by Carlos Bueno
“No, I’m sure Tortoise is right,” said Laurie. “Don’t you see? Every piece he adds is half as long as the one before. That leaves just enough room left over. Even if he adds an infinite number of pieces, the string will never reach two inches.”
“Well, hardly ever,” said Tortoise.
Achilles grimaced. “It appears you’ve proven the impossible again, Tortoise. But just to make sure, I will check the arithmetic myself.” He continued to scribble in his notebook:
+ 1/512 inch
+ 1/1,024 inch
+ 1/2,048 inch
+ 1/4,096 inch
+ 1/8,192 inch
+ 1/16,384 inch
+ 1/32,768 inch
+ 1/65,536 inch
+ 1/131,072 inch
+ 1/262,144 inch . . .
“That should keep him busy. If anyone has the patience to actually count to infinity, it’s Achilles. Thank you for your assistance, Miss Ipsum.”
“You’re welcome, Mister Tortoise,” said Laurie. “I didn’t know something so big could be so small.”
“That’s the Power of Two,” said Tortoise. “If you cut a number into two halves, then cut it in two again, and so on, very soon it will be too small to see. But there will always be something left over.”
“Mister Tortoise, do you know how long this road is? It feels like it goes on forever. I’m trying to get to Symbol.”
“This road is quite long,” he replied. “In fact, it is infinite.”
“Oh, no! How do I get to the end?”
“You can do it in two simple steps.”
“How?”
“How do you think? A step with your right foot, then a step with your left foot,” said Tortoise. “Your point of view is what’s important. It’s integral.”
Of course! If an infinite string could be less than two inches long, then an infinite road could certainly be less than two steps, if you looked at it in the right way. Laurie closed her eyes and took a deep breath. She tried to imagine an infinite road. That was a little too much to handle, so instead she imagined a really, really, really, really long road. Then she imagined folding it in half. Then she folded it in half again, and again, and again, and again . . .
When Laurie opened her eyes, Achilles and Tortoise were gone. The infinite road was a tiny, short thing now, hardly more than a stepping stone. She stepped forward with her right foot. Then she stepped again with her left foot. In front of her was a road sign that read . . .
Chapter 5. Welcome to Symbol
The town of Symbol was surrounded by high, perfectly smooth walls. A large stream split into two and flowed around either side of the town.
“Do you know anything about this place?” Laurie asked.
“The people are kind of strange,” said Xor.
“What does that mean?”
“You’ll see.”
The path led to a gateway with a turnstile. It looked like this: ╞
Two boys about Laurie’s age were guarding the gateway. One wore a bright white suit and a black shirt. The other had a bright black suit and a white shirt.
Maybe they’ll ask me a riddle, Laurie thought. Or I’ll need to figure out which one is a liar and which one tells the truth! Laurie had read a lot of fairy tales, you see. She believed she was figuring out how this place worked.
“Name?” asked one.
“Laurie Ipsum.”
“Password?” asked the other.
“Why, I don’t know the password,” Laurie said.
“Then you may not enter!” the boys said in unison.
“That’s not much of a riddle.”
“A riddle? Ha ha, no, begging your pardon, miss. That’s not how it works. There are no riddles, no bets, no liars, and no truth-tellers. We’ve read those books too, haven’t we, Tollens?”
“Yes, we have, Ponens. There’s none of that mythic fairy-tale stuff in our System, miss. Word games and clever riddles, ha! That’s just bad security!”
“But that’s not fair!” cried Laurie. “How do I get inside?”
“It’s very simple,” said Ponens. “If you have a password, then that means you can pass through the semantic turnstile.”
“If you can’t pass through the turnstile, then that means you don’t have a password. It’s only logical,” Tollens said. “Do you have a password?”
“No, I don’t know what it is,” Laurie said.
“Then you may not enter!” they said again, together.
“Can you give me a hint?” Laurie was sure she could guess the password, given some kind of clue.
“Yes, hints are part of the System,” said Ponens.
“Oh, good.”
“Did you set up a hint with your account?” asked Tollens.
“Well, no,” she said. “This is my first time here.”
“Then it’s hard luck for you, miss,” said Ponens. “Once you are inside, do make sure to set up a hint for the next time.”
“And remember to change your password to something memorable, but hard to guess,” said Tollens. “It’s just good security.”
“But I’m trying to get inside! I don’t know what to do!”
“It’s very simple,” said Ponens. “If you have a password, then that means you can pass through the semantic turnstile.”
“And if you can’t pass through the turnstile, then that means you don’t have a password. It’s only logical,” Tollens said.
“That’s not completely true, is it?” Laurie said. “What if the turnstile is broken? I wouldn’t be able to enter even if I had a password.”
“Um . . .” Tollens looked a little unsure of himself.
“Or what if I fooled you into believing I had the password, even if I didn’t? Then I could enter without it.”
“Hmm.” Ponens considered Laurie’s argument, trying to find a flaw.
Laurie rushed on. “Or what if I had the password, but I didn’t want to give it to you?”
