It Doesn’t Add up: Mathematics in Wonderland

We’re probably a little off-topic here, but with the renewed interest in Alice’s Adventures in Wonderland (the proper title), due to the Tim Burton film, people are starting to pick up Lewis Carroll’s books again, which I can only see as a good thing (being an Alice fan myself and all). The New York Times is running an interesting article about an aspect of the Alice books you won’t see in most adaptations: the mathematical one.

Alice is the creation of writer Charles Dodgson, who used the pen name Lewis Carroll to publish the Alice books. Contrary to popular belief, there are two Alice books, and neither of them are named “Alice in Wonderland”. The first book, published in 1865, is called “Alice’s Adventures in Wonderland”, and has a decidedly card deck theme to it all. The second book, titled “Through the Looking-Glass, and What Alice Found There”, was published in 1872, and shifts to a chess theme. In other words, no, the Red Queen (chess theme) and the Queen of Hearts (card theme) are not one and the same.

Most films based on the two Alice books mix and match themes, characters, and locations from the two books, completely disregarding the fact that the two books are quite different from one another. The feeling I get watching any modern interpretation of the Alice books can more or less be described as the feeling people will have if someone were to mix and match various locations and characters from the Lord of the Rings trilogy, mangling them together at random into some sort of storyline that doesn’t align itself at all with the three books. It just doesn’t feel right.

Even something as basic as forcing an actual storyline atop the Alice books is something that appears wholly alien to me. The beauty – and power – of the Alice books is exactly that it doesn’t have much of what you would traditionally call a storyline, with basic elements like goals, motivations, and the likes. Wonderland, and the events that take place there, as well as Alice’s role in those events are entirely the construct of Alice’s own mind.

This is crucial.

Alice is a young girl, round and about seven years old, which means her mind isn’t the same as that of adults. It makes less sense, has less structure, and tends to jump from here to there without much logic. Anyone who has ever seen a young child play with a whole bunch of toys can attest to this; multiple different types of toys are played with interchangeably, seemingly without much structure or logic behind it all.

Since Wonderland is a construct of Alice’s own mind, it reflects this chaotic nature. Events in the books seem quaintly disjointed, and there’s no real reason why one event follows the other. This is the beauty of it all; it doesn’t have to make sense. It doesn’t have to have a point. It doesn’t have to have a moral. This is what created the books’ lasting appeal.

I know I’m a bit off the topic I wanted to talk about here, but hey, Alice is a passion of mine. I love the books, the ideas behind them, the dark undercurrent, the absurdity of it all. In a way, Lewis Carroll is the 19th century version of Pixar; they, too, have that uncanny ability to thoroughly entertain both children and adults within one and the same work of art.

Anyway – to get back on topic. Another important and oft-neglected aspect of the Alice books is math. Charles Dodgson was a mathematics tutor at the University of Oxford, and his love for math can most surely be seen in the Alice books in countless ways. In lieu of the Tim Burton film, The New York Times decided to run a piece on this particular aspect of the books, which contains some interesting examples (although there’s one oft-made error in there too; contrary to what the article states, Alice was not based on Alice Liddell specifically).

Melanie Bayley, the author of the piece, explains that Dodgson’s work contains a lot of satire about then-modern ideas in the world of mathematics. “Dodgson found the radical new math illogical and lacking in intellectual rigor,” Bayley writes, “In ‘Alice’, he attacked some of the new ideas as nonsense – using a technique familiar from Euclid’s proofs, reductio ad absurdum, where the validity of an idea is tested by taking its premises to their logical extreme.”

Dodgson was able to achieve this because Wonderland, being a construct of Alice’s mind, did not have to adhere to the laws of physics and science as would the real world; this gave him the freedom to explore what the world would look like if the logical extremes of the mathematical ideas he disagreed with were truth. I especially like the example of chapter 6, “Pig and Pepper”.

Chapter 6, “Pig and Pepper”, parodies the principle of continuity, a bizarre concept from projective geometry, which was introduced in the mid-19th century from France. This principle (now an important aspect of modern topology) involves the idea that one shape can bend and stretch into another, provided it retains the same basic properties – a circle is the same as an ellipse or a parabola (the curve of the Cheshire cat’s grin).

Taking the notion to its extreme, what works for a circle should also work for a baby. So, when Alice takes the Duchess’s baby outside, it turns into a pig. The Cheshire Cat says, “I thought it would.”

The article contains many more interesting examples that seem to make a lot of sense – but do remember that it is impossible to know exactly what Dodgson meant with all the symbolism in his books. He never explained himself, so all we can do is look back and place his work in the proper context, and go from there.

And this is yet another important aspect of the books: the mystery. You can read just about anything into the strange things that happen to Alice and her Wonderland companions, meaning that child and adult alike can lose themselves in the experience.

Tim Burton

I was really looking forward to the Tim Burton film, by the way, but I have this sneaking suspicion that it won’t be a good film. You see, both Burton and Dodgson are incredibly good artists, with their own unique styles, and while Wonderland itself might present Burton with a wonderful canvas for his creative genius, I’m afraid that the actual Alice stories do not.

Tim Burton makes Hollywood films. He does this incredibly well – I’m a huge fan – but being a Hollywood director, working for Disney (of all places), means the film must adhere to basic story telling, with goals, motivations, and most likely, a love interest of some sort squeezed in somewhere. This kind of thinking just doesn’t become Alice. I saw some previews that showed Alice running around with swords and armor, fighting in some Helm’s Deep-esque battle.

With Wonderland being a construct of Alice’s mind, I wonder just how mentally messed up Burton’s “real” Alice has to be in order to have her running around like that. American McGee got it right; his “real” Alice was in a psychotic and catatonic state in an old-world insane asylum, rendering McGee’s twisted Wonderland rendition totally believable.

For some reason, I doubt Disney would allow the film to start with a psychotic and mentally deranged Alice – leading me to the scary conclusion that Burton had to violate the most important aspect of the Alice mythos: Wonderland is Alice.

If you severe the two, it’s no longer Alice, now, is it?


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