Dark Tales Of Lost Civilizations (41 page)

Leading historians and cosmologists squealed like stuck pigs.

“Impossible!” swore one historian. “Sumerians barely had their base 60 sexigesimal number system by then. How could anyone in Larsa, one hundred twenty years before the city was captured in 1762 B.C., possibly have had calculus or that crazy string theory stuff?”

“Ludicrous, and certainly a hoax!” insisted a cosmologist. “How could mud-daubed primitives in the Fertile Crescent, thirty-seven and a half centuries ago, have known about Clebsch-Gordan coefficients in nuclear physics, let alone mass ratios of leptons and baryons?”

Duncan J. Melville of St. Lawrence University pointed at the first photograph on the wall-sized projection screen with his laser pointer.

“YBC 7289 is a small clay disc containing the rough sketch of a square and its diagonals. Across one of the diagonals is scrawled 1, 24, 51, 10—a sexagesimal number that corresponds to the decimal number 1.4142129, an approximation of the square root of two. Below is the answer to the problem of calculating the diagonal of a square whose sides are 0.5 units. This bears on the issue of whether the Babylonians had discovered Pythagoras’s theorem some 1,300 years before Pythagoras did.”

Dr. Dugan Dwamish, sitting next to me, whispered in sushi-stinking breath: “They knew a hell of a lot more than the Pythagorean theorem.”

Melville went to the next slide. “No tablet bears the well-known algebraic equation, that the squares of the two smaller sides of a right-angled triangle equal the square of the hypotenuse. But Plimpton 322 contains columns of numbers that seem to have been used in calculating Pythagorean triples, sets of numbers that correspond to the sides and hypotenuse of a right triangle, like three, four, and five.”

Melville then showed and explained a photograph of the University of Pennsylvania’s excavations at Nippur in 1899. The photo looked like a cross between a
Raiders of the Lost Ark
set, an M.C. Escher etching, and a Piranesi print of uncanny towers beside a pit with descending staircases. Nippur was the principal center of scribal training in the Old Babylonian period. The tablets excavated there provided the basis for research through about 2015 A.D. on mathematical education and curriculum, and now the new, almost unbelievable finds.

“Institute for the Study of the Ancient World,” said Melville to the packed audience of scientists, educators, and reporters, “assures us that, since the second half of the nineteenth century, thousands of cuneiform tablets from the Old Babylonian period have been found at various sites in ancient Mesopotamia.”

The map showed a mix of familiar and exotic names. The Tigris and Euphrates rivers, with Mari to the northwest, Babylon and Sippar, Uruk, and Ur farther southeast. There was Larsa, where the strangest tablets had been found, down from Nippur and Kish.

Melville put up a photo of a thin-faced man with hair cut short on the sides, wireframe eyeglasses, and pursed lips. “The considerable mathematical knowledge of the Babylonians was uncovered by the Austrian mathematician, Otto E. Neugebauer,” he said, “who died in 1990. Scholars since then have turned to the task of understanding how the knowledge was used.”

“What about the Calabi-Yau data?” whispered Dr. Dwamish.

“Ssshhh,” I said. “He’s getting to it.”

Melville pointed to Neugebauer’s hand drawing, of two sides of YBC 4713. “In the 1920’s, he became aware that hundreds of Babylonian mathematical tablets lay unstudied in European and American museums.”

“YBC?” said Dugan.

“Yale Babylonian Collection,” I whispered back. “Keep up.”

“YBC 4713 is a tablet showing a series of abstract problems,” continued Melville, sweating, perhaps not only from the spotlight. “While some mathematical techniques learned in scribal schools were intended for use in scribes’ later careers, many would never have been applied in practical situations.”

He jumped to the next slide, with a much simpler tablet. “This is a school tablet with an incomplete calculation. The early training of scribes consisted of copying lists of units of measure and arithmetic tables. Later, they practiced calculations and simple problem solving.”

“Simple? He calls Calabi-Yau manifold simple?” Dugan asked.

“Ssssshhh.”

“Sumerian math was a sexagesimal system, meaning it was based on the number 60. Why the Sumerians picked 60 as the base of their numbering system is not known. The idea developed from an earlier, more complex system known from 3200 B.C. in which the positions in a number alternated between 6 and 10 as bases.”

