by Guy Murchie
From here, ascending the size scale in jumps, the garnet is a ferromagnetic crystal with oppositely magnetized sublattices so independently responsive to temperature that, by warming or cooling it a mere degree or two, its polarity can be reversed, making it useful for storing information in computer memory cells. Another big jump takes us to the size of a lava field containing quartz, such as a large one on the North Idu peninsula in Japan where an earthquake in November 1930 was accompanied by spectacular lightning in a clear sky, a phenomenon already becoming well enough recognized to have disseminated the theory that piezoelectric crystals subjected to disruptive mechanical pressure over many square miles of territory can generate voltage sufficient to cause a thunderstorm.
In all such widely different cases, it seems appropriate to cite the increasing evidence that, in magnetically ordered crystals, the spins of the atoms periodically do something that could be called a flip, reversing their direction and polarity, just as happens much more slowly to the magnetic polarity of the Earth, the sun and stars and probably the galaxies. And this change actually advances from atom to atom like a row of falling dominoes in a microprogression physicists call a spin wave that sweeps through the material like a swell on the ocean. Indeed it is a vital factor in the melodic reality behind the wave nature of all matter - the still-mysterious truth discovered in France in 1922 by Louis de Broglie after the merest of hints in ancient times by Pythagoras, who had listened and listened for, and eventually heard, the music of the spheres.
LIQUID CRYSTALS
While one naturally thinks of a crystal as a solid substance, the stability and order that are its essence also extend into water, other liquids and even into the realms of gases and plasmas. Looking and flowing almost like honey of varying viscosities, liquid crystals have optical properties that were recognized as crystalline as early as 1888 and by now their occurrence adds up to nearly one percent of all new organic compounds synthesized in chemical laboratories. The skin of a soap bubble is a simple kind of liquid crystal, in which flexible eelshaped soap molecules regiment themselves like soldiers in stretchable double layers that form both the inner and outer surfaces of the bubble, confining a sheet of water between them. These parallelly aligned molecules stand perpendicular to their layer, holding their "heads" and "tails" in phase and, if the bubble stretches, extra "free" molecules in the interstitial water automatically slip between them to increase the area of the layer.
A more complex liquid crystal is the kind that harbors cholesterol and comes in thousands of one-molecule-thick layers, with the molecules lying flat like sleeping soldiers, their long axes parallel to the plane of the layer and to each other but with each layer's axes rotated 15 minutes of arc relative to the next (the very same angle, by the way, that the radii of both sun and moon subtend from Earth), therefore cumulatively forming a helical progression of polarity that is common to all genes and protoplasm. The potent effect on penetrating light of this 15-minute twist of polarity per molecule can be judged slightly by the fact that it adds up to a spin of 18,888° or fifty full revolutions for every millimeter the light travels!
As a poetic chemist might explain it, the water molecule (H2 O) loves to dance with its oxygen neighbors, and this is shown by the fact that its two hydrogen atoms always reach out their "arms" to any oxygen atoms they meet. At the same time it has a philanderer's habit of latching on to all other hydrogen atoms that come within reach through similar bonds that keep joining them to its oxygen and, because the effective angle between the twin hydrogen atoms averages somewhere near 120°, the rolling, milling water molecules tend to join hands in momentary rings of six, while tied to other rings by zigzag chains, all the members excitedly grouping and regrouping as the turbulent units of water create their hexagonal chicken-wire-like crystalline lattices.
Something comparable has been known to take place in the forming of half a dozen small tornadoes around a central mother vortex, a similar number of eddies around a whirlpool, thunder cells about a thunderstorm, or even five or six clouds of space dust around the solar system. But the water molecule is also incurably congenial with carbon and a good many other common elements like nitrogen, phosphorus, magnesium, iron, etc. In fact it socializes with the cellulose as well as the carbohydrates of plants and trees and flows through all animals - certainly all multicelled ones - not just as plasma in their blood but by permeating their cells and the interstitial channels between cells, including organs and tissues of every sort, uniting and reuniting with them in its unceasing but remarkably orderly motion.