“No, you still couldn’t encroach on our premises,” Ponens said with more confidence. “You have to give us a password that matches the name you give us.”
“Is it ‘Laurie’?” she guessed.
“No!” they shouted together.
“Is it ‘November First’?” That was Laurie’s birthday.
“No!” they shouted again.
“Only one try left, miss,” said Tollens.
“Oh, no! Really?”
“You can try only so many times. It’s just good security.”
“Do I even have an account?” she wondered.
“We can’t confirm or deny that,” answered Ponens. “That would be bad security.”
“Do I . . . not have an account?”
“We can’t deny or confirm that, either!” answered Tollens.
“It’s very simple—” began Ponens.
“—yes, yes,” Laurie interrupted. “If I have it, I know it, but if I don’t, you can’t tell me. And you can give me a hint only if I set it up earlier!” This certainly was very good security. Think, Laurie, think!
“Okay, let’s start over,” she said.
“Very well, miss,” said Tollens. “Name?”
“I told you, it’s Lau—” she stopped herself. “Actually . . . my name is Eponymous Bach.”
“Password?” said Ponens.
It couldn’t be that simple, could it?
“Bach’s Password.”
“Welcome!” said Ponens and Tollens, waving her through the turnstile.
Of course Bach would name her password after herself!
Chapter 6. A Tinker’s Trade
When Laurie and Xor were safely inside the town walls, the little lizard popped his head out of Laurie’s pocket.
“See what I mean? Let’s hope they don’t figure out what you did to get in here,” Xor said. “So, why are we here?”
“We’re looking for information that could help me get home. Maybe we can find a map or something.”
“Oh,” said Xor. “I was hoping you were
going to say food. Why don’t we try this place?”
In front of them was a storefront with a very fancy sign painted on the window:
“Al-go-rith-ms. That sounds like a kind of fruit.”
“Are you always hungry, Xor?”
“Time flies like an arrow, and fruit flies like a banana. Let’s see if there’s a fruit fly problem I can help them solve.”
A bell jingled as Laurie opened the door. “Hello, hello!” the shopkeeper said. “And welcome to my shop. I’m Tinker, and you are looking for a finely crafted algorithm, am I right?”
Laurie looked at the items listed on the chalkboard, but they didn’t make any sense.
“I’m not sure. What is an algorithm? Can you eat it?” asked Laurie.
“What? No, it’s just a fancy way of saying ‘how to do something.’ But Algorithm looks more impressive on the sign,” said Tinker.
Xor turned orange with disappointment.
“How to do something,” repeated Laurie. “In that case, I want to find a sensible way to visit every town.”
“That sounds like an interesting problem. What have you been doing so far?”
Laurie told Tinker about her adventure in the Red-Black Forest and her visit with Eponymous Bach.
“A Hamiltonian path, eh?” said Tinker. “That’s a tough one. I hate to say it, because he sounds like a nice person, but the Wandering Salesman might take a long, long time to finish his tour of all the towns.”
“Oh, no! But why?”
“If you always go to the nearest town you haven’t visited yet, you might miss a town that’s just a little farther away. Then you go to another town that’s closer to you but still farther from the one you missed, and so on. You can end up crisscrossing the whole country to get to the last few towns.”
“That sounds exhausting,” said Laurie. The Wandering Salesman wasn’t so sensible after all! “So how do I find the shortest path?”
“I’ll see what I have in stock. But it might be expensive.”
“I don’t have much money with me,” Laurie said. She took a few quarters from her pocket and showed them to Tinker.
He looked at them with surprise. “Quarter Dollar? I don’t know what a Dollar is, never mind a quarter of one. Is this money where you come from?”
“Of course it’s money! That’s seventy-five cents,” she said.
“Cents? We use Fair Coins here.”
“What’s a Fair Coin?”
“Well, they are a bit bigger than these Quarter Dollars of yours, but not nearly as pretty! You can tell genuine Fair Coins because they always flip heads or tails, fifty-fifty.”
“But you can flip quarters fifty-fifty, too!”
“That may be true, but I can’t just take your word for it, can I? Here, all Fair Coins must be certified Fair.”
Laurie was crestfallen.
“Don’t look so sad! I do want to help you,” said Tinker. “Maybe we can do a trade. It so happens I’m in the market for a particular algorithm.”
“But I don’t have any algorithms, either,” said Laurie.
“That’s not a problem,” said Tinker. “You can compose new ones any time you want, with a little bit of thinking.”
“I can? How?”
“Well, everyone develops their own style. You can put little ideas together to make big ideas. Or you put two ideas side by side and compare them. Or you start with big ideas and take them apart.”
“You mean like Eponymous does?”
“Yes, just like her. She’s a great Composer.”
Laurie had never thought that she could do things like that herself. But Tinker seemed to think it was normal.
“So what do I do?”
“The algorithm I’m looking for is how to draw a circle,” Tinker said. “It’s a tough one, so you’ll have to use your imagination. I’ve asked all the adults and even Ponens and Tollens already, but all they do is mutter about x squared plus y squared and never get anywhere.”
