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Pi and the Meandering Paths of Rivers

Einstein, fluid dynamics, and the beautiful mathematics of nature.

Pi and the Meandering Paths of Rivers

“Rivers run through our civilisations like strings through beads,” Olivia Laing wrote in her stunning meditation on life, loss, and the meaning of rivers. “There is a mystery about rivers that draws us to them, for they rise from hidden places and travel by routes that are not always tomorrow where they might be today.” But this chaotic, civilization-strewing mystery may be underpinned by one of the most elemental mathematical truths of nature.

In her splendid ode to pi, the Nobel-winning Polish poet Wisława Szymborska extolled it as “the admirable number… nudging, always nudging a sluggish eternity to continue.” On the second day of spring in 1996, the Cambridge University earth scientist Hans-Henrik Stølum published a paper announcing his astonishing finding that pi is also nudging, always nudging, the bendy paths of the world’s rivers to continue their seemingly chaotic meanderings — in a mathematically predictable pattern. His simulation, using empirical data and fluid dynamics modeling, found that the oscillating paths of rivers — their sinuosity, calculated by dividing the river’s actual meandering length by the length of the direct line drawn from source to sea — average 3.14.

The Tetons and the Snake River by Ansel Adams, from Ansel Adams in the National Parks.

Although crowdsourced data failed to replicate Stølum’s average, the crowdsourced sample was too small to prove or disprove a theoretical model addressing the total average of all the world’s rivers. What is more interesting than the finding itself is the very notion that pi — or any number — can undergird seemingly chaotic patterns in nature. Simon Singh takes up this question in his endlessly fascinating book Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (public library), which also gave us Pythagoras on the purpose of life and the meaning of wisdom.

Citing Stølum’s finding, Singh writes:

The number π was originally derived from the geometry of circles, and yet it reappears over and over again in a variety of scientific circumstances. In the case of the river ratio, the appearance of π is the result of a battle between order and chaos. Einstein was the first to suggest that rivers have a tendency toward an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which will in turn result in more erosion and a sharper bend. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist, and so on. However, there is a natural process that will curtail the chaos: increasing loopiness will result in rivers doubling back on themselves and effectively short-circuiting. The river will become straighter and the loop will be left to one side, forming an oxbow lake. The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth.

Complement with the fluid dynamics of Van Gogh’s “Starry Night” and mathematician Lillian Lieber on what Euclid can teach us about social justice, then revisit the story of how a Hungarian teenager revolutionized mathematics and equipped Einstein with the building blocks of relativity, without which he would not have been able to make his prediction about the curvature of rivers.

BP

Little Panic: A Literary Laboratory Exploring What It Is Like to Live in the Stranglehold of Anxiety and What It Takes to Break Free

“This terrible truth binds us all: fear there’s a single, unattainable, correct way to be human.”

Little Panic: A Literary Laboratory Exploring What It Is Like to Live in the Stranglehold of Anxiety and What It Takes to Break Free

“Life and Reality are not things you can have for yourself unless you accord them to all others,” Alan Watts wrote in the early 1950s, nearly a quarter century before Thomas Nagel’s landmark essay “What Is It Like to Be a Bat?” unlatched the study of other consciousnesses and seeded the disorienting awareness that other beings — “beings who walk other spheres,” to borrow Whitman’s wonderful term — experience this world we share in ways thoroughly alien to our own.

Today, we know that we need not step across the boundary of species to encounter such alien-seeming ways of inhabiting the world. There are innumerable ways of being human — we each experience life and reality in radically different ways merely by our way of seeing, but these differences are accentuated to an extreme when mental illness alters the elemental interiority of a consciousness. In these extreme cases, it can become impossible for even the most empathic imagination to grasp — not only cerebrally but with an embodied understanding — the slippery reality of an anguished consciousness so different from one’s own. Conversely, it can become impossible for those who share that anguish to articulate it, effecting an overwhelming sense of alienation and the false conviction that one is alone in one’s suffering. To convey that reality to those unbedeviled by such mental anguish, and to wrap language around its ineffable interiority for others who suffer silently from the same, is therefore a creative feat and existential service of the highest caliber.

