61 pages • 2 hours read
Norton JusterThe Phantom Tollbooth
Fiction | Novel | Middle Grade | Published in 1961A modern alternative to SparkNotes and CliffsNotes, SuperSummary offers high-quality Study Guides with detailed chapter summaries and analysis of major themes, characters, and more.
Chapters 14-15Chapter Summaries & Analyses
Chapter 14 Summary: “The Dodecahedron Leads the Way”
As the travelers near Digitopolis, they encounter a road sign at an intersection that indicates the city is five miles—or 8,800 yards, or 26,400 feet, or 316,300 inches—away. The road splits into three; they don’t know which to take.
From behind the sign steps a strange creature with a 12-sided head, each side containing a face with a different expression. This Dodecahedron explains, “I have one for smiling, one for laughing, one for crying, one for frowning, one for thinking, one for pouting, and six more besides” (173). He wonders how Milo can do all that with only one face and asks if everyone with one face is named Milo. Milo says people have different names; the Dodecahedron says that, in Digitopolis, every triangle is called a triangle, and circles are simply called circles, and so on. Anything else would be hopelessly confusing.
They ask the Dodecahedron which road to take; he answers with a complicated word problem involving cars, speeds, and road lengths. Milo and Humbug can’t figure it out, and Milo admits that he does poorly at thought problems. Tock figures out that all three roads arrive at the city at the same time. Milo reasons that any of the roads is the correct path, but the Dodecahedron insists that they’re all wrong: “Just because you have a choice, it doesn’t mean that any of them has to be right” (175-76).
The Dodecahedron spins the sign, and the three roads are replaced with a single one. He joins them, and they drive forward, bouncing along on the bumpy road until they arrive at a cave entrance. Milo asks if numbers are made; the Dodecahedron replies that they’re mined; he chides Milo for not knowing that. Embarrassed, Milo retorts that numbers aren’t very important. The Dodecahedron becomes livid and lectures the boy on the importance of numbers: “If you had high hopes, how would you know how high they were? And did you know that narrow escapes come in all different widths?” (177), and so on.
They enter the mine: It’s honeycombed with passages and hung with large, glowing stalactites. Everywhere, short men strike at the walls and fill carts with stones. They meet the owner, the Mathemagician, who wears a pointed cap and a robe embroidered with equations. The Mathemagician shows them some mined stones, each one a number. Workers polish the stones; they’re then sent out across the world. Milo accidentally drops one; the Mathemagician says not to worry because broken numbers are used as fractions.
The Humbug asks if there are any precious stones. The Mathemagician shows them a giant pile of diamonds, emeralds, rubies, and other precious gems. They’re considered useless and a nuisance.
Chapter 15 Summary: “This Way to Infinity”
A huge cauldron arrives, and the workers ladle out lunch. The three hungry travelers fill their bellies and then have seconds, thirds, fourths, and more, but the more they eat, the hungrier they get. The Dodecahedron explains that they’ve consumed “subtraction stew” and that, in Digitopolis, people begin full and eat until they’re hungry. Milo says he prefers to do it the other way around. The Mathemagician supposes that Milo next will claim he only sleeps when tired.
The Mathemagician’s staff is topped with a large eraser, which he rubs against the cave’s ceiling. Suddenly, he and the travelers are standing in his study. He explains that “the best way to get from one place to another is to erase everything and begin again” (187). The study is circular, with 16 windows, one for each main compass point. All the furniture pieces are labeled with their dimensions and distances from one another. Hanging from walls and ceilings are all types of measuring devices.
Milo asks if the Mathemagician always travels via eraser. The man draws a line with the pencil end of his staff, steps through the line, and suddenly stands at the other end of the room: “Most of the time I take the shortest distance between any two points” (188). He then writes “7 × 1 = 7” on a giant note pad (188), and suddenly there are seven Mathemagicians. These are for when he needs to be in several places at once. He agrees that his staff is simply a big pencil, but with practice, it can do wonderful things.