“That’s nuts.”

Melville showed a slide of a tablet, bearing rows and columns of squares, with complicated combinations of wedges. “A 59 x 59 multiplication table is too large to memorize, so tablets were needed to provide essential look-up tables. Cuneiform numbers are simple to write because each is a combination of only two symbols, those for 1 and for 10.”

“Look-up tables. Now he’s talking like a computer guy,” said Dugan with a smile.

“This tablet shows a list of practical problems, worked out step-by-step, to calculate width of a canal, given its other dimensions. These calculations,” he pointed with the laser, “are the cost of digging the canal, under different assumptions about a worker’s daily wage.”

This made perfect sense to me. We have vestiges of sexagesimal. “1:23:45” on a digital hologram computer display means 1 (times 60-squared) seconds + 23 times 60 seconds + 45 seconds. 60 seconds in a minute, 60 minutes in an hour.

“The students doing these homework problems spoke Akkadian, a Semitic language unrelated to Sumerian. The Sumerian in their problem sets was in a language already extinct during their time. But both languages were written in cuneiform, meaning wedge-shaped, because of the marks made by punching a reed into softened wet clay.”

So the notation was a unifying force, like the alphabets that swept the modern world, or the 0 and 1 in the molecular memory of a computer.

“Here,” the soundtrack music swelled, “is YBC 8886. These tables are what scientists today call
Clebsch-Gordan coefficients
. These are mathematical symbols used to integrate products of three spherical harmonics. Clebsch-Gordan coefficients commonly arise in applications involving the addition of angular momentum in
quantum mechanics
.”

“Quantum mechanics!” whispered Dugan, loudly enough that someone in the row in front of us turned around and shushed him.

“If products of more than three spherical harmonics are desired,” said Melville, “use a generalization known as Wigner 6j-symbols or Wigner 9j-symbols. These,” he pointed with the sparkling dot of the laser, “are undoubtedly Wigner 6j-symbols or Wigner 9j-symbols, although in base 60 numbers. Gentlemen, the ancient Babylonians either knew quantum mechanics, or were in communication with someone who did.”

The audience jumped to their feet and bombarded Melville with questions. He held up his hands. “Please wait until I’ve shown all the slides. Then I’ll take questions, one at a time.”

Melville continued, pointing at the projection screen as the slides shuffled through. “Here is the most recent dig at Larsa. This next slide is of the principal investigator supervising the
in situ hologrammetry
. Next is YBC 8886. Tables in this slide are look-up tables of the most sophisticated math we’ve ever seen in any archeological artifact.”

Melville squinted at his glowing NotePad, as if still not familiar with the terminology. “Calabi-Yau spaces are important in string theory. One model posits the geometry of the universe to consist of a ten-dimensional space of the form M x V, where M is a four-dimensional manifold of space-time and V is a six-dimensional compact Calabi-Yau space. Although the main application of Calabi-Yau spaces is in theoretical physics, they’re also interesting from a purely mathematical standpoint.

“Also called Calabi-Yau manifolds, they have interesting properties. One is that the symmetries in the numbers forming the Hodge diamond are a compact Calabi-Yau manifold. It is surprising that these symmetries, called
mirror symmetry
, can be realized by another Calabi-Yau manifold, the so-called mirror of the original Calabi-Yau manifold. The two manifolds together form a mirror pair, through the
Supercompactification Triality Theorem of 2023
. Although the definition can be generalized to any dimension, they are usually considered to have three complex dimensions. Since their complex structure may vary, it is convenient to think of them as having six real dimensions and a fixed smooth structure. The ancient tables are very similar to these.”

He advanced to a page of last year’s Nobel Prize acceptance speeches from Stephen Hawking, Roger Penrose, and Jacob Bekenstein.

“The authors of these tables knew results in
black holes
,
string theory
,
M-theory
, and
multiverse cosmology
3,750 years ago that our greatest minds just rediscovered last year.”

The room exploded in hub-bub. Melville couldn’t get them to settle down. I left as several beefy security guys came into the auditorium. Good time to hit the men’s room.

And then I needed a stiff drink.