Putting it in less biological terms, the sea is not merely around us. It is inside us. For life on Earth, and perhaps life mostwhere, is primarily organized water. Thales intuitively understood this in the fifth century a.c. when he voiced his extraordinary surmise that "water is the one essential element of the world." And today we find that water literally constitutes most of ourselves - that a new-conceived human embryo averages 97 percent water, a newborn baby 77 percent, a grown man 60 percent. In judging a body's water content, of course one tends to think of fat as a fluid, so I'll point out that fat is chemically very different from water and, although water is two thirds of a thin cat, it is only one third of a fat pig, while the average woman is both fatter and drier than her husband; Moreover the proportion of water in all healthy bodies is regulated with remarkable exactitude, any excess being automatically eliminated as urine, any shortage (even 1 percent) demanding its prompt replenishment through thirst.
On this note of intimacy between life and Earth's most characteristic substance then, we shall pause for a chapter change before venturing even deeper into the dynamics and geometry of life's elusive nature.
Chapter 17
Living Geometry and Order
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WHILE WE ORBIT our way dynamically around them, Earth's creatures, from armadillos to spiders, live their own dynamic lives and display every kind of geometry from their studded hides to their symmetric webs. Even flowers, as we will soon see, have a flowing geometry closely analogous to the dynamic motion of the simplest of forms - forms that are naturally fluid and alive, and that epitomize the symmetry that seems to be the beginning of being.
THE SPHERE
First the sphere, the most elementary of shapes and which encloses the greatest volume with the least surface of any imaginable body. It is the very plinth of primordial life and the germ of evolving complexity. Everything fluid from bacteria to bubbles to stars tends to this centrosymmetry, plus many a virus much smaller or a galaxy much larger, not to mention a supergalaxy or the knowable universe. Raindrops, for example, have to be nearly perfect spheres to produce rainbow colors (by refraction through spherical segments), yet, as they grow larger by accretion and fall faster, they begin intermittently to flatten out at the bottom from air friction, momentarily taking on a shape something like a hamburger bun, the while oscillating from bun to egg form hundreds of times a second until, as they approach a quarter inch in diameter, they break up from their own aerodynamic turbulence. Thus generation succeeds generation in raindrops, as in amebas, by division and division, each drop "breathing" air and carbon dioxide and in its rainy way "eating" and digesting dust, germs and spores and, on landing, spitting out microbes, nitrogen and sometimes carbonic acids strong enough to etch rock. Like the rain, the living cell also runs to the spherical or (when crowded) the polyhedral, presumably mostly because of its fairly continuous storing and spending of energy in a flexible bag that is under almost equal pressure on all sides. The cell's sensitive feedbacks of metabolism, moreover, normally ensure a remarkable net stability over the days or months, indeed a sort of steady state sometimes termed the vital equilibrium and which largely explains why sphericity is so universal.
If you still wonder how the trend to roundness began, consider the curious suggestion of some of the older geometers (and perhaps a few cosmologists) that spherical stability evolved out of primordial instability or waviness that somehow blossomed into tu
rbulence. Liquid flowing from a bottle, to take a homely example, begins by limiting its body to a smooth, straight stem nearly cylindrical for a little way, but, by the time its length exceeds its circumference, it thins down, whereupon it can't help but weave and get weavier and wavier until it sort of lets go, bursting out in turbulent blobs and separate drops that take
the shapes of spheres. If there ever could have been a celestial analogue of this sequence starting in a stable, smooth universe, I can't help but wonder whether such a universe really might have developed a corresponding instability and waviness, veering into turbulence, eddies, galactic bubbles and spiral world systems in space-time? Or is cosmic turbulence as we see it in the sky today simply continuous, eternal and on the whole unchanging in God's creation, of which Baha'u'llah, Prophet of the Baha'i Faith (page 614), said, "Its beginning hath had no beginning, and its end knoweth no end"?
