Math is everywhere, it’s in the nature, engineering, and the basis of modern civilization – keeping that in mind it won’t come as a surprise that mathematics also inspires many artists and has its solid place in art. Many graphical artists use it in their paintings and some of them even paint the mathematical principals themselves. We discussed one such example of a mathematically (and not only) astounding painting by M.C. Escher in our article “The Infinite Painting“, but visual artists are not alone, many writers are inspired by mathematics as well. One of the best- and well-known examples is, of course, Alice’s Adventures in Wonderland (commonly shortened to Alice in Wonderland), by Charles Lutwidge Dodgson.
Yes yes, Charles Lutwidge Dodgson and not Lewis Carroll, because “Lewis Carroll” was just a pseudonym, Charles’ pen name, although even that fact is not so clear. You see, Charles was a mathematician and taught math at the university. As a teacher, his classes were known to be dull and boring, lacking any charisma or imagination. As a writer, well, that’s a different story. Charles/Lewis himself tried to distinguish between these two identities, even going so far as to return letters addressed to Lewis Carroll but received at Christ Church, Oxford, where Charles worked – he returned the letters unanswered, since they didn’t reach their destination.
There is a famous anecdote about Carroll’s life: Queen Victoria enjoyed Alice’s Adventures in Wonderland so much that she requested the first edition of Carroll’s next book. Naturally, Carroll sent her a copy of the next book he published – a mathematical work titled “An Elementary Treatise on Determinants”. Unfortunately, that story isn’t true, Carroll himself refuted it. However, such a story does highlight the oddness of Carroll’s double life.
It seems to be reasonable to assume, that his complicated life and character, full of contradictions and creativity, are reflected in his writing. Alice in Wonderland is full of nonsense, absurd, puns, crazy ideas, and unforgettable characters. And, of course, math! Lewis Carroll embedded mathematics in the book so elegantly, as Easter Eggs, that many young and old readers won’t pay any attention to it, but taking a close look, re-read the book once or twice, and a completely different picture opens in front of us. While the book is full of references to Lewis’ days’ culture, it’s also full of references to mathematical ideas and concepts.
Calculus
It starts with the very first chapter (“Down the Rabbit-Hole”): In this episode, Alice desperately wants to enter a magical garden. The problem is, you see, Alice is just too big to enter the small door. She finds a bottle, and hopes that drinking it will make her smaller, she tastes it and then…
And so it was indeed: she was now only ten inches high, and her face brightened up at the thought that she was now the right size for going through the little door into that lovely garden. First, however, she waited for a few minutes to see if she was going to shrink any further: she felt a little nervous about this; “for it might end, you know,” said Alice to herself, “in my going out altogether, like a candle. I wonder what I should be like then?” And she tried to fancy what the flame of a candle looks like after the candle is blown out, for she could not remember ever having seen such a thing.
Why Alice was afraid to disappear (“going out altogether”)? Normally, when one shrinks down, he just changes its size and becomes smaller and smaller, but does not disappear, right? Well, math has a different idea and it’s described by the concept of a limit.
Maybe some of you still remember it from the calculus lessons: say we have a function f, that like a machine or a computer, receives input and produces an output. The function f receives an input x, and produces an output of f(x) (for example f(x) = x2, and for x = 2 this means f(2) = 4). Sometimes, a function has a limit L. We say the function has a limit L at an input p if the output f(x) gets closer and closer to L as the input x moves closer and closer to p. You see, in mathematical limits, a function may never reach L. It may get closer and closer to L, forever approaching it, but never actually reaching in. Nevertheless, we can say that its limit is L and normally use it as if the function actually reaches it at some point. So does Alice, who shrinks and becomes smaller and smaller and maybe at some point she will go out altogether, like a candle.
Numeral Systems
Another fascinating example comes in the second chapter (“The Pool of Tears”). Alice is no longer sure who is she – Alice, or maybe now she is someone else. She tests her memory by declaiming a poem, but the words come out a bit differently than they should. She also tries math, and probably without realizing it, or maybe on the contrary, she comes to some very interesting conclusions:
Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate!
This phrase, like all the others, is brilliant because it can be understood in many different ways. The multiplication produced results are definitely odd, and at that rate it indeed will take a while (“forever”, in children’s terminology) to get to twenty – most of the readers will stop here, laugh, and move on to the next sentence, but we will pause here a bit. Those of you with a sharp eye already noticed that: Why she will never be going to get to twenty?
