Guides who lead tours of caves painted by our prehistoric ancestors like to suggest that visitors are in the presence of masterpieces. Often it isn't true. As with other art, true Cro-Magnon masterpieces are the exception. There are some in Chauvet Cave of southern France, but tourists can't see them: Discovered in 1994, the cave is still reserved for scientists. And then, of course, there is the Lascaux cave, a couple of hundred miles to the west in Dordogne. That's not open to the public either--but a faithful replica is, and as the Michelin guidebook says, it's worth a detour.
Lascaux was discovered in September 1940 by several teenage boys who, while chasing their dog, had tumbled down a hole made by a fallen tree. When they got back on their feet they were in what is now called the Hall of the Bulls. They did not at first notice the herds of animals around them. Twenty-two yards farther, however, they entered what's now called the Axial Gallery, a passage so narrow that the flickering light of their oil lamp bounced off potbellied horses and a leaping cow. Those two chambers, which contain 90 percent of the paintings in Lascaux, have been reproduced in the concrete bunker called Lascaux II. It is set into the hillside a couple of hundred yards from the real cave.
In the first of two antechambers, the guide points to a map of the real thing, explaining why visitors are not allowed to see it: The carbon dioxide in their exhalations would react with the cave wall and cover the paintings with a layer of white calcite. More than a million people visited Lascaux between 1948 and 1963 before the cave was closed to tourists. In Lascaux II, the guide says reassuringly, "you are allowed to breathe."
The second antechamber exhibits the painters' paraphernalia--the sandstones they hollowed out and filled with reindeer tallow to use as lamps, the animal-hair brushes, the pigments. At Lascaux, unlike at most other caves, the artists did not limit themselves to simple line drawings in black or red; they colored in the animals with red, brown, and yellow. A small model of their oak scaffolding stands near the exit. Passing through that doorway, one plunges into a half-light alive with bulls.
Beyond lies the Axial Gallery, with its leaping red-and-black cow and its horses marching single file. But it is here in the Hall of the Bulls that Lascaux II creates an indelible impression. It seems like a real cave, albeit a well-groomed one. The people who built it in the 1970s and 1980s reproduced the shape of the walls, inch by inch, with cement molded onto a metal frame, and the artists who painted the replica used the same materials as their Cro-Magnon predecessors.
The first impression is one of richness: The walls are covered with paintings. Then a sense of motion takes over, motion on a large scale combined with fine detail--a prehistoric pageant. An enormous bull leads a herd of small horses along the left wall. Near the rear they encounter a group of stags and another bull. Two more giant bulls march up the right wall, each different.
The best bull is the last. It's all black, but its face is fully rendered; the guide calls attention to the eyebrows. There is a hint of sadness in its expression. The line protruding from its snout may represent a spear--but because Lascaux is littered with enigmatic symbols, one can't be sure.
Prehistoric paintings don't have to be masterpieces to be moving; the mystery's the thing. Two hours southeast of Lascaux, a cave called Pech-Merle is open to the public, and it contains, among other things, a very nice pair of ponies. More striking than the animals themselves, however, are the half-dozen hands that surround them--negative handprints of the artists, outlined in black. What were they trying to say with those? What were Cro-Magnon painters ever trying to say? Maybe they were reaching through the cave wall to the spirit world beyond. But today, looking at the Pech-Merle handprints, one gets the spooky sense of people reaching out as they fall away into the chasm of time. One hears their voices calling, but can't make out the words.
The French Ministry of Culture page
dedicated to Lascaux Restaurants
NEW YORK CITY
Mars 2112 has landed in Times Square, an alien invading New York City's community of theme restaurants. Like the Hard Rock CafŽ, Planet Hollywood, and the Harley-Davidson Cafe, these establishments can often be identified by their eclectic menus, big frozen drinks, and crowds of out-of-towners.
With two curious visiting nieces to entertain, Mars 2112 seemed heaven-sent. The adventure proved long-lived, but my notes on the evening are easily condensed: Waited 1 hour, 15 minutes for the "ride" to Mars, where costumed staff members strapped us in jostling chairs to watch a brief virtual reality show; deplaned near the restaurant's arcade and lost $3 at the claw game (60 seconds) while waiting for our table (30 minutes); dined amid lots of red stuff, strolling aliens, and glittering ceilings (1 hour, 30 minutes). Total costs were $65 and 3 hours and 30 minutes, relatively painless for a night out in one of the world's most expensive cities.
Aurora, age 12, and Sophie, 8, rated the evening highly entertaining, but I wondered how more-skeptical adults who know a thing or two about the Red Planet might find the experience. So I returned with Vivien Gornitz, a Columbia University planetary geologist and Goddard Institute fellow, and Corey Powell, a senior editor at Discover
. Skipping the ride to Mars, we descended the stairs, which are lined with crimson faux-rock walls dripping with cornflower-blue liquid. "Does it look like Mars?" I asked Gornitz.
"Well, it's red and they're rocks," she said. "That's like Mars. If we ever did colonize Mars, it's conceivable that we'd excavate to form underground habitats like this."
