corners of the isle

By the seaside, and draw his diagrams

With a long stick upon the sand, and thus

Did oft beguile his sorrow.


Wordsworth freely adapts elements from the life of John Newton—a slaver who developed, by shivering fits and involuntary starts, into an abolitionist.

Shipwreck, you see—though often on voyages it threatened him—, never stranded Newton anywhere. Deserting the navy (he had been press-ganged into it), Newton joined the slave trade. Beginning in 1745, for a period lasting eighteen months, his boss in the business, Amos Clowe, angry with him, condemned him to the chains and nakedness of any one of their African captives. Newton clutched his only possession, a copy of Euclid. Off the Guinea coast, on an island two kilometres in circumference—it went by the name of the Platanes—the enfeebled young man gouged out geometric proofs at the tideline. The episode savours, sea-changed, of Shakespearean romance. Ariel might have sung, “Come unto these yellow sands/Then shackle hands.”

Euclid preserved Newton. Sun-blistered, wasting away on a pint of rice a day, he wielded a doddering wand, diagrammatized a beach. Prospero underwent the pains of Caliban, who also hid, when he could, in “spots remote and corners of the isle.”

Wouldn’t you guess Newton’s eighteen months of atrophy and torment would inspire an instant revolt against the practice of slavery? For that matter, does it dumbfound you that Meno doesn’t free the young slave? Neither he nor Socrates denies their geometrician an immortal soul. Laying down his stick, the old philosopher seems to teach us to omit to draw the inference, liberty.

By 1787, John Newton did at last stand against the slave trade. Yet, languid in coming to a firm resolution, he had lagged. Euclid, mage of parallels and parities, served in 1745 to assuage Newton’s sea-salted hungers and lesions. But geometry was only instrumental in keeping madness in abeyance for the sake, we may say, of a procrastinated renunciation. Some problems take time to figure out. Escaped the Platanes, he even returned to slaving. The implications of his life dawdled in making themselves felt: as slow as the tempering, seasoning, intensification undergone by a Wordsworthian spot of time.


Born in 1743, Governor Toussaint L’Ouverture came to administer an island much greater than the Platanes: Haïti. He helped end slavery there in 1793. In 1802, however, the French General Jean-Baptiste Brunet, after issuing a hospitable invitation to L’Ouverture, arrested him instead, near the Haïtian town of Gonaives. The prisoner, detained on the frigate Créole, was deported to Europe. Governor L’Ouverture had declared his country a republic.

Dismayed by news of L’Ouverture’s removal, Wordsworth wrote a sonnet promising “there was not a breathing of the common wind that would forget him.” Pleasant to think so. Does a spot of time recall us? The difficulty of equations can stump us longer than the length of prison sentences. L’Ouverture (Wordsworth insists) had great allies on his side, including earth and air.

His wardens at the Fort-de-Joux—a castle more than a thousand metres up in the Juras, close to the Swiss border—used a piece of L’Ouverture’s skull as a paperweight. It secured some excuses on the mantel of the room where he died in 1803. A draft might otherwise send them flying. Toussaint L’Ouverture, you would have to believe, had laid up a vast treasure in Haïti. Not just that. Proportionality, after all, is the essence of the law. He conspired to hide it—so they lied—in the mountains.


Among the axioms that Euclid allocates to the category of a “common notion” (κοινή εννοια), he offers a very familiar one: “The whole is greater than the part.”

The difficulty lies in that adjective, “greater.” What is greatness? Is not a part the equal of a whole? “Things which are equal to the same thing,” says Euclid, “are equal to each other.” That is not as clear a statement as the geometrician seems to think it is.

Fractal geometry teaches us that, in nature, certain figures no sooner form than they are copied. The process of replication goes on. The parts are similar, owing their common origin to chaos. Yet few of us could easily conjecture, in a given case, the character of the prodigious whole from our study of the part. Greater than! Less than! Who does not know the fickleness, the insecurity of the difference? Who can measure it?

“A boundary,” says Euclid, “is an extremity of anything.” Extremity! There are complacent as well as desperate extremities.

“A figure is what is contained by any boundary.” Where are the bounds? What binds us? This much feels sure: copying ourselves, something else comes up with us. Though I descended from its top so many years ago, the Albertan mountain onto which I once decanted a libation of effervescent wine still keeps me company. I agree with Dorothy Wordsworth, William’s sister. She walks by my side now, as truly as any of my companions of those days. (The immigration policy observed by spots of time is lax in the extreme.) “Wherever we looked,” she says, “there was a delightful feeling of something beyond.” But “beyond” is ambiguously distant, is it not?


Cornelius Varley painted the illustration, “Mountain Lake in the Sunlight,” in 1802.