That is what author, Happy Ending Music & Reading Series host, and my dear friend Amanda Stern accomplishes in Little Panic: Dispatches from an Anxious Life (public library) — part-memoir and part-portrait of a cruelly egalitarian affliction that cuts across all borders of age, gender, race, and class, clutching one’s entire reality and sense of self in a stranglehold that squeezes life out. What emerges is a sort of literary laboratory of consciousness, anatomizing an all-consuming yet elusive feeling-pattern to explore what it takes to break the tyranny of worry and what it means to feel at home in oneself.

Art by Catherine Lepange from Thin Slices of Anxiety: Observations and Advice to Ease a Worried Mind

Part of the splendor of the book is the way Stern unspools the thread of being to the very beginning, all the way to the small child predating conscious memory. In consonance with Maurice Sendak, who so passionately believed that a centerpiece of healthy adulthood is “having your child self intact and alive and something to be proud of,” the child-Amanda emerges from the pages alive and real to articulate in that simple, profound way only children have what the yet-undiagnosed acute anxiety disorder actually feels like from the inside:

Whenever I am afraid, worry sounds itself as sixty, seventy, radio channels playing at the same time inside my head. Refrains loop around and around my brain like fast jabber and I cannot get any of it to stop. I know there is something wrong with me, but no one knows how to fix me. Not anyone outside my body, and definitely not me. Eddie [Stern’s older brother] says a body is blood and bones and skin, and when everything falls off you’re a skeleton, but I am air pressure and tingly dots; energy and everything. I am air and nothing.

[…]

My breath flips on its side, horizontal and too wide to go through my lungs.

The grave paradox of mental illness and mental health is that, despite what we now know about how profoundly our emotions affect our physical wellbeing, these terms sever the head from the body — the physical body and the emotional body. A century after William James proclaimed that “a purely disembodied human emotion is a nonentity,” Stern offers a powerful corrective for our ongoing cultural Cartesianism. Her vivid prose, pulsating with a life in language, invites the reader into the interiority of a deeply embodied mind that experiences and comprehends the world somatically:

A burning clot of dread develops under my ribcage. One hundred radios are trapped in my head, all playing different stations at once.

Art from Emotional Anatomy: The Structure of Experience

“I was born with a basketball net slung over my top ribs, where the world dunks its balls of dread,” she writes as she channels her young self’s budding awareness that something is terribly, fundamentally wrong with her:

The kids around me are carefree and happy, but I’m not, and life doesn’t feel easy for me, ever, which means I’m being a kid in the wrong way.

You can’t see the wrong on my outside, but I wish you could because then my mom would get me fixed. My mom can fix anything; she knows every doctor in New York City.

And so Amanda is put through a series of tests. Although she is so small and slight as to be literally off the height and weight distribution chart for children her age, the medical tests fail to find the locus of her anguish:

I am a growing constellation of errors. I don’t know what’s wrong with me, only that something is, and it must be too shameful to divulge, or so rare that even the doctors are stumped.

Psychological tests follow. “Amanda equates performance with acceptability,” one clinician reports in the original test results punctuating the book like some ominous refrain of wrongness. Then there are the IQ tests. Growing up in an era well before scientists came to understand why we can’t measure so-called “general intelligence,” well before Howard Gardner revolutionized culture with his theory of multiple intelligences, the young Amanda does poorly on the tests — lest we forget, test-taking itself is an immensely anxiety-inducing act even for the average person unafflicted by a panic disorder. Deemed learning-disabled and held back a grade, she reanimates that first school day of her second second year in sixth grade:

The air is fresh, the slight coolness in front of each breeze carrying the smell of change and beginning, except I’m not changing; my worries keep repeating themselves, just like the rest of my life.

Looking back on this disorienting and rather punitive experience, Stern writes:

There was a version of me that felt out of alignment with who I really was. The adults’ version had me learning disabled, and the other version — mine — had me devoured by mental anguish.

It would be more than a decade until that mental anguish is finally correctly diagnosed as a severe panic disorder. But the intervening time — those formative years when one’s sense of self sets in as the child morphs into a young adult — is filled with a growing, gnawing shame of otherness. It takes root in the child’s conscience as she finds herself unable to learn to tell time. Her world is governed not by clocks and calendars but by countdowns tolling her acute separation anxiety — the suffocating dread of being away from her mom:

Away is what time is made of; away is counted in fear-seconds, not number-seconds.