As to whether he can make things disappear, the Mathemagician writes a long arithmetic problem on an easel. Milo reckons that it sums to zero. “Precisely,” says the Mathemagician, and the numbers all disappear. Milo asks what the biggest number is, and the Mathemagician shows him a number 3, dug from the mine that’s twice his height. Milo says he meant the longest, not the largest, number, so the Mathemagician pulls out a very wide eight that’s as long as the three was high.
Tock figures out that Milo wants to know “the number of greatest possible magnitude” (189). The Mathemagician asks Milo to name the greatest number he can imagine; Milo offers one just shy of 10 trillion. The Mathemagician tells him to add the number 1 to it and then do so again, and again, and again. Milo asks when he can stop; the Mathemagician says, “Never.” Similarly, Milo’s best guess for a tiny number, a millionth, can be divided in half over and over and over and never get to the smallest possible fraction.
Milo asks where such a tiny number would be stored. The Mathemagician says, “In a box that’s so small you can’t see it—and that’s kept in a drawer that’s so small you can’t see it, in a dresser that’s so small you can’t see it” (191), and so on. He takes Milo to a window where an attached line begins and heads off into the distance. If Milo follows that line forever and then turns left, he’ll be in the land of Infinity, where all infinite numbers are kept.
Milo says that’ll take too much time. The Mathemagician suggests instead that he simply climb the nearby stairs, which also lead to Infinity. Milo says he’ll be right back and bounds up the stairs.
Chapters 14-15 Analysis
Milo and his companions arrive in Digitopolis for a short visit, where they intend to receive permission from the city’s king, the Mathemagician, to bring back the princesses Rhyme and Reason. They meanwhile learn unusual and fascinating facts about math.
While Dictionopolis is the capital of words, Digitopolis is where numbers matter. The two cities share several symmetries. In Dictionopolis, letters grow on trees, whereas in Digitopolis, numbers are mined; both cities thus produce their main product from the earth. The city’s rulers, brothers who resemble each other in size, intelligence, high energy, and the arrogant assumption that each is the more important ruler, become a matched pair of sorts. Both also agree that Rhyme and Reason must return to the realm.
Milo’s visit to each metropolis serves him as an introduction to the wonders, respectively, of words and numbers. The Mathemagician’s explanation of infinitely large and small numbers wows the boy. The point of the scene is to inspire in readers awe of the marvels of mathematics and the endless possibilities of the human mind, pointing again to The Wonders of Learning.
Besides his writing career, the author was an architect, and his knowledge of geometry and math gave him plenty of intriguing and amusing ways to play with the logic of numbers in The Phantom Tollbooth. One of the citizens of Digitopolis is the Dodecahedron, its giant head a 12-sided object, like a die with 12 sides instead of six. (In its name, “do” means two, and “deca” means 10, for a total of 12.) Each side or face is a pentagon; in Digitopolis, the Dodecahedron has a human face on every side. As the author puts it, it’s “somewhat like a cube that’s had all its corners cut off and then had all its corners cut off again” (172). The dodecahedron’s well-known cousin is a soccer ball, a shape with 32 sides called a truncated icosahedron.
As in Dictionopolis, with its strange use of words, Milo learns many things about the possibilities hidden in math during his visit to Digitopolis. This is part of the book’s theme on The Wonders of Learning, especially those of numbers and arithmetic. He can, for example, get to the land of Infinity, but only if he travels forever. In Chapter 16, Milo meets .58 of a boy, a fractional “average” child who can still enjoy life but in ways somewhat different from whole kids. These silly story moments stimulate the reader’s curiosity about the depth and breadth of math, and perhaps of knowledge in general, and how playing with unusual ideas can confer perspective and even wisdom.
Along with the wit of each city’s denizens is their immense silliness, which the story attributes to the absence of Rhyme, Reason, and Wisdom. Each place thus inspires, befuddles, and motivates Milo as he discovers his love for learning and his great purpose: to retrieve the two ladies of wisdom from banishment.