THE FLOW IN FLOWERS
At least a hint of an answer may be gleaned, if you're interested, from a close look at the dynamics of splashes, which D'Arcy Thompson among others began to study last century. When a round pebble falls into calm water, he observed, its downward pressure after impact pushes a "filmy cup of water" upward all around, which "tends to be fluted in alternate ridges and grooves, its edges ... scalloped into corresponding lobes and notches, and the projecting lobes ... into drops or beads ... " Although this creation lasts but a tiny fraction of a second, photographs show it to have a beautiful, symmetrical, flowerlike form put there by the same sort of dynamic forces that genetically infuse the flow into flowers, only many millions of times faster. And, curiously, the lateral speed of the splash has been measured to be several times faster than the impact speed that caused it. Of course it was a rather new concept in those days, but the recent proliferation of time-accelerated and time-decelerated movies has made it much clearer, demonstrating dramatically the extraordinary parallels between fluid turbulence and the relatively solid forms of life, between a lacteal crown of two dozen beadlike points tossed up in a splash of milk and certain polyps in the sea with two dozen vertical tentacles surmounting a similar cuplike body, between columns of ink sinking in water or fusel oil in kerosene and various medusae jellyfish, even between the furrowed torsos of protozoans, the fluting of instable sleeves of plasma in a jet engine and the gadrooned blossoms of gentians and lilies.
"The drop, the bubble and the splash," as Thompson says, "are parts of a long story," and the actual flows of material in all of them must have close counterparts with such rotund forms as eggs, eyes, honeycomb cells, hurricanes and volcano eruptions. Though it seems farfetched, comparable rhythms enable softwood shingles on a barn to "breathe" their nails outward, a tide of earthworms to bury stones, frost to heave highways, galaxies to spawn satellites, atoms to invent molecules. In every case, form is closely related to function, through which flow-force kinships link clams to cobras, campanulas to comets, a bursting bubble to the upshoot of a mushroom, the gastrulation of an embryo to a rolling, in-folding snowstorm. Acoustics, in a similar sense, is a volatile version of hydraulics. When sound enters a room and is absorbed by rugs, furniture and windows, it behaves like invisible water filling a tank containing baffles. Oil gushing through a pipeline so closely emulates traffic on a turnpike that highway engineers use data from fluid dynamics in drafting the shapes of intersections, islands, curves and grades. With the latest computers and supersonic wind tunnels they research turbulent flow as a general theory, subprogramming it with coefficients of oscillation and crystal coordinates in space-time, noting that, with increasing speed of flow, the "vortex street" (pioneered by aerodynamicist Theodor von Karman early this century) exuberates progressively from prim, regular eddies toward random, vagabond turbulence, whence, if you can overlook a modest allowance for gremlins, it may be safe to say that the feedback-tempered stability qualifies in the long run as one of life's most characteristic factors.
THE GEOMETRY OF LIFE
In a calmer, optic aspect of the same vital geometry the rainbow bespeaks a conic skeleton engendered abstractly by the refractive angles of 42° and 51° between the kindred spheres of sun, raindrop and eye. If this be an etherialization of geometry, there are a thousand times more species of fluent "organisms" in the liquid and gaseous realms of the uncounted transsolar worlds, from volatile planets of the Jupiter breed to the starry arms of galaxies, which may represent much more than populated waves or residues of what some astrophysicists think are reversible (others irreversible) processes. Some of these "organisms" may be shaped by tensions in the manner of bubbles, which probably are more disciplined than ballet dancers and far more "single-minded" in their body control, as anyone knows who has dipped geometric wire frames into soapy water and looked thoughtfully at the resulting cubes, prisms, graceful spirals and tetrahedrons ... For one has to admit that mysterious but rigorous laws are in full control here: that it is forbidden for more than three films to meet in a common line, or for more than four edges or six films to touch at a single point. And all angles of intersection must be balanced and equal, like 120° between each of three planes that together add up to a full 360° circle.
Of course when a batch of lively bubbles is freshly spawned, as in the washtub or a frog pond, many will momentarily flout the law: four carefree young bubbles, say, meeting in one common line, each perforce limited to but a 90° quarter of the circle. But, as the illustration shows, this arrangement is so pinched and uncomfortable, it is highly unstable - and the four-bubble line, itching to burst loose, will soon split in two like an ameba, giving birth to twin offspring lines joined by a new film that now enables each new line to relax and spread its young angles from 90° to 120°, as one of the four bubbles slips away, leaving only three. This is precisely how the stress in fresh-whipped lather or foam eases off as its myriad bubbles settle into more congenial configurations - a peace in effervescence you can literally feel and hear in the barber shop or the beer hail. And it says something, however indefinable and mysterious, about how life springs out of bubbles through the media of seeds and eggs into amebas, bacteria, caviar, frogs and man.
