# The Math Behind Wonderland: Understanding the Insanity

With the advent of Spring, book lovers are often reminded of the sun-shiny, lovey-dovey worlds of romance novels. If you spend your Spring in and out of mental wards or under the influence of opiates, you’ll probably be reminded of *Alice’s Adventures in Wonderland* (*Alice in Wonderland* for short) by Lewis Carroll.

For those who aren’t familiar with *Alice’s Adventures in Wonderland* or the movie popularized by Disney (titled* Alice in Wonderland*), the story is about a young girl named Alice who follows a talking rabbit down its rabbit hole and ends up in a strange land full of anthropomorphic abominations and strange happenings. As Alice traverses through the odd landscape, she encounters many problems that involve her size (she is restricted from exploring the strange world because she is either too tall or too short to fit through crevices and oddly placed, multi-sized doors). In order to overcome these obstacles, she ingests questionable potions and pastries that change her size drastically. Alice soon finds herself on a journey that will mostly lead her on a search for these size-changing foods in order to learn more about the strange world around her. Along the way, Alice meets many strange characters, mostly talking animals and other cross-bred “things”, that tell her more about her eccentric surroundings. She eventually finds herself in less than favourable situations which inevitably lead to a cliché ending. If you haven’t guessed by now, this is, in fact, a children’s story. A really messed up children’s story…

Indeed, many would consider *Alice’s Adventures in Wonderland* the workings of a madman or the hallucinations of a Victorian commoner enjoying new and exotic intoxications. Others may simply see the story as something to please the easily-distracted minds of young children at the expense of literary nonsense. While these are quite viable hypotheses, there is one connection that few people make. That connection is to math—the one thing that makes nonsense even more nonsensical.

Let’s rewind to the time the book was written, 1865, the Victorian era. Charles Lutwidge Dodgson, who wrote under the pen name Lewis Carroll, was a young English mathematician and logician (one who studies reasoning and logic). Yes, a mathematician decided to write a children’s book. That explains a lot doesn’t it? Dodgson was a very conservative man; he was conservative religiously, politically and even mathematically. The mid-19th century was a revolutionary time for math. More and more abstract mathematical concepts started to emerge. These new concepts contested standard Euclidian math (the branch of geometrical math that is used to explain the natural world) that Dodgson swore by. Obviously, Dodgson didn’t like this new abstract math very much, so it is said that he used hidden meanings within his literature to attack abstract math and other mathematicians, as well as to promote traditional Euclidian math. In fact, these underlying messages seemed so blatant to other mathematicians that Dodgson became quite the talk of the science editorials during that time.

Before looking at the math behind *Alice’s Adventures in Wonderland*, let’s try to make sense of the story as “normal” people….if that’s even possible. As this story is intended for children, there should at least be some sort of moral or lesson. If we look at Alice herself, along with her decisions and actions, we may be able to gain some insight on the message that Dodgson was trying to convey to children. Staying true to his roots as a logician, it is possible that Dodgson was trying to teach children about logic and reasoning. In the story, Alice mostly finds herself either too tall or too short to proceed further in the world. When she comes to her first obstacle (right after falling down the rabbit’s hole), she realizes that she needs to somehow become much shorter to fit through a tiny door that leads to a marvelous garden. After searching her surroundings, she comes across a bottle that is labelled “drink me”. This is when the rather weak lesson-teaching comes in.

Before drinking the contents of the bottle she checks to make sure there isn’t another label that reads “poison”. After making sure it’s safe, she downs the rather pleasantly flavoured drink and experiences after-effects. Obviously, she begins to shrink. Before proceeding to the door, Alice makes the smart to decision to wait until the effects fully wear off just in case something else was to happen. After realizing she left the key to the door on the table where the bottle was, which would now be far out of her reach, she realizes that she somehow needs to grow taller again. She searches around again and finds a cake that is labelled “eat me”. As this is a cake and not some strange liquid decanted into a fancy bottle, there will not likely be any other label that would provide any other description. Given what just happened with the drink, she reasons that the cake must have some affect on her size and is probably safe to consume. This is the logic that is implemented throughout the rest of the story, “if something gives a certain effect, then something similar will give a similar effect”, or more specifically “if something appears to be safe, then something similar must also be safe”. Alice uses this reasoning to eat questionable food products on impulse in order to get out of sticky situations. It’s a shame this kind of reasoning would probably have a tragic end for a child in real life. Yeah, you hide behind that pen name Dodgson, you sicko. Don’t despair however, this lesson is perhaps masked by the fact that Alice turns out to be a weirdo that likes to talk to two different sides herself in an attempt to reason unreasonable things. Oh, that and starfish-baby-pig hybrids along with a few other monstrosities.