One explanation is that: back then, the multiplication table reached twelve (12 x 12) and not ten (10 x 10) as it is now. Lets continue Alice’s weird “multiplication” pattern: 4 x 5 = 12, 4 x 6 = 13, 4 x 7 = 14, …, 4 x 10 = 17, 4 x 11 = 18, 4 x 12 = 19, and… here the multiplication table ends – there is no 13 in the table, so there is no point to ask how much is 4 times 13 – it’s just outside the table! Therefore, by using this odd method, she will never reach 20.
There is more – what if Alice realized something deeper, or at least Lewis Carroll hints us that there is something deeper in that thought. We can look at it from a different angle: What if we are not using a decimal base at all? A base (or radix) is the number of different symbols (normally digits but sometimes also letters) we use to represent numbers. For example, we normally use a decimal base, which has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. What we do if we want to get above 9? We take the first two digits, 0 and 1, and combine them together: 10, then 11, then 12 and so on. Eventually we get to 19, and again we are out of digits, so we take the second digit, 2, and the first one, 0, and combine them together to get 20, then 21, then 22 and so on. But what if we had only 4 digits: 0, 1, 2, 3. It is called base 4, or Quaternary numerical system. We don’t have the symbol “4”, although we do want to get to the next number after 3. So, as previously we took the first two digits: 10, 11, 12, 13 to continue counting, we will use the same trick now: 10 in base 4 (also denoted as 104) is 4 in base 10 (decimal), 114 is 5, 124 is 6 and so on. 20 in base 4 (204) is 8 in base 10. We developed a system to represent any number with only four symbols!
Computers use a binary base, base 2, which has only two digits: 0 and 1. There are no “2” or “3” – these symbols are absent for the binary system, so after exceeding all the available digits, we take the first two and combine them together: 10 (more exactly: 102), then 112 and then we are out of digits again, so we go to 1002, which is 4 in decimal notation, 1012 is 5 and 1102 is 6.
Another very useful base, frequently used in the technological world, is base 16, also known as hexadecimal base. It has 16 symbols to represent digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Indeed, the “letters” A-F are used here as digits. A in base 16 (A16) is 10 in the decimal base, B16 is 11, C16 is 12, D16 is 13, E16 is 14, F16 is 15. If we want to continue beyond 15 (or F), we use the first two digits and get 10 in base 16 (1016), which is 16 in decimal base. 1116 is 17, 1216 is 18, …, 1916 = 25, and now we don’t get to 20 because we still have a few digits left (A-F), so instead of 20 we get 1A16 which is 26, 1B16 which is 27, …, 1F16 which is 31 and since know we are truly out of digits, we finally get to 2016 which is 32 in decimal base. A bit confusing at first, but eventually you got used to it.
Back to the Alice’s multiplication problem: 4 × 5 = 12 is true in base 18 (so it’s actually 418 x 518 = 1218), while 4 × 6 = 13 is true in base 21, and 4 × 7 could be 14 in base 24 notation. Continuing this sequence, going up three bases each time, the result will continue to be less than 20 in the corresponding base notation. After 4 × 12 = 19 in base 39, the product would be 4 × 13 = 1A in base 42 which is a “weird” number, and not the nice “20” as Alice wants. Then, 4 x 14 = 1B in base 45, 4 x 15 = 1C in base 48, 4 x 16 = 1D in base 51, 4 x 17 = 1E in base 54, 4 x 18 = 1F in base 57 and then… the next value is 4 x 19 = 1G in base 60! The multiplication sequence is 12, 13, 14 and so on, incremented by 1, but the base sequence, 54, 57, 60, is incremented by 3 – it’s a diverging sequence. Yes, Alice will never get to 20, because the base also goes up, so the number of available symbols (that represent digits) goes up well, and it happens faster than the multiplication. We will get 1H, 1I, 1J and so on, and after all the letters will exceed, we will have to use different symbols, but never reach 20.
Logic
All the previously discussed mathematical concepts are nice, but without any doubt, from the mathematical point of view, mathematical logic is the queen of the book. Although Lewis Carroll tried his best to develop geometry and algebra, as a mathematician he is best remembered as a logic innovator. Thus, his book is full of mathematical logic references.
