The occupation of floating bodies
katastroph: from Greek; an overturning, ruin, conclusion (American Heritage Dictionary)
By B. Frank |
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Another legend credits our hero with running naked through the streets yelling, “Eureka!” because he’d just discovered how to judge the gold content of a statue while sitting in a bathtub. I’m a little dubious on that one, though I’ve discovered many delightful things after stripping down and getting in the water. Flotation, for instance — some bodies (and body parts) are more buoyant than others. As old Archimedes laid out rather exhaustively in his “On Floating Bodies — Book I,” it depends on the comparative weight of the solid vs. the liquid in which it rests. As applied to the human body, this “Law of Buoyancy” is pretty simple. Fatty tissue floats, while bone and muscle sink. Left to my own devices, most of me sinks; but some of the last two thousand years have been spent inventing and improving flotation devices for purposes other than conquest. Air mattresses, inner tubes, good old “personal flotation devices” (PFDs), canoes, kayaks, rafts and “duckies” (inflatable kayaks) have saved me whenever watery catastrophe has come acourtin’ so far. It’s part of why I love science.
There are certain parts of a place known as The Canyon where almost anyone can try to ride the frothing turbulence clad only in a PFD and bravado, once reticence releases its death-grip on your sense of the impossible. Back in “The Day,” before sunfearing/ drug-testing/family-friendly purveyors of “safe adventure” clamped suffocating rules onto commercial river-running, our fearless trip leader, “The Lady,” announced an opportunity to swim a rapid. “The Swamper” ignored her, since he was working his passage rowing the bucketbottomed baggage raft on the 95- percent of The River that is flatwater, while “The Baggage Boatman” lounged like a barefoot Captain Nemo and deigned to conduct a clinic on canyon geology and water hydraulics. Then “The Cook” looked up from her sunning perch on the dry-bags and said, “You’re going, too; aren’t you?”
The Swamper pointed his feet downstream and steered with the arms, just like The Lady had taught us. It feels a lot like riding a very small inner-tube down a very large surf wave off the edge of the world. It was a good cruise for awhile, with time for quick breaths and a little sightseeing between dunkings. Then he saw a raft bearing down on him, turned in the river and raised his arms for an expected hand up. (Note: When swimming a rapid in a personal flotation device, never face upstream, especially when only half-way through the rapid.) The cruise became a rinse-and-spin cycle, with the theme of “20,000 Leagues Under the Sea” running through his panicking brain. When he was dragged into a raft like a gaffed fish at the rapid’s end, he expected a little sympathy.
Instead, the crew was trying to catch The (suddenly bloodyfaced) Cook (she’d hit the only surface rock in the rapid with her head; but it was “just a scalp wound,” as John “The Duke” Wayne would say back in The Day), and searching the waves for the guy we’ll call “Lefty.” Oh, did I mention Lefty? He was a cop on disability leave, on a buddy trip with two former brothers-in-arms, trying to regain the sense of wholeness you lose when a dirtbag’s lucky shot rips off your gun arm. The three amigos slept in a protective row in the sand, and bonded with manly (not quite) full-frontal hugs. Lefty scrambled up cliffs, took his turn in the paddle-boat and even tried rowing a ducky single-handed (literally). Of course, he joined in the PFD-permitted rapid. Lefty had disappeared in the top half, and now as the swimmers coughed Colorado Plateau silt all over the raft floor, The Boatmen were getting a little worried about him. I peered over the tube just as somebody pointed to an orange PFD eddying out below the turbulence. Lefty was still in it, and he was waving.
Archimedes wrote another book on floating bodies, creatively called “Book II,” that further explored buoyancy and other theories, including the “Law of the Lever.” Not long ago, a guy named Dr. Chris Rorres combined these ideas with two thousand years of scientific calculations, and used computer modeling to create a visual aid that got me up to speed with what the old Greek was saying with his geometric equations. It’s available at http://mail.vet.upenn.edu/~rorres/ if you care. Two of Rorres’ models: “Turning a paraboloid upside down” and “Righting and energy arm curve,” could be used to demonstrate a kayak roll, and might have convinced Lefty that it is catastrophic for a one-armed man to float a rapid in a PFD, had he studied them. This is the Achilles’ heel of following laws and rules unquestioningly. They discourage attempts at the impossible. Still, a measured dose of knowledge (and of fear) usually helps more than it hurts, so here goes: To define a floating paraboloid you can say, “A homogeneous solid convex object bounded by a surface obtained by rotating a parabola about its axis.” (Dr. Rorres’ written translation of the text), or you can visualize sitting on an object that has a rounded bottom and a flat top, floating toward the edge of the visible world. To avoid a flip, all you’ve got to do is control the tilt of your paraboloid, so that neither edge of the flat surface dips below the surface of the liquid. Simple in 95- percent of a pool-and-drop river; but it’s the other 5-percent that brings in the application of the “Righting and energy arm curve” to control the rotation of your paraboloid, as you change the center of gravity by thrusting a butt cheek into the rising flat surface. If this fails, you flip; but if your paraboloid is a hard-shell kayak, proper application of the energy arm and reliance on the Law of the Lever (don’t drop your paddle!) will produce a roll, and may look like you meant to do it all along.