There is something surprisingly enlightened (I almost said "mental") about intrabarm relations too: the way two or more bubbles will apparently consult each other and instantly make the most mutually agreeable of compromises. It seems to be a mystic nonverbal language of touch and pressure. Although each bubble, when alone, remains spherical in form because its elastic soap-skin yearns to be as small as possible (a yearning that is the raison d'etre of sphericity), two bubbles that join together somehow agree to a modified form such that their outer surfaces, plus that of their shared interface film, add up to the smallest total area that can hold the air in both bubbles separately. This double bubble is actually an automatic, simple and beautiful solution to a complex mathematical problem. And triple bubbles even more so with their three interfaces all curved yet meeting only at 120° angles and all obediently, patiently, holding their invisible centers of curvature, for some secret reason, in a single straight line.
EFFICIENCY UNDER PRESSURE
Cells in a honeycomb are something like bubbles, but even more regimented, being controlled by animals as well as the less animate forces, and their visible ends form the well-known hexagonal crystalline pattern that comes from subjecting circles to pressure. Johannes Kepler, the great astronomer, who enjoyed geometric, puzzles, was perhaps the first to realize this and, logically advancing his inquiry into the third dimension, he observed that the honey cells are cylinders pressed so tightly together from six sides that they are laterally molded into hexagonal prisms. And the ends of the cylinder-prisms, which bulge outward, naturally settle into hollows between three neighbor cells, whose triangular funnel effectively pinches them into three-sided pyramids.
If the honey cells were shorter, wrote Kepler, say as stubby as pomegranate seeds, the same pressure would still give them the same angles and twelve equivalent faces where twelve neighbor cells touched and squeezed them:
six surrounding at the same level, three just above and three below. That would make each cell a twelve-sided figure with diamond-shaped faces known to a geometer as a rhombic dodecahedron or, to a jeweler, the crystal form of the garnet.
Until the mid-nineteenth century, it seems, the few scientists who thought about it believed the reason such garnet shapes are so common among compressed eggs and cells, indeed among any figures that completely fill space, is that this shape is the most economical since it packs together in such a way as to divide a given volume with relatively less partitional area than any other configuration. Then Lord Kelvin appeared on the academic scene in that legendary bastion of frugality, Scotland, to make the astonishing discovery that a fourteen-sided figure called a tetrakaidekahedron is even more ideal for a living cell than the dodecahedron, actually offering a slightly smaller
percentage of partitional surface. Of course he had to measure his areas very carefully because the fact was not judgeable just from looking at the thing, bounded as it was by three pairs of equal and opposite quadrilateral faces and four pairs of equal and opposite hexagonal faces. And in time there arose an aura of ancient mystery about it, because, unbeknownst to Kelvin, his brain child had once been studied by Archimedes, who wrote a detailed description of it in the third century B.C., and even before Archimedes, as someone archly speculated, it could as easily have been known to the "illiterate Pythagoreans."
Undeterred by such thoughts, Kelvin did some high-precision research on his fourteen-faced construct of life and was a little awed to find that, to achieve absolute minimal area, its edges must be slightly curved and its hexagonal faces warped just enough to become perceptibly anticlastic, like nearly flattened saddles which would be barely convex on one axis and barely concave on the other, with "equal and opposite curvatures" at every point. And although, as with the rhombic dodecahedron, a mass packed tight with such figures must so partition space that wherever three faces meet in one edge they can do it only at coequal angles of 120°, unlike the dodecahedron, wherever four edges meet in one corner, they will do so at coequal angles that ideally amount to 109° 28'16", this being the so-called Maraldi angle (first approximated by the astronomer J. P. Maraldi), at which lines from the four corners of a perfect tetrahedron meet at its center.
