Lying on the grass looking at tundra flowers, one couldn’t help thinking about life. In the sunlight the little flowers stood on their stems glowing with their anthracyins, dense with color. Ideograms of order. They did not look like a mere difference in entropic levels. Such a fine texture to a flower petal; drenched in light, it was almost as if it were visible molecule by molecule: there a white molecule, there lavender, there clematis blue. These pointillist dots were not molecules, of course, which were well below visible resolution. And even if molecules had been visible, the ultimate building blocks of the petal were so much smaller than that that they were hard to imagine— finer than one’s conceptual resolution, one might say. Although recently the theory group at Da Vinci had begun buzzing about developments in superstring theory and quantum gravity they were making; it had even gotten to the point of testable predictions, which historically had been string theory’s great weakness. Intrigued by this reconnection with experiment, Sax had recently started trying to understand what they were doing. It meant foregoing sea cliffs for seminar rooms, but in the rainy seasons he had done it, sitting in on the group’s afternoon meetings, listening to the presentations and the discussions afterward, studying the scrawled math on the screens and spending his mornings working on Riemann surfaces, Lie algebras, Euler numbers, the topologies of compact six-dimensional spaces, differential geometries, Grassmannian variables, Vlad’s emergence operators, and all the rest of the mathematics necessary to follow what the current generation was talking about.
Some of this math concerning superstrings he had looked into before. The theory had existed for almost two centuries now, but it had been proposed speculatively long before there was either the math or the experimental ability to properly investigate it. The theory described the smallest particles of spacetime not as geometrical points but as ultramicroscopic loops, vibrating in ten dimensions, six of which were compactified around the loops, making them somewhat exotic mathematical objects. The space they vibrated in had been quantized by twenty-first-century theorists, into loop patterns called spin networks, in which lines of force in the finest grain of the gravitational field acted somewhat like the lines of magnetic force around a magnet, allowing the strings to vibrate only in certain harmonics. These supersymmetrical strings, vibrating harmonically in ten-dimensional spin networks, accounted very elegantly and plausibly for the various forces and particles as perceived at the subatomic level, all the bosons and fermions, and their gravitational effects as well. The fully elaborated theory therefore claimed to mesh successfully quantum mechanics with gravity, which had been the problem in physical theory for over two centuries.
All very well; indeed, exciting. But the problem, for Sax and many other skeptics, came with the difficulty of confirming any of this beautiful math by experiment, a difficulty caused by the very, very, very small sizes of the loops and spaces being theorized. These were all in the 10-33 centimeter range, the so-called Planck length, and this length was so much smaller than subatomic particles that it was hard to imagine. A typical atomic nucleus was about 10-13 centimeter in diameter, or one millionth of a billionth of a centimeter. First Sax had tried very hard to contemplate that distance for a while; hopeless, but one had to try, one had to hold that hopelessly inconceivable smallness in the mind for a moment. And then remember that in string theory they were talking about a distance twenty magnitudes smaller still— about objects one thousandth of one billionth of one billionth the size of an atomic nucleus! Sax struggled for ratio; a string, then, was to the size of an atom, as an atom was to the size of . . . the solar system. A ratio which rationality itself could scarcely comprehend.
Worse yet, it was too small to detect experimentally. This to Sax was the crux of the problem. Physicists had been managing experiments in accelerators at energy levels on the order of one hundred GeV, or one hundred times the mass energy of a proton. From these experiments they had worked up, with great effort, over many years, the so-called revised standard model of particle physics. The revised standard model explained a lot, it was really an amazing achievement, and it made predictions that could be proved or disproved by lab experiment or cosmological observations, predictions that were so varied and had been so well fulfilled that physicists could speak with confidence about much of what had gone on in the history of the universe since the Big Bang, going as far back as the first millionth of a second of time.
String theorists, however, wanted to make a fantastic leap beyond the revised standard model, to the Planck distance which was the smallest realm possible, the minimum quantum movement, which could not be decreased without contradicting the Pauli exclusion principle. It made sense, in a way, to think about that minimum size of things; but actually seeing events at this scale would take experimental energy levels of at least 1019 GeV, and they could not create those. No accelerator would ever come close. The heart of a supernova would be more like it. No. A great divide, like a vast chasm or desert, separated them from the Planck realm. It was a level of reality fated to remain unknown to them in any physical sense.
Or so skeptics maintained. But those interested in the theory had never been dissuaded from studying it. They searched for indirect confirmation of the theory at the subatomic level, which from this perspective now seemed gigantic, and from cosmology. Anomalies in phenomena that the revised standard could not explain, might be explained by predictions made by string theory about the Planck realm. These predictions had been few, however, and the predicted phenomena very difficult to see. No real clinchers had been found. But as the decades passed, a few string enthusiasts had always continued to explore new mathematical structures, which might reveal more ramifications of the theory, might predict more detectable indirect results. This was all they could do; and it was a very chancy road for physics to take, Sax felt. He believed in the experimental testing of theories with all his heart. If it couldn’t be tested, it remained math only, and its beauty was irrelevant; there were lots of bizarrely beautiful exotic fields of mathematics, but if they weren’t modeling the phenomenal world, Sax wasn’t interested.