"What about the blue paint?"
"I'm not sure," Gornitz said, "but if you dug below ground on Mars, you might find water. There are pictures showing what look like dried-up steam channels on Mars. The air is so thin that any water on the surface would evaporate immediately."
A hostess seated us in the cavelike dining room, under a massive set of pale-orange tentacles with red blood vessels. "What about that?" I asked.
Powell and Gornitz shrugged, too busy enjoying a thin vertical beam of light on the front wall. If you turn your head fast enough, you can see it project an image of a hamburger, a fish, a martini glass, or the Mars 2112 logo. None of us could explain its significance, but we liked it.
A waiter in a costume vaguely resembling a Star Trek
uniform offered us menus. Powell immediately noticed Schiaparelli's chicken delight: "Isn't Schiaparelli the astronomer who marked the channels on Mars?"
"That's right, Giovanni Schiaparelli," said Gornitz. When the Italian astronomer discovered the grooves he called them canali
, which was improperly translated as canals
, leading to the mistaken belief that they had been dug by intelligent beings.
"And here's Stickney's sirloin," added Gornitz. "That's the maiden name of the wife of Asaph Hall. He's the one who discovered Mars's two moons--Phobos and Deimos."
After ordering, our attention turned again to the front wall where, high up, inside a Plexiglas bubble built into the wall, a costumed alien danced and blew kisses at diners. "If there is life on Mars, it would be microbial," offered Gornitz.
To the left of the alien, Gornitz identified gigantic lighted images of what looked like the surface of a foreign planet as false: "That's some kind of model. What they should have done is projected real pictures from the Pathfinder."
When the waiter brought my lobster man from Mars ($13.95), I had to be a pathfinder, too, digging deep into the wrap sandwich to locate the one small piece of lobster buried beneath a spill of salty, can-flavored shrimp. Gornitz and Powell, however, were pleased with their Schiaparelli's chicken (penne and smoked chicken in a garlic cream sauce, $15.95) and Astral tuna (broiled tuna steak with coconut rice, $19.95).
Next came visits from various Martians and a gentleman with bright red hair combed straight up several inches high and lacquered into place. He seemed like a cross between an alien jester and a human refugee from open-mike night. "Anyone mating here tonight?" he asked. "Let me read your biorhythms."
After scanning me with some sort of intergalactic Palm Pilot, he announced: "I'd say you're 85 percent human and 15 percent Saturnite, on your mother's side. You're OK."
That established, we paid the $145 bill, which included the entrŽes, the sub-space sampler appetizer tray, three cocktails (among them a "Mars'tini"), a Mercury Oreo miracle, Venus banana vista, and polar ice caps sorbet. All in all, better than a hot night out on Venus.
For more about Mars 2112 Books A History of the Circle: Mathematical Reasoning and the Physical Universe
Ernest Zebrowski Jr.
RUTGERS UNIVERSITY PRESS, 1999, $28 The Nothing That Is: A Natural History of Zero Robert Kaplan
OXFORD UNIVERSITY PRESS, 1999, $22.
The circle and zero have a couple of things in common beyond roundness: Both are crucial for mathematicians and scientists, and both are abstractions.
Although circular wheels, orbits, pennies, and pizzas may be mere approximations, the circle is so important to math that Ernest Zebrowski Jr. has built a 200-page history of the subject around it. In A History of the Circle
, he begins with a chapter devoted to pi. Zebrowski points out that dividing the circumference of a circle by its diameter yields a number that never terminates or repeats for the same reason that a true circle doesn't exist: because measurement can never be exact. Every digit added to pi takes us a step further toward unachievable accuracy. But wait a minute. If no measurement of the circumference or diameter of a circle can be exact, how was pi calculated? In the third century B.C., Zebrowski tells us, Archimedes approximated a circle with various polygons, whose sides and perimeter he could easily calculate. The more sides a polygon has, the closer it is to a circle. "[U]ltimately, he decided that a 96-sided polygon . . . would be a reasonable approximation to the circumference of a circle. And, in fact, if you build a 96-sided polygon and roll it on a flat surface, it will indeed roll along reasonably well."
That observation leads easily into the next chapter, on wheels and rollers. Long before the advent of the wheel, people rolled their heavy burdens on logs. But because rollers need to be uniform in size and shape to work well, Zebrowski suggests the Egyptians opted for a different technique to schlepp stone blocks up the pyramids--by adding "circular segments" to each side, creating rounded-edged rectangular cylinders that could be rolled up the pyramid. The celebrated wheel, he writes, wasn't much use until ball bearings were invented. The Romans, for instance, used chariots only for short distances and special occasions, because the crude bearings they used on axles quickly wore out.
Later chapters stray further from the circle. From the difficulties of inventing the clock and the development of our understanding of the universe, conics, and sine waves, to the discovery of neutrinos, Zebrowski explains the major advances of physics and math simply and understandably, eventually leaving the circle behind. The same book with a different name, minus the first chapter or two, could easily be mistaken for a review of the math you should have learned in high school.