“Take a look at this.” He handed Laurie a wind-up toy animal. It had a Shell, and was Round and Green. “This turtle can do three things: it can move forward or backward, it can turn, and it can draw a little dot on the paper.”
“Hey, that’s pretty neat!”
“Yes, but the thing is, it doesn’t know how to do anything else. That’s where the algorithm comes in.” Tinker took out a piece of paper and wrote what looked like a little poem:
Go forward one inch,
make a mark,
repeat five times.
Then he wound up the turtle and placed it on the poem. It went zzzrbt bzzaap whuzzzsh, and so on. Then it drew a line of dots, just like the poem said:
“You see? If you put little ideas together, you can make bigger ones,” Tinker said. “And you can compose those ideas into even bigger and bigger ones.”
“How do you do that?” asked Laurie.
“By giving them a name. You can use the name like a handle: you’d carry a pot of soup by the handle, and you can move around an entire idea just by writing its name. Here, let’s call the first idea LINE. Then you can put four lines together to make a square.”
LINE:
Go forward one inch,
make a mark,
repeat five times.
SQUARE:
Make a LINE,
Make a right turn,
repeat four times.
Make a SQUARE.
The little turtle zzzrbted and whuzzzshed and bzzaaped, then drew this:
Laurie was amazed. It was like magic, but every step made sense.
“So, knowing what the turtle can do, can you teach it how to draw a circle?” Tinker asked.
“I don’t know,” Laurie said, “but I want to try!”
“That’s good enough for me. Here, you can work at my desk. There’s plenty of paper and compasses and things like that.”
Laurie sat down at Tinker’s desk. She doodled with the compass and played with the turtle for a while, trying to remember what she knew about circles.
A circle is round. No, not just round—perfectly round. You put the pin in the center, and the pencil spins around. To make a bigger one, you open the compass; to make a smaller one, you close the compass. If you change the width of the compass when it’s spinning, it doesn’t make a circle . . .
Suddenly an idea, or maybe a memory, popped into her head: a circle is all of the points that are exactly the same distance from the center. Hmm, what if you . . .
Go forward one inch,
make a mark,
go back one inch,
turn right a tiny bit,
then repeat!
After Laurie wrote out her poem, she wound up the little turtle again and placed it on the paper. It buzzed and burbled for a moment, then drew this:
“It’s working!” she called to Tinker. “Hey, it’s not stopping.” The turtle was drawing over dots it had already drawn.
“I think it’s because you told it to repeat, but not how many times,” said Tinker.
“Well, it should stop when the circle is done,” Laurie said.
“It doesn’t really understand circles,” Tinker said. “It’s just a toy turtle, remember? You have to teach it.”
Laurie thought a little more, then rewrote her poem:
CIRCLE:
Go forward one inch,
make a mark,
go back one inch,
turn right one degree,
repeat three hundred sixty times.
Then she realized that she could make circles of any size she wanted. It was just like opening the compass wider.
TWO-CIRCLE:
Go forward two inches,
make a mark,
go back two inches,
turn right one degree,
repeat three hundred sixty times.
“This is interesting. You’re working really hard!” Tinker scratched his head. “But as it is, it’s no good.”
“Why?”
“People want
to make lots of different circles,” he said. “I’ll have to keep a lot of algorithms of different sizes, just in case someone wants three-and-nine-thirteenths inches or four-and-three-quarters inches.”
“Well, what if you tell the turtle how big to make the circle?” she said. “Maybe like this.”
ANY-CIRCLE (how-big?):
Go forward how-big? inches,
make a mark,
go back how-big? inches,
turn right one degree,
repeat three hundred sixty times.
“And then,” she said, “instead of ONE-CIRCLE or TWO-CIRCLE, you can say ANY-CIRCLE(one), or (two), or even (one-and-eleventy-sevenths)!”
“Good idea, Laurie. That’s a lot simpler,” said Tinker. “I was worried you were going to fill my shop with circles!”
“You know, the turtle is drawing really slowly. Not like when it was drawing the square,” she said.
It was true. The turtle would crawl all the way to the edge of the circle, then make a mark, then crawl all the way back to the center, 360 times. With small circles it wasn’t too bad, but big circles took a lot longer.
“Hmm,” Tinker said. “It spends a lot more time running back and forth than it does making marks. Do you think you can reduce the running time?”
It makes sense, but it isn’t sensible. Laurie thought and doodled, and doodled and thought, but she couldn’t figure out how to make it more sensible. The turtle has to go back to the center, right? How else could it know where the edge of the circle was?
Laurie let her eyes wander around the room. Xor was staring at a moth that was flying in lazy loops around a lightbulb. His skin was slowly fading from red to yellow and back to red. The moth went around and around. It was hypnotic. Around and around and around and . . .
Oh! If the moth doesn’t have to go to the center of the lightbulb to fly around it in a circle, then why does the turtle need to go back to the center to draw one?
Laurie reached for a fresh piece of paper before the idea got away. Don’t let a new thing out of your sight without a name.