If, however, your paraboloid is a raft or ducky, it’s best that you have a fuller understanding of the “Toppling Structure” model, which depicts an object with a flat base topped by a mass rising above the supporting surface. Think of a skyscraper. An earthquake has liquefied the soil under it, and now it’s sinking. Archimedes’ equation “proved” that, even when the surface becomes a liquid, “this structure cannot topple until its base is partially exposed above the soil level” (Rorres again). If, from a reasonable fear of skyscrapers and the other toppling detritus of civilization, you too have taken to The River, the model still applies. If you become the skyscraper straining upward to see what’s ahead as the flat base of your paraboloid catches air, you are aiding your own catastrophe. High-side (now!), diving toward the rising tube to lower your center of gravity. Remember that a paraboloid will float an incredible amount of weight, if you follow Archimedes’ advice. On the aforementioned River Trip, one of The Boatmen busted an oar at the top of a rapid. The current pushed his raft into a rocky cove on the wrong side of the river, downstream of the V-shaped, glassy-smooth bulge of the entry line. Lateral waves crashed off The Canyon’s fluted black schist, creating a scene that, even to my inexperienced eyes, looked like a flip in the making. The two frightened faces of The Passengers showed a good sense of their situation. As The Boatman put his spare oar in the lock, I noticed The Lady, paddling the trip’s ducky upstream in a lesser current along the near shore. When she reached the quiet pool above the rapid’s entry point, she crossed and hiked the raft’s passengers back upstream. On a ducky built for one, the three people looked like a sinking skyscraper; but The Lady made The Passengers lie on the tubes while she ran the rapid. They arrived wet and shaky, but helped hoot and holler as the lightened raft bucked and spun its way down to us. No flips, no foul words. See, gang, applied science can be fun!
Unfortunately, all hands-on applications must end, and then language must transmit the knowledge gained, to avoid the need to “re-invent the wheel” every time you want to float a river. Archimedes wrote his theories in the lingua franca of his day, Greek; and for over a thousand years, transcriptions survived in that language. In the 12th Century, a Greek Orthodox priest found himself in possession of one of the last copies containing “On Floating Bodies,” written in Greek. With exalted purpose, the priest scraped off the old ink as best he could and overwrote it with prayers. Most of the Archimedes research since has been based on the inexact art of translations of translations, and though we think these theories of geometry are based on Archimedes’ writings, we could be mistaken.
Hidden under its layer of prayers, the Greek transcription passed through the hands of various monasteries and museums, until the 1920s, when the book disappeared. Not knowing they had stolen the last known copy of Archimedes’ ideas transcribed in his native language, forgers painted over some of the pages to increase its value in the thriving “stolen prayer-book” black-market. Just a few years ago, the book was re-discovered at an auction; and now imaging technology may show that Rorres’ models (and my new-found confidence in my understanding of the geometry of flotation) are based on flawed translations of Archimedes’ theories. Tune in again when the imaging is complete in 2008, says William Christens-Berry, the physicist in charge of the recovery project. I’ll try to remember, though, in the meantime, I hope to be busy proving geometric theories on any river I can find.
Another thing I love about science is that flawed understanding carries only the penalty of failing to predict a natural reaction to physical facts. Say you think that prayer can replace flotation as you swim off the edge of your known world. If you’re wrong, you sink. It’s scientific. I don’t know about you, but I snug my PFD tight and learn to live with any discomfort this reaction may cause, before slipping into The River. The Lady proved the pleasure of following all natural reactions with the same gusto she brought to river rescue and swimming rapids, long before the film “Brokeback Mountain” redefined the “Code of The West.” On that River Trip back in The Day, she ogled the semi-clad babes of other river parties with a collegial nudge to those of us with a weakness for the ladies, and for the traditional after-trip dinner, she chose a sushi bar. The Swamper didn’t know the first thing about eating raw fish; but he was seated by The Lady, and she gave gentle advice with a knowing grin.
As is usual with River Trips, we all exchanged addresses after dinner, and then eventually lost touch. I haven’t seen The Passengers, The Boatmen, The Cook, Lefty, The Baggage Boatman or The Lady in years, and The Swamper’s naiveté is a fading memory. But in honor of The Baggage Boatman’s ironic shouted salute to Disney-fied adventure as he steered into The Canyon’s biggest waves, I still think “E Ticket ride!” when I take the big water line through a rapid; and when I see lessons re-discovered by occupants of bodies floating on the rising flow of each new river season, I can imagine the legendary Archimedes stripping off his clothes and jumping into The River as he repeats his exultant cry, “Eureka!”
Contributing editor B. Frank’s last article for the Gazette — if you don’t count the Obituary that appears in this issue — was his strangeas- shit review of some shit dealing peripherally and parenthetically with Kafka and, well, ice. He lives in southwest Colorado, well off of some grids, and well on others.