Although the concept of the circle was very real for thinkers throughout history, the idea of zero took much longer to come into its own, explains Robert Kaplan in The Nothing That Is
. There was a time when arithmetic was conducted without it, albeit painstakingly (imagine long division with Roman numerals).
Nowadays most people take the number for granted. It has some special rules--everyone knows you can't divide by zero--but on the whole, it seems to function just like any other digit. Nonetheless, it makes our entire system of numerals possible, allowing us to tell the difference among, say, 32, 302, and 30,200,000. It also serves as the boundary between positive and negative integers.
There's not much historical evidence telling us when zero first appeared. Did the Babylonians invent it first, when they used a pair of diagonal wedges to show an empty column in their addition? Perhaps the Greeks beat them to it, using a bar over a circle to indicate nothing. However it started, by the Middle Ages, the use of the zero was widespread in Europe, though viewed with some distrust; centuries passed before the new number was fully accepted. "[T]echnical difficulties, combined with the slow spread of knowledge before books were printed and writing in the vernacular was common, added to the reputation that the Arabic numerals already had for being dangerous Saracen magic . . . So in Florence the city council passed an ordinance in 1299 making it illegal to use numbers when entering amounts of money in account books: Sums had to be written out in words."
The story of the slow acceptance of zero is the most interesting part of Kaplan's book. But as the text continues, his use of a literary quote here and there to sweeten the prose degrades into paragraphs that groan under passages from the likes of Henry James, Samuel Beckett, and Fyodor Dostoyevski, making this tome at times seem more like a version of Bartlett's Familiar Quotations
than a history of math.
Both books provoke amazement at our ability to understand the world through abstractions. But both also acknowledge the probability of limits to that understanding. Near the end of A History of the Circle
, Zebrowski comments: "[W]hat is not detectable cannot be considered to be real. At some level--and perhaps this is it--science may be destined to merge once again with philosophy, the discipline it split off from many centuries ago." It's a reunion both authors seem to anticipate.--Michael M. Abrams Order The Nothing That Is from Amazon.com.
Order A History of the Circle from Amazon.com. God's Equation: Einstein, Relativity, and the Expanding Universe
Amir D. Aczel.
FOUR WALL EIGHT WINDOWS, 1999, $22.
Cosmology, say astronomers, is the search for two numbers: Hubble's constant, which measures the rate at which the universe is expanding, and total cosmic mass, which determines whether the universe will eventually slow to a stop or expand forever. Find these two numbers, and all the mysteries of the cosmos will be revealed. It's a cold view, reducing space and time, matter and energy, past and future, to a single equation, but it suggests a faith in the utter simplicity of creation. We should be able to explain everything from quasars to lightbulbs with the same set of rules.
This universe-as-an-equation model has many parents, but none as prominent as Albert Einstein, who made it his life's work to write the concept in a single line. The closest he got was his field equation of gravitation, which describes the way space curves under the influence of gravity. Astronomers recently made the first solid stabs at finding all the terms in the equation--even to the point of resurrecting a factor called the cosmological constant, which Einstein inserted and then disavowed. It is thus a wonderful time to glance back over Einstein's path in developing the field equation.
It's not an easy road to follow: Although Einstein's beginnings as a dropout and patent clerk are well-known, he was also a highly trained physicist who taught himself all the higher mathematics he would ever need. Fortunately, we have a fabulous guide in Amir D. Aczel, the author of Fermat's Last Theorem
. In Aczel's latest book, God's Equation
, Einstein is not an eccentric cartoon but a young, compulsive, competitive genius who saw beyond the ornaments of the universe to its underlying structure.
Einstein began his work at the turn of the last century, at the very beginning of an explosion in our understanding of the cosmos and our place in it. At the time, there was only one known galaxy--ours--and theories of cosmology were limited by the belief that all the stars in the universe were bound by gravity to the same center. Einstein's theoretical progress was both dependent on quality observations and a catalyst for them. For example, his prediction that space will curve in the presence of a large mass sent astronomers scrambling to photograph the stars near the sun during a solar eclipse. As the moon covered the face of the sun, they found that the field of stars nearest the sun looked slightly distorted; the sun's heft had warped space enough to change the path traced by starlight.
Einstein, writes Aczel, "had come to the conclusion that his theory alone would not be worth much without physical verifications. He seems to have taught himself a great deal about astronomy in a relatively short time. . . . It seems that Einstein was so determined that everything work right that he was not going to leave to astronomers the details of their everyday work."
But God's Equation
is not just a retelling of breakthroughs in early twentieth century physics. Aczel makes it a study of the intersection of mathematics and the physical world. The geometry of space has real consequences. Enough mass concentrated into a small enough point can create a pucker in space-time so deep that nothing can ever escape. An ever-so-slight curvature in a universe that appears perfectly Euclidean over millions of light-years can spell the difference between eternal expansion and cosmic collapse. At a time when so many popular physics books avoid equations and fudge mathematical explanations, Aczel wants to delve deep into the mathematics. He believes--as Einstein did--it is in fact the underlying mathematics that makes the universe elegant.
--Jeffrey WintersOrder God's Equation from Amazon.com.