The road suddenly splits into three and a sign appears. There are arrows pointing down all three roads and the sign says how far away Digitopolis is in various units of measurement. The Humbug suggests they travel by miles, while Milo suggests it’d be shorter to go by half inches. Tock doesn’t care what units they use; he thinks they need to pick a road. As they argue, a small, strange figure steps out. He’s “constructed” of lines and angles that comes together into a many-sided shape. (The narrator directs readers to look at the picture to see what they mean.)
Here, Tock essentially encourages Milo and the Humbug to not get lost in specifics. Whatever units they use to measure their travel, the bigger question is which road to take. Milo’s suggestion that it’d be “shorter” to use half inches betrays how little he knows about math—choosing a smaller unit of measurement doesn’t make a measurement any shorter.
The figure introduces himself as the Dodecahedron, which is a shape with 12 faces. He shows off his 12 faces; each has a different expression. When Milo introduces himself, the Dodecahedron frowns—Milo only has one face, and his name is odd. The Dodecahedron thinks it’s strange that other people with only one face have names like John or George—around here, “everything is called exactly what it is.” This is Digitopolis, and everything is very precise.
The Dodecahedron’s many faces are funny—but his inability to understand that a human face isn’t the same as a shape’s face (meaning one of its sides) shows how lacking Digitopolis is in figurative “rhyme or reason.” He doesn’t allow for the possibility that everything isn’t “called exactly what it is.” A person’s name or appearance, for instance, aren’t the full story of who they are.
Milo asks the Dodecahedron which road they should take. In response, the Dodecahedron gives a math problem asking which of the three cars going at different speeds along different routes will arrive at their destination first. The Humbug shouts that the answer is 17, while Milo can’t figure it out and admits he’s bad at math problems. The Dodecahedron says that’s a shame. With math, you know that if a small beaver can build a small dam in two days, a massive beaver could build Boulder Dam. The Humbug mutters that beavers don’t come that big, but the Dodecahedron notes that if you could find one, “you’d certainly know what to do with him.” Milo insists that’s absurd, and the Dodecahedron says it’s accurate nevertheless—if you want the questions to be right, you have to find some sense.
Absurdity can be fun—the Dodecahedron’s anecdote about what a massive beaver can do is ridiculous and funny. But, as Humbug and Milo note, this isn’t the most useful way of looking at things if one wants to solve a practical problem—like building such a large dam. For that matter, beaver dams differ wildly from manmade dams, so this is also a language misunderstanding (as when the Dodecahedron got confused about Milo’s name having nothing to do with his single face). Digitopolis’s focus on only numbers makes it difficult for the Dodecahedron to effectively and meaningfully communicate with the newcomers.
Tock has been working on the original math problem, and he says the cars will arrive at the same place at the same time. Milo asks if the roads are all the right way then, but the Dodecahedron says they’re actually all the wrong way—not all choices are right. He spins the sign and the three roads converge into one. It’s a bumpy road, and it leads to a rocky land. Milo asks if the numbers are made here. The Dodecahedron says they’re mined and asks if Milo knows anything about numbers. Milo admits he’s never thought of them as important, which infuriates the Dodecahedron—you need numbers to have tea for two, or to know how high your hopes are. At this, the Dodecahedron leads the party into a cave. This is the numbers mine.
Because of Milo’s incomplete grasp of math and logic, he discovers again that he needs to ask for and accept help. His willingness to ask the Dodecahedron questions shows he’s learning and becoming more engaged. Keep in mind that before this, Milo hasn’t considered numbers important because he hasn’t had anyone tell him why he should care. The Dodecahedron, then, fills this role and notes that numbers are a part of everyday life—and even figures of speech. His figures of speech, though, show again that words and numbers are intimately connected.
Milo squints and sees that they’re in a big cavern. Men his size are digging and pushing carts. Milo asks who the mine belongs to—and a booming voice answers that it’s his. The figure coming toward Milo is clearly the Mathemagician: he wears a flowing robe covered in equations and carries a staff that’s a giant pencil. Milo asks if there are precious stones in the mine, and the Mathemagician says there are. He picks a small object out of a cart, which turns out to be a sparkly five. Tock says the numbers are exceptional. As Milo gives a handful of numbers to the Dodecahedron, he drops several. The Mathemagician says they use the broken ones for fractions, and then he shows them a pile of emeralds, rubies, and diamonds. They’re a nuisance, and nobody knows what to do with them. He whistles; it’s lunchtime.
Milo and the Mathemagician clearly take “precious stones” to mean different things. Milo is talking about the rubies and diamonds, which to the Mathemagician are useless. The Mathemagician is talking about the numbers that come out of the mine, which to Milo aren’t as valuable as the precious stones. This becomes another lesson in perspective—because of their differing backgrounds, Milo and the Mathemagician look at the Numbers Mine very differently.