After looking at the absurdity of the possible morals behind *Alice’s Adventures in Wonderland*, the mathematical explanation must make more sense right? Well, not really. The supposed underlying concepts of math in the story are actually very, very complex and were only discovered by mathematicians or those who work in the field of math and science. Mathematicians have suggested that *Alice’s Adventures in Wonderland* contains such subjects as complex numbers, projective geometry and even quaternions (geometry involving 4 dimensions, a groundbreaking topic of the Victorian Era). As to not leave your brains spattered on your keyboard, I’ll only explain a few concepts so you can get the gist of it.

One of the most popular suggested concepts of the story is counting in different number systems (if you’re familiar with computer science you’ll know what I’m talking about). After growing taller again, Alice starts to ponder if she is truly still herself and tests her knowledge of different school subjects to make sure she is in fact herself and not another unintelligent girl. She does simple multiplications in her head but comes up with strange answers. According to her, 4 x 5 = 12, 4 x 6 = 13 and when she gets to 4 x 7, she exclaims that she will never reach 20 by multiplying the numbers the way she has been. Upon closer inspection, and with a little number know-how, you can see Alice is multiplying the numbers by increasing the base by 3 each time starting at base 18 (we use the base 10 number system, meaning we use 10 digits and our place value increases by powers of ten). 4 x 5 in base 10 is 20. 20 converted to base 18 is 12 using 1 group of eighteen and 2 groups of one ( (1 x 18) + (2 x 1) = 20) ). 4 x 6 in base 10 is 24. 24 converted to base 21 (18 + 3 = 21) is 13 using 1 group of twenty-one and 3 groups of one ( (1 x 21) + (3 x 1) = 24) ). Finally, if Alice were to continue, 4 x 7 in base 10 is 28. 28 converted to base 24 (21 + 3 = 4) is 14 using 1 group of twenty-four and 4 groups of one ( (1 x 24) + (4 x 1) = 28) ). As you can see, if she continued to use this method of multiplication, it would take Alice a long time to reach 20. This is supposed to show the reader that in this new world it is so insane that not even normal math makes sense. Mathematicians view this as Dodgson’s way of saying that this new suggested form of number-counting is quite ridiculous. Little did he know that this new concept would make one of the greatest inventions of all time possible with the creation of the binary (base 2) number system.

It looks like I failed. I can already smell those freshly spattered brains. If you had trouble following that last concept, you can learn more about number systems here.

Okay, let’s look at something a little easier to grasp. In the iconic scene where Alice comes across a hookah-smoking caterpillar in search of a way to make herself taller again, we are introduced to the concept of proportions. The rather cryptic caterpillar warns Alice to keep her “temper” in the middle of their conversation. Alice is obviously confused at this abrupt remark because she didn’t do anything to suggest that she was angry. At the end of the conversation the caterpillar tells Alice that eating a piece of one side of the mushroom that he was sitting on will make her taller and the other side will make her shorter. Taking a piece from each side of the mushroom, Alice quickly finds out that one side of the mushroom causes only her neck to get longer and the other side causes only her torso to get shorter. Alice realizes that she needs to eat a little bit of both sides to maintain her shape but change her overall height. Now, if we go back to what the caterpillar said, “temper” does not only mean the capacity of one’s anger, it also refers to an object’s *proportions*. Alice managed to change her height but kept her bodily proportions intact. The idea of proportions is an important concept in the Euclidian geometry Dodgson loved so much. Seeing as this proportional conservation is the only sane thing Alice seems to do, Mathematicians speculate that Dodgson is trying to say that traditional Euclidian math is the only math that makes sense.

As I said before, these are just a few of the mathematical concepts embedded in *Alice’s Adventures in Wonderland* but judging from what we know now, it’s safe to assume that everything that seems absolutely mad in the story (and there is a lot of madness) relates to Dodgson’s view on some form of complex math.

So there you have it, some ammunition to help you make sense of a nonsense book, or something to confuse you even more because in reality, any mathematical explanation always leaves nothing but a scattering of miscellaneous words and symbols on a page. However, if you’re somebody who enjoys math and odd literature, or somebody who likes drug-induced hallucinations or even if you’re just plain insane, then *Alice’s Adventures in Wonderland *is just the book for you. Everybody else, go and enjoy the funny colours in the Disney film…

Enlightening… thanks.