Now, however, after all the decades of work, they were beginning to make progress in ways that Sax found interesting. At the new supercollider in Rutherford Crater’s rim, they had found the second Z particle that string theory had long predicted would be there. And a magnetic monopole detector, orbiting the sun out of the plane of the ecliptic, had captured a trace of what looked to be a fractionally charged unconfined particle with a mass as big as a bacterium— a very rare glimpse of a “weakly interacting massive particle,” or WIMP. String theory had predicted WIMPs would be out there, while the revised standard did not call for them. That was thought provoking, because the shapes of galaxies showed that they had gravitational masses ten times as large as their visible light revealed; if the dark matter could be explained satisfactorily as weakly interacting massive particles, Sax thought, then the theory responsible would have to be called very interesting indeed.
Interesting in a different way was the fact that one of the leading theorists in this new stage of development was working right there in Da Vinci, part of the impressive group Sax was sitting in on. Her name was Bao Shuyo. She had been born and raised in Dorsa Brevia, her ancestry Japanese and Polynesian. She was small for one of the young natives, though still half a meter taller than Sax. Black hair, dark skin, Pacific features, very regular and somewhat plain. She was shy with Sax, shy with everyone; she even sometimes stuttered, which Sax found extremely endearing. But when she stood up in the seminar room to give a presentation, she became quite firm in hand if not in voice, writing her equations and notes on the screen very quickly, as if doing speed calligraphy. Everyone in these moments attended to her very closely, in effect mesmerized; she had been working at Da Vinci for a year now, and everyone there smart enough to recognize such a thing knew that they were watching one of the pantheon at work, discovering reality right there before their eyes.
The other young turks would
interrupt her to ask questions, of course— there were many good minds in that group— and if they were lucky, off they would all go together, mathematically modeling gravitons and gravitinos, dark matter and shadow matter— all personality and indeed all persons forgotten. Very productive exciting sessions; and clearly Bao was the driving force in them, the one they relied on, the one they had to reckon with.
It was disconcerting, a bit. Sax had met women in math and physics departments before, but this was the only female mathematical genius he had ever even heard of, in all the long history of mathematical advancement, which, now that he thought of it, had been a weirdly male affair. Was there anything in life as male as mathematics had been? And why was that?
Disconcerting in a different way was the fact that areas of Bao’s work were based on the unpublished papers of a Thai mathematician of the previous century, an unstable young man named Samui, who had lived in Bangkok brothels and committed suicide at the age of twenty-three, leaving behind several “last problems” in the manner of Fermat, and insisting to the end that all of his math had been dictated to him by telepathic aliens. Bao had ignored all that and explained some of Samui’s more obscure innovations, and then used them to develop a group of expressions called advanced Rovelli-Smolin operators, which allowed her to establish a system of spin networks that meshed with superstrings very beautifully. In effect this was the complete uniting of quantum mechanics and gravity at last, the great problem solved— if it were true. And true or not, it had been powerful enough to allow Bao to make several specific predictions in the larger realms of the atom and the cosmos; and some of these had since been confirmed.
So now she was the queen of physics— the first queen of physics— and experimentalists in labs all over were on-line to Da Vinci, anxious to have more suggestions from her. The afternoon sessions in the seminar room were invested with a palpable sense of tension and excitement; Max Schnell would start the meeting, and at some point call on Bao; and she would stand and go to the screen at the front of the room, plain, graceful, demure, firm, pen flying over the screen as she gave them a way to calculate precisely the neutrino mass, or described very specifically the ways strings vibrated to form the different quarks, or quantized space so that gravitinos were divided into three families, and so on; and her colleagues and friends, perhaps twenty men and one other woman, would interrupt to ask questions, or add equations that explained side issues, or tell the rest of them about the latest results from Geneva or Palo Alto or Rutherford; and during that hour, they all knew they were at the center of the world.
And in labs on Earth and Mars and in the asteroid belt, following her work, unusual gravity waves were noted; in very difficult delicate experiments; particular geometric patterns were revealed in the fine fluctuations in the cosmic background radiation; dark-matter WIMPs and shadow-matter WISPs were being sought out; the various families of leptons and fermions and leptoquarks were explained; galactic clumping in the first inflation was provisionally solved; and so on. It seemed as if physics might be on the brink of the Final Theory at last. Or at least in the midst of the Next Big Step.