For example, in the fifth chapter (“Advice from a Caterpillar”) a pigeon suspects Alice to be a serpent, which make some sense because during that time she had an enormously long neck:
“I’ve seen a good many little girls in my time, but never one with such a neck as that! No, no! You’re a serpent; and there’s no use denying it. I suppose you’ll be telling me next that you never tasted an egg!”
“I have tasted eggs, certainly,” said Alice, who was a very truthful child; “but little girls eat eggs quite as much as serpents do, you know.”
“I don’t believe it,” said the Pigeon; “but if they do, why then they’re a kind of serpent, that’s all I can say.”
This was such a new idea to Alice, that she was quite silent for a minute or two.
That is definitely a new idea, and an incorrect one. The pigeon says that since all serpents like to eat eggs, and Alice likes to eat eggs, then it must be true that Alice is a serpent – and that, according to mathematical logic is an invalid statement (you can read about it in detail in “Logic in Wonderland: An Introduction to Logic through Reading Alice’s Adventures in Wonderland” by Nitsa Movshovitz-Hadar and Atara Shriki).
Of course, there is more. We won’t write here all the wrong and funny logical assertions, but here are a few:
Chapter 6 (“Pig and Pepper”):
“In that direction,” the Cat said, waving its right paw round, “lives a Hatter: and in that direction,” waving the other paw, “lives a March Hare. Visit either you like: they’re both mad.”
“But I don’t want to go among mad people,” Alice remarked.
“Oh, you can’t help that,” said the Cat: “we’re all mad here. I’m mad. You’re mad.”
“How do you know I’m mad?” said Alice.
“You must be,” said the Cat, “or you wouldn’t have come here.”
Alice didn’t think that proved it at all; however, she went on: “And how do you know that you’re mad?”
“To begin with,” said the Cat, “a dog’s not mad. You grant that?”
“I suppose so,” said Alice.
“Well, then,” the Cat went on, “you see a dog growls when it’s angry, and wags its tail when it’s pleased. Now I growl when I’m pleased, and wag my tail when I’m angry. Therefore I’m mad.”
Chapter 7 (“A Mad Tea-Party”) gives us some of the best examples:
“The Hatter opened his eyes very wide on hearing this; but all he said was, “Why is a raven like a writing-desk?”
“Come, we shall have some fun now!” thought Alice. “I’m glad they’ve begun asking riddles.—I believe I can guess that,” she added aloud.
“Do you mean that you think you can find out the answer to it?” said the March Hare.
“Exactly so,” said Alice.
“Then you should say what you mean,” the March Hare went on.
“I do,” Alice hastily replied; “at least—at least I mean what I say—that’s the same thing, you know.”
“Not the same thing a bit!” said the Hatter. “Why, you might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”
“You might just as well say,” added the March Hare, “that ‘I like what I get’ is the same thing as ‘I get what I like’!”
“You might just as well say,” added the Dormouse, who seemed to be talking in his sleep, “that ‘I breathe when I sleep’ is the same thing as ‘I sleep when I breathe’!”
“It is the same thing with you,” said the Hatter, and here the conversation dropped, and the party sat silent for a minute, while Alice thought over all she could remember about ravens and writing-desks, which wasn’t much.”
And of course:
“Take some more tea,” the March Hare said to Alice, very earnestly.
“I’ve had nothing yet,” Alice replied in an offended tone, “so I can’t take more.”
“You mean you can’t take less,” said the Hatter: “it’s very easy to take more than nothing.”
Tip of the Iceberg
Of course, that was only the tip of the iceberg. There are many more mathematical Easter Eggs in the book. Can you spot them? There is more mathematical logic, modulo and abstraction, but we will leave it for some other time, and will finish with a quote from chapter 9 (“The Mock Turtle’s Story.”):
“And how many hours a day did you do lessons?” said Alice, in a hurry to change the subject.
“Ten hours the first day,” said the Mock Turtle: “nine the next, and so on.”
“What a curious plan!” exclaimed Alice.
“That’s the reason they’re called lessons,” the Gryphon remarked: “because they lessen from day to day.”
This was quite a new idea to Alice, and she thought it over a little before she made her next remark. “Then the eleventh day must have been a holiday?”
“Of course it was,” said the Mock Turtle.
“And how did you manage on the twelfth?” Alice went on eagerly.
“That’s enough about lessons,” the Gryphon interrupted in a very decided tone: “tell her something about the games now.”
Featured image by Sir John Tenniel – “Alice’s Adventures in Wonderland” (1865) with additions of a few mathematical symbols.