Alice in Wonderland got its start as a simple story, told by a mathematics professor to a colleague's daughter. It's a strange story that seems to be the result of a drug trip, but is actually a scathing satire of the new-fangled math that the professor was seeing invade his area of study.

Most of us just enjoy the White Rabbit and the hookah-smoking caterpillar. But now you can understand the math in Alice without needing to be a math whiz.

Some people who read Alice in Wonderland find it a whimsical adventure into a world of fun little paradoxes. Other people consider it a creepy march through a world of characters who seem to be set on making life as frustrating as possible as manically as they can. Which side you see might possibly have some bearing on your view of the world. Alice isn't just fun and games. Charles Dodgson — the real name of Lewis Carroll — added all those paradoxes and puzzles as he was poring over the new math that was springing up in the middle of the 1800s.

Carroll liked good old-fashioned no nonsense algebra and Euclidean geometry — areas of study that could prove things about the natural world. Suddenly math students, and even teachers, were using different mathematical methods to prove things like one and one not equaling two. It seemed to Carroll that they were just being difficult on purpose, so he skewered them in prose.

Alice's Mathematical Attempts at Control

Alice herself isn't the focus of Carroll's ire, so while she thinks circuitously about mathematics, and make mistakes, she's mostly the straight man for the characters of Wonderland. She gets us started with mathematical concepts early on in the proceedings, when she's still shrinking down. She wonders if she can shrink forever, getting smaller and smaller, or if she'll eventually reach the point of nothingness. Where, exactly, is the mathematical partition between a very small something, and nothing at all?

Later, when she gets bigger and attempts to do math, she gets mixed up. She tries simple multiplication, but comes up with four times five equaling twelve, four times six becoming thirteen, and four times seven turning into fourteen.

In regular math, of course, this doesn't work. If, however, you mess around with the base systems, things change. We work in base ten, meaning we have zero-through-nine digits, and then when we get to ten we move over and put a one in the next column. Alice was calculating in base ten, but her answers slipped into higher base systems. Four times five is twenty, which in base eighteen is one (1) group of eighteen, and two (2) extra singles, making 12. Four times six is twenty-four, got changed to a base twenty-one system, with one (1) group of twenty-one, plus three (3) extra singles, or 13. Four times seven is twenty-eight, but if you change that to a base twenty-four system, that's one (1) group of twenty-four, and four (4) extra singles, or 14. When you change the system of measurement, but keep thinking of it as the original standard, you can pile on the numbers and never get anywhere, leaving you as lost as Alice.

Alice also has to work on her sense of grace and Euclidean proportion. Alice meets the caterpillar on his mushroom and asks to be made larger. He tells her that one side of the mushroom makes her smaller and one makes her larger, then warns her to "keep her temper." Temper, in this case, means correct proportions. This is hard when the same object can have exactly opposite effects. This is Carroll's jab at a new style of math.

For most of us, math has an analog in the real world. That's the point of it, to use this system of symbols as a translation to figure out problems that would be ungainly if expressed in language. Around Carroll's time, math became something different. Instead of a translation, it became a language in and of itself, or rather several different languages. If someone set up certain rules for a problem, and then opposite rules for another problem, they could prove opposite conclusions to be true. Each set of rules has opposite effects, but as long as each proof remains true to its internal rules, each is considered correct.

Alice tries to remain in proportion, despite the inconsistency. In the movie she simply grows and shrinks. In the book, at one point her neck grows long like a snake's, which is even worse for her than being either big or small. She has to figure out how to keep gracefully geometric, having the same proportions at any size, before she can go on.

Hallucinations and Reality

A part that was left out of the movie, and shakes a lot of readers off the book, is an encounter with the Duchess. Alice meets her in her house, eating soup with her baby. When the soup is too peppery, her baby sneezes. What follows is a delightful little poem that starts with the lines, "Speak roughly to your little boy and beat him when he sneezes." Alice grabs the baby and, when she looks, the baby has retained many of his original features, but has turned into a pig. It is at that point that many people put down the book and quietly go to find some Nancy Drew mysteries.

This section is actually Lewis Carroll ridiculing the work of Jean-Victor Poncelet, who talked about how geometric figures transform. He stated that a geometric figure undergoing a continuous transformation, without any sudden changes or subtractions, will retain certain features. However, it won't retain those features in a way that can be understood physically. It only manages to keep them on paper, using things like imaginary numbers. The baby-to-pig transformation is Carroll's commentary on how absurd and grotesque he found that idea. It's either a baby or a pig, and an infinite series of tiny changes can't truly make it both at the same time, argues Carroll.

Perhaps the most iconic scene in the Wonderland story is the tea party in the garden. This is where Carroll really starts grinding his axe. William Rowan Hamilton had come up with a new thing called quaternions. This is a sort of coordinate system based on four terms, three that designate place, and one that designated, or so Hamilton decided, time. With these four terms, Hamilton could describe rotation in a three dimensional universe. He could only do this, though, if he added that fourth component. Without it, they rotated in a plane, like the hands of a clock.

Carroll was miffed that someone had appropriated all of time just so they could have a fourth component to allow them to rotate things properly, so in the tea party he took it away. In the movie, no one explains why the Doormouse, the Mad Hatter, and the March Hare are all going in a circle around a table in a perpetual tea time during a perpetual unbirthday. In the book, Time had been the fourth member of their party, but had gotten fed up and walked out. That left the other three to keep going around in circles forever, like an incomplete quaternion. Alice, free of the madness of requiring an extra dimension in order to be of any use, leaves the tea party.

The Lord of Wonderland, much more than the absurd royalty and the sad, manic Mad Hatter, is the Cheshire Cat. He goes where he wants, and does as he pleases, and it's all because of his grin. At the end of a loopy conversation with Alice, the cat disappears, but the grin remains. Alice remarks that she's seen a cat without a grin, but she's never seen a grin without a cat.

This is the heart of the story. Sure, anyone can have a smile on their face, but if people remove the face — the reality of equations — they have to remove the grinning, silly, maddening numbers as well. Wonderland is just fine for the Cat, who has fully embraced and embodied this crazy new math, but what good, Carroll argues, is it to anyone else? The Cat voices Carroll's opinion. Alice complains that she doesn't want to "go among mad people." The Cat tells her that she has no choice: "Oh, you can't help that. We're all mad here. I'm mad. You're mad. . . . You must be, otherwise you wouldn't have come here."

Via NPR, New Scientist, and the NY Times.