2. Nippur and Nanotechnology

“I mentioned today’s
Nanotech Molecular Memories
,” said the Director of the ultra-secret Interstellar Intelligence Directorate to the President of the United States, “which use no science beyond what was known in the late twentieth century. But how could anyone in what’s now Iraq know mid-twentieth century nuclear physics and early twenty-first century black holes, string theory, M-theory, and multiverse cosmology?”

The President responded, “My science adviser told me black holes are an eighteenth century idea. He briefed me that the concept of a body so massive that even light could not escape, was first put forward by English geologist, John Michell.”

“That’s true, ma’am,” said the IID Director, glancing at his NotePad, “And in 1796, French mathematician and astronomer Pierre-Simon Laplace promoted the same idea in his book,
Exposition du système du Monde
. His work was pivotal to the development of mathematical astronomy and statistics. However such
dark stars
were largely ignored in the nineteenth century, since it was not understood how a massless wave such as light could be influenced by gravity.”

“Hair-splitting,” said the President. She sipped coffee from an oversize mug designed by Teddy Roosevelt. “What does that tell us about the clay tablets dug up in Nipple, or whatever the damned place was.”

“The first batch were from Larsa, south of Nippur. The point is, if they were about John Michell’s idea, we could chalk that up to ingenuity, but it still would be qualitative reasoning. Problem is, it wasn’t eighteenth century stuff, which would be amazing, but not such a top priority national security matter. Not even early twentieth century stuff.”

“How so?”

“It was in 1915,” he continued, “that Albert Einstein developed his
Theory of General Relativity
, having earlier shown that gravity does influence light’s motion. A few months later, Karl Schwarzschild, a German Physicist, gave the solution for the gravitational field of a point mass and a spherical mass. Schwarzschild accomplished this triumph using
tensor calculus
, while serving in the German army during World War I. Unfortunately, he died the following year from disease contracted while at the Russian front, so his work was not completed. We see no tensor calculus as such in the Nippur tablets, but there
are
tables of results that we don’t know how else could be computed.”

“Schwarzschild,” said the President, “as in the
Schwarzschild radius
?”

“Exactly. This solution had a peculiar behavior at what is now called the Schwarzschild radius, where it became singular, meaning that some of the terms in Einstein’s equations became infinite.”

“Physicists, unlike mathematicians,” she said, “seem to hate infinity.”

“That’s right. It hurts them when their theories blow up. Monsignor Georges Henri Joseph Édouard Lemaître proposed what became known as the
Big Bang Theory of the Origin of the Universe
, which he called his
Hypothesis of the Primeval Atom
.”

“I don’t like to hear the words
atom
and
bang
, except from my Secretary of Energy, or the Joint Chiefs,” said the President.

“I quite understand, Madam President, but this is potentially a greater threat than nukes. In 1931, Subrahmanyan Chandrasekhar calculated, using general relativity, that a non-rotating body of electron-degenerate matter above 1.44 solar masses would collapse.”

“I’m not familiar with this Chandrasekhar.”

“Subrahmanyan Chandrasekhar was an Indian-born American astrophysicist who, with William A. Fowler, won the 1983 Nobel Prize in Physics for key discoveries on late evolutionary stages of massive stars. Chandrasekhar was the nephew of Sir Chandrasekhara Venkata Raman, who won the Nobel Prize for Physics in 1930.”

“I’ve heard that Nobel prizes run in families like seats in Congress,” she said.

“Yes. Chandrasekhar served on University of Chicago faculty from 1937 until his death in 1995. Arguments were opposed by many contemporaries such as Lev Landau, who argued that some yet unknown mechanism would stop the collapse.”

“Landau, the one from the former USSR?”

“Yes, ma’am. Lev Landau was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics. But I digress.”

“I’m not going anywhere,” she said. “I have a meet-and-greet with some Girl Scouts in a half hour.”

“Okay, I’ll move it along. In 1932, Landau proposed that every star has a condensed core consisting of
one gigantic nucleus
that does not behave in accord with
the ordinary laws of quantum mechanics
. Later he suggested all stars have a neutron core that generates energy as nuclei and electrons condense onto it. He received the 1962 Nobel Prize in Physics for developing a mathematical theory of superfluidity that accounts for the properties of liquid helium II at a temperature below -270.98° Celsius.”