Caitlin knew LensCrafters’ slogan from the ubiquitous commercials: glasses in about an hour. She thought it would be the longest hour of her life. She felt her Braille watch as she, Kuroda, and her mom walked through the mall to the food court — for the first time, without the use of her white cane. Everything was still blurry, and that was giving her a headache. Still, in a way, it was relaxing. To see the people coming toward her! To not bump into things! She hadn’t realized it until now, but she’d always used to walk with her shoulders tensed, preparing for an impact. But now — well, now she had a bounce in her step, something else she’d never thought could happen literally.

Still, all the visual input was disorienting, and she found herself taking a look, then closing her eyes for five or six paces, then looking again. When they got to the food court, Kuroda went to the sushi place — which, Caitlin suspected, would disappoint him — and she and her mom went to Subway. Caitlin was amazed to see how colorful the sandwich fillings were, and, somehow, seeing the food made it taste even better.

The three of them sat together at a little red table with chairs attached to it. Dr. Kuroda used chopsticks to dip a piece of sushi in sauce.

Caitlin couldn’t resist. “Do they tell you in Japan that it’s raw fish?”

Kuroda smiled. “Do they tell you what’s in the special sauce on a Big Mac?”

She laughed. At last the hour was up and they headed back to LensCrafters. Caitlin took a seat on the stool, and the nice woman placed the glasses on her face—

And Caitlin didn’t wait. She got up, and turned around, and looked — really looked — at her mother.

“Wow,” Caitlin said. She paused, trying to come up with a better word, but couldn’t. Her mother’s face was so detailed, so alive! “Wow!”

“Here, let me adjust how they sit…” said the clerk.

Caitlin sat back down and swiveled to face her.

“I’m sorry,” the woman said, “but your ears go up a bit when you smile like that. If you want me to get the frames adjusted properly, you’ll have to stop grinning…”

“I’ll try,” Caitlin said, but she doubted she’d have much success.

Suddenly everything became sharp. The images I was seeing were now…

I struggled for an analogy, found one: just as when I thought intently about things they seemed more focused, so the images I was looking at seemed now.

And, with this greater clarity, I started having revelations about the nature of the other realm. Unlike the lines in my world that flickered in and out of existence, objects in the other realm were permanent. And when objects disappeared for a time it didn’t mean that they had ceased to exist; rather, they were extant but not currently visible and might be encountered again. In a way, that was similar to my own experience: when I’m not making a line to a particular point, the point is still there, and I can connect to it again at a later time.

But my next breakthrough was without precedent in the realm in which I existed. I had a sense of space, of a volume that I encompassed, but the points I connected to were all the same arbitrary distance away, or whole multiples of that same distance. I could link directly to a point, meaning it was one unit away, or get to it through intermediate points, putting it two or more units away. But in this other realm objects could recede in infinitely fine increments, becoming apparently smaller in size, a fact I only belatedly recognized after originally thinking they were actually shrinking. And objects could pass behind each other. Most were opaque, but some were transparent or translucent — and those had been instrumental in letting me at least start to figure out what was going on.

Bit by bit, I was learning to decode this other universe.

When Caitlin, her mom, and Dr. Kuroda returned from the mall, they saw that Caitlin’s father’s car was here, meaning he’d come home surprisingly early on a weekday. Caitlin hurried into the house to see him — to really see him. She came to the open den doorway, Kuroda behind her, while her mom went off to do something else. Blondie’s “Heart of Glass” was playing on his stereo.

The detail Caitlin was perceiving now was overwhelming, and her father’s face was … harder now that she saw it crisply. “Hi, Dad,” she said.

He was sitting at his desk, looking at his LCD monitor. He didn’t meet her eyes. “Hi.”

Still, he’d come home early from work, presumably to see Caitlin, and that made her happy. “Um, whatcha doin’?”

He tilted his head. Caitlin didn’t know what to make of it, but Kuroda seemed to think it was an invitation to come see. He tapped her on the shoulder, urging her to move into the room. She did so, and was pleased that she could make out the characters on the monitor clearly from several feet away, although she still couldn’t read the text.

“I had an idea,” her dad said, “so I came home to check it out.”

“Yes?” said Caitlin.

He didn’t look at Kuroda, but he did address him: “This is more your field than mine, Masayuki,” he said. “I thought I’d look again at the data set we did the Zipf plots on.”

“The secret spook communiqués?” said Caitlin, hoping to get a rise from her dad.

But her father shook his head. “I don’t think that’s what they are anymore.”

He gestured at the monitor.

Kuroda moved in and peered at the screen. “Shannon entropy?”

Caitlin smiled. Sounds like the name of a porn star. “What’s that?”

Kuroda looked at her father, as if giving him first chance to explain, but he said nothing, so Kuroda did: “Claude Shannon was the father of information theory. He came up with a way of gauging not just whether a signal contained information — which is what Zipf plots show — but how complex that information is.”

“How?” asked Caitlin.

“It’s all about conditional probabilities,” said Kuroda. “If you’ve already got a string of information chunks, what’s the likelihood that you can predict what the next chunk will be? If I say, ‘How are,’ you’ve got a really high probability of correctly predicting what the next word will be: ‘you,’ right? That’s what Shannon called third-order entropy: you’ve got a great shot at predicting the third word. In English, Japanese, and most other languages, you actually have a shot — progressively slimmer, but still better than just a random guess — up to the eighth or ninth word, so we say those languages have eighth — or ninth-order Shannon entropy. But after that — after the ninth word — it really is just a random guess what’s coming next, unless the person happens to be quoting poetry or something else that has a fixed form.”

“Cool!” said Caitlin.

There was a black leather couch in the den. Kuroda sat on it, and it made a poof sound. “It is indeed. Mindless communication systems — like the chemical signals employed by plants — have just first-order entropy: knowing the most recent signal gives no clue what the next one might be. Squirrel monkeys show a Shannon entropy of the second or third order: their language, such as it is, has a little predictability, but is really mostly just random noise.”

“What about dolphins?” asked Caitlin, who was now leaning against a bookcase. She loved reading about dolphins, and had already bugged her parents to take her to MarineLand in Niagara Falls as soon as it opened up again in the spring.

“The best studies to date show dolphins have fourth-order entropy — complex, yes, but not as complex as human language.”

“And now, Dad, you’re making one of these plots for the stuff that’s in the background of the Web?”

He still wasn’t used to the fact that she was seeing, Caitlin thought. He could have saved himself a word by just nodding, but instead he said, “Yes.”