• • •
Given the significance of what Bao was doing, Sax felt shy about speaking to her. He did not want to waste her time on trivial things. But one afternoon at a kava party, out on one of the arc balconies overlooking Da Vinci’s crater lake, she approached him— even more shy and stumbling than he was— so much so that he was forced into the very unusual position of trying to put someone else at ease, finishing sentences for her and the like. He did that as best he could, and they stumbled along, talking about his old Russell diagrams for gravitinos, useless now he would have thought, though she said they still helped her to see gravitational action. And then when he asked a question about that day’s seminar, she was much more relaxed. Yes, clearly that was the way to put her at ease; he should have thought of it immediately. It was what he liked himself.
After that, they got in the habit of talking from time to time. He always had to work to draw her out, but it was interesting work. And when the dry season came, in the fall helionequinox, and he started going out sailing again from the little harbor Alpha, he asked her haltingly if she would like to join him, and they stuttered their way through a deeply awkward interaction, which resulted in her going out with him the next nice day, sailing in one of the lab’s many little catamarans.
When day sailing, Sax stayed in the little bay called the Florentine, southeast of the peninsula, where Ravi Fjord widened but before it became Hydroates Bay. This was where Sax had learned to sail, and where he still felt best acquainted with the winds and currents. On longer trips he had explored the delta of fjords and bays at the bottom end of the Marineris system, and three or four times he had sailed up the eastern side of the Chryse Gulf, all the way to Mawrth Fjord and along the Sinai Peninsula.
On this special day, however, he confined himself to the Florentine. The wind was from the south, and Sax tacked down into it, enlisting Bao’s help at every change of tack. Neither of them said much. Finally, to get things started, Sax was forced to ask about physics. They talked about the ways in which strings constituted the very fabric of space-time itself, rather than being replacements for points in some absolute abstract grid.
Thinking it over, Sax said, “Do you ever worry that work on a realm so far beyond the reach of experiment will turn out to be a kind of house of cards— knocked over by some simple discrepancy in the math, or some later different theory that does the job better, or is more confirmable?”
“No,” Bao said. “Something so beautiful as this has to be true.”
“Hmm,” Sax said, glancing at her. “I must admit I’d rather have something solid crop up. Something like Einstein’s Mercury— a known discrepancy in the previous theory, which the new theory resolves.”
“Some people would say that the missing shadow matter fills that bill.”
“Possibly.”
She laughed. “You need more, I can see. Perhaps some kind of thing we can do.”
“Not necessarily,” Sax said. “Although it would be nice, of course. Convincing, I mean. If something were better understood, so that we could manipulate it better. Like the plasmas in fusion reactors.” This was an ongoing problem in another lab at Da Vinci.
“Plasmas might very well be better understood if you modeled them as having patterns imposed by spin networks.”
“Really?”
“I think so.”
She closed her eyes— as if she could see it all written down, on the inside of her eyelids. Everything in the world. Sax felt a piercing stab of envy, of— loss. He had always wanted that kind of insight; and there it was, right in the boat beside him. Genius was a strange thing to witness.
“Do you think this theory will mean the end of physics?” he asked.
“Oh no. Although we might work out the fundamentals. You know, the basic laws. That might be possible, sure. But then every level of emergence above that creates its own problems. Taneev’s work only scratches the surface there. It’s like chess— we might learn all the rules, but still not be able to play very well because of emergent properties. Like, you know, pieces are stronger if they’re out in the center of the board. That’s not in the rules, it’s a result of all the rules put together.”
“Like weather.”
“Yes. We already understand atoms better than weather. The interactions of the elements are too complex to follow.”
“There’s holonomy. Study of whole systems.”
“But it’s just a bunch of speculation at this point. The start of a science, if it turns out to work.”
“And so plasmas, though?”
“Those are very homogeneous. There’s only a very few factors involved, so it might be amenable to spin-network analysis.”
“You should talk to the fusion group about that.”
“Yes?” She looked surprised.
“Yes.”
Then a hard gust hit, and the
y spent a few minutes watching the boat respond, the mast sucking in sails with a bit of humming until they were reset, and running across the strengthening breeze, into the sun. Light flaked off the fine black hair gathered at the back of Bao’s neck; beyond that, the sea cliffs of Da Vinci. Networks, trembling at the touch of the sun— no. He could not see it, with eyes open or closed.
Cautiously he said, “Do you ever wonder about being, you know. Being one of the first great women mathematicians?”
She looked startled, then turned her head away. She had thought about it, he saw. “The atoms in a plasma move in patterns that are big fractals of the spin-network patterns,” she said.
Sax nodded, asked more questions about that. It seemed possible to him that she would be able to help Da Vinci’s fusion group with the problems they were having engineering a lightweight fusion apparatus. “Have you ever done any engineering? Or physics?”
Affronted: “I am a physicist.”
“Well, a mathematical physicist. I was thinking of the engineering side.”
“Physics is physics.”
“True.”
Only once more did he push, and this time indirectly. “When did you first learn math?”
“My mom gave me quadratic equations at four, and all kinds of math games. She was a statistician, very keen about it all.”