[…]

Time moves everyone forward, but it’s always forgetting to bring me.

Art by Harvey Weiss from Time Is When by Beth Youman Gleick

Perhaps the most savaging aspect of anxiety is how it kidnaps its victims from the present moment and hurls them into the dungeon of a dread-filled future. Channeling the early experience that becomes an overtone of her young life, she writes:

Sometimes I feel like I’m watching a movie about myself. I am always in the future somehow, separated from my body, and it’s from there I feel sad for the moment I’m living. Soon this moment will be gone; it will turn into another moment that will go, and I think I must be the only person who feels life as though it’s already over. This is the weight I feel every time the sun goes down. No matter how hard I try to stop the feeling, I can’t. Even if I run from it, it meets me wherever I land.

At night, when I’m in bed, I try to hear the house sounds that comfort me: the low mumblings of my siblings, the tamped down warble of the radio, the needle’s skipped return over scratches inside a song, the ceramic clatter of plates being rinsed, and the first turbulent bumps of the dishwasher before it coasts into its varoom lulling hum. My mother’s voice talking on the phone curls its way to my room, and I pull it toward me, past the other sounds, and try to swallow it inside me.

Anxiety warps time and space for this young mind trying to navigate the world’s topography of dread:

When people try to explain that uptown is not far, or that a weekend isn’t long, it makes me feel worse, more afraid that my worries are right, and that the world I live in is different from the world everyone else lives in. That means I’m different, something I don’t want other people to figure out about me. Something is wrong inside me; I’ve always known that, but I don’t want anyone to ever see that I’m not the same as they are.

This sense of being a problem to be solved becomes the dominant overtone of young Amanda’s life, until it swells into the aching suspicion that there may be no solution to it at all — that she is doomed to a life marked by the wrong way of being human:

There is a way to be and I’m not being it, and I don’t know how to change. Is there someone I should be the exact copy of, and they’ve forgotten to introduce me? Or maybe a person is supposed to be a fact, like an answer that doesn’t change, and I’m more like an opinion, which the world doesn’t want?

This terrifying suspicion seeps into the fabric of her being, permeating every aspect of her life. It leads her into confused and conflicted relationships that distort her understanding of love and leave her with a version of the same question:

Is this what real life is then? An endless effort to match the story of yourself someone else tells?

Art by Lisbeth Zwerger from a rare edition of Alice in Wonderland

When she is finally diagnosed with a panic disorder that gives shape and validity to her lifelong experience, she meets her diagnosis with elated relief. (A century earlier, Alice James — Henry and William James’s brilliant sister — had articulated that selfsame elation in her extraordinary diary: “Ever since I have been ill, I have longed and longed for some palpable disease, no matter how conventionally dreadful a label it might have, but I was always driven back to stagger alone under the monstrous mass of subjective sensations, which that sympathetic being ‘the medical man’ had no higher inspiration than to assure me I was personally responsible for, washing his hands of me with a graceful complacency under my very nose.”) Stern writes:

I feel weirdly solid, like I’m a valid human being. I didn’t even realize my feelings were categorizable as symptoms. Panic disorder. The air is softer, expansive, as though the world has suddenly opened and is unfolding every opportunity my panic had once ruled out. Every single thing in my life now makes perfect sense: the connections I couldn’t bridge; the choices I couldn’t make; the strange switches the natural world and all its sunsets turned on and off in me.

From this deeply personal experience emerges the universal assurance that what doesn’t kill you makes you more alive. Stern writes:

Over my life I’ve worried so much and feared so many things, and though many of those things actually happened, here I am, still alive, having survived what I thought I couldn’t. I didn’t turn out the way I thought I would: I didn’t get married and I didn’t have kids, and the not-having didn’t kill me either.

[…]

We are all just moments in time, a blink in a trillion-year history, even if our existence sometimes feels endless.

Art by Derek Dominic D’souza from Song of Two Worlds by Alan Lightman

With an eye to the centrality of anxiety in her own blink of existence, she telescopes to a larger truth about this widespread yet largely invisible affliction that seems a fundamental feature of being human:

When did it start? It started before I was born. It started before my mother was born. It started when friction created the world. When does anything start? It doesn’t, it just grows, sometimes to unmanageable heights, and then, when you’re at the very edge, it becomes clear: something must be done.

Left untreated, anxiety disorders, like fingernails, grow with a person. The longer they go untended, the more mangled and painful they become. Often, they spiral, straight out of control, splitting and splintering into other disorders, like depression, social anxiety, agoraphobia. A merry-go-round of features we rise and fall upon. Separation anxiety handicaps its captors, preventing them from leaving bad relationships, moving far from home, going on trips, to parties, applying for jobs, having children, getting married, seeing friends, or falling asleep. Some people are so crippled by their anxiety they have panic attacks in anticipation of having a panic attack.

I’ve had panic attacks in nearly every part of New York City, even on Staten Island. I’ve had them in taxis, on subways, public bathrooms, banks, street corners, in Washington Square Park, on multiple piers, the Manhattan Bridge, Chinatown, the East Village, the Upper East Side, Central Park, Lincoln Center, the dressing room at Urban Outfitters, Mamoun’s Falafel, the Bobst library, the Mid-Manhattan Library, the main library branch, the Brooklyn Library, the Fort Greene Farmer’s Market, laundromats, book kiosks, in the entrance of FAO Schwartz, at the post office, the steps of the Met, on stoops, at the Brooklyn Flea, in bars, at friends’ houses, on stage, in the shower, in queen-sized beds, double beds, twin beds, in my crib.

I’ve grown so expert at hiding them, most people would never even know that I’m suffering. How, after all, do you explain that a restaurant’s decision to dim their lights swelled your throat shut, and that’s why you must leave immediately, not just the restaurant, but the neighborhood? If you cannot point to something, then it is invisible. Like a cult leader, anxiety traps you and convinces you that you’re the only one it sees.

In a sentiment that calls to mind poet Nikki Giovanni’s remark to James Baldwin that “if you don’t understand yourself you don’t understand anybody else,” Stern adds:

For better or worse, we can only teach others what we understand… Each person begins, after all, as a story other people tell. And when we fall outside the confines of our common standards, we will assume our deficits define us.

[…]

My fear and my conviction were the same: that I was the flaw in the universe; the wrongly circled letter in our multiple-choice world. This terrible truth binds us all: fear there’s a single, unattainable, correct way to be human.

Little Panic stands as a mighty antidote to that universal fear. Complement it with Catherine Lepange’s illustrated meditation on anxiety and Seneca’s millennia-old, timeless wisdom on how to tame this psychic monster, then revisit William Styron’s classic masterwork accomplishing for the kindred monster of depression what Stern accomplishes for anxiety.

BP

Marcus Aurelius on How Meeting Reality on Its Own Terms Helps Us Live Through Our Difficulties

“Accept everything which happens, even if it seem disagreeable, because it leads to this, the health of the universe.”

Marcus Aurelius on How Meeting Reality on Its Own Terms Helps Us Live Through Our Difficulties

“At bottom the whole concern of both morality,” William James wrote in contemplating the human search for meaning, “is with the manner of our acceptance of the universe. Do we accept it only in part and grudgingly, or heartily and altogether? … If we accept the whole, shall we do so as if stunned into submission… or shall we do so with enthusiastic assent?” The pioneering psychologist and philosopher was reaching across time, space, and cultures to perch on the shoulders of another giant of thought: the Roman emperor and great Stoic philosopher Marcus Aurelius (April 26, 121–March 17, 180), who had articulated this selfsame idea nearly eighteen centuries earlier.

James quotes Aurelius in The Varieties of Religious Experience: A Study in Human Nature (public library | free ebook) — his 1902 masterwork, which gave us his insight into science and spirituality and the four qualities of transcendent consciousness. The passage does not appear to endure in any translations of Aurelius’s writings currently in print, leading me to suspect that James himself may have translated it — like every intellectual of his generation, he had mastered Latin at a young age.

Aurelius, quoted by James, writes:

It is a man’s duty to comfort himself and wait for the natural dissolution, and not to be vexed, but to find refreshment solely in these thoughts — first that nothing will happen to me which is not conformable to the nature of the universe; and secondly that I need do nothing contrary to the God and deity within me; for there is no man who can compel me to transgress. He is an abscess on the universe who withdraws and separates himself from the reason of our common nature, through being displeased with the things which happen. For the same nature produces these, and has produced thee too. And so accept everything which happens, even if it seem disagreeable, because it leads to this, the health of the universe.

Art from Philographics by Genis Carreras — a visual dictionary of philosophy.

Complement with Marcus Aurelius on how to motivate yourself to get out of bed each morning, the key to peace of mind, what it takes to live fully, and what his father taught him about kindness, decency, and humility, then revisit Emily Levine on meeting reality on its own terms.

BP

Amanda Palmer and The Decomposers Cover Joni Mitchell’s “Big Yellow Taxi” in Tribute to Rachel Carson

Art and science meet resistance in a modern reimagining of a classic anthem for the protection of nature.

Amanda Palmer and The Decomposers Cover Joni Mitchell’s “Big Yellow Taxi” in Tribute to Rachel Carson

I dedicated the 2018 edition of The Universe in Verse to one of my great heroes, Rachel Carson (May 27, 1907–April 14, 1964), who catalyzed the modern environmental movement with her epoch-making 1962 book Silent Spring. Carson — a biologist who never relinquished her first love of literature — launched a courageous crusade against the deadly impact of pesticides and DDT in particular on nature. Conveying her unassailable science through exquisite literary prose, she awakened millions of lay people to the chemical industry’s ruthless assault on nature — not with mere facts, but with a larger poetic truth about our relationship and responsibility to this beautiful, fragile planet we call home. The creation of the first Earth Day and the founding of the Environmental Protection Agency were both direct consequences of her work. She never lived to see either — like Copernicus, Carson died shortly after the publication of her paradigm-shifting book. But she left behind a novel understanding of nature as a complex and beautiful interleafing of relationships, of which we are only a small part — a small part with a great responsibility for stewarding the whole.

It is hard, with our pathological cultural amnesia, to fully appreciate today just how far Silent Spring reached — beyond science, beyond policy. For years after its publication, after Carson’s death, the book’s message rippled and rippled across the groundwaters of popular culture. New Yorker cartoons and Peanuts strips celebrated Carson and her legacy, which touched a young musician only just making her name.

Rachel Carson (left) and Joni Mitchell

In 1970, Joni Mitchell composed “Big Yellow Taxi” — a song that would become a sort of bittersweet anthem of the environmental movement. It features this stanza inspired by Carson’s exposé of how pesticides, long marketed as harmless, were killing the birds and the bees:

Hey farmer, farmer —
Put away the DDT now.
Give me the dots on my apples
But leave me the birds and the bees.

In putting together The Universe in Verse — a labor-of-love celebration of science and nature through poetry, and a voice of resistance against the current assault on nature, with all proceeds benefiting the Natural Resources Defense Council — I realized that among the lovely humans who had donated their time and talent to read poems were four stellar musicians. So I asked one of them — my frequent collaborator and dear friend Amanda Palmer — to reimagine “Big Yellow Taxi” in a cover dedicated to Carson. She kindly did, enlisting the accompaniment of the other three — cellist Zoë Keating, Hedwig and the Angry Inch co-creator John Cameron Mitchell, and singer, songwriter, and guitarist Sean Ono Lennon. In a lovely burst of spontaneity, this makeshift band christened themselves The Decomposers and proceeded to deliver a stunning rendition of Mitchell’s masterpiece, emanating the timelessness and growing urgency of Carson’s message.

Amanda Palmer and The Decomposers at The Universe in Verse, April 28, 2018. (Photograph: Molly Walsh for Brain Pickings)

Prior to the show, they made a studio recording of the song at Pioneer Works, where The Universe in Verse was hosted. It is now released as a record, with cover art generously donated by Pioneer Works founder Dustin Yellin. All proceeds from the downloads benefit the Natural Resources Defense Council — please enjoy, >download, and join this small but significant act of resistance against the destruction of our Pale Blue Dot.

Below is the live performance with my prefatory contextualization, courtesy of Kickstarter Live and Bridgeside Productions, who contributed to this many-peopled project of goodwill by donating the livestream and the recording:

For more of The Universe in Verse, see poet Marie Howe’s stunning tribute to Stephen Hawking, astrophysicist Janna Levin’s reading of Maya Angelou’s cosmic clarion call to humanity, and other highlights, then revisit the full recording of the inaugural 2017 show.

To become a patron of Amanda’s music, a great deal of which benefits various humanitarian and environmental causes, join me in supporting her on Patreon.

BP

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