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Hi I'm Toby and welcome to this video where i'm going to take a little look
at the famous story Alice in Wonderland written by Lewis Carroll. It's a very creative story,
it's a bit nonsensical and at times bizarre but behind some of this writing there supposedly hides
a little bit of mathematics too. You see, Lewis Carroll is actually the pen name for 19th century
mathematician Charles Dodgson and he initially dreamed up the story for 10 year old Alice Liddell
the daughter of his university college dean and he told the story to her and her sisters on a
boat trip in 1864. The fact that Carroll is much more famous as a writer than as a mathematician
perhaps speaks to the fact that his mathematical work was comparatively unnoteworthy. He did write
several books on mathematics including an elementary treatise on determinants.
His work was in the fields of geometry, linear and matrix algebra and mathematical
logic so let's take a look at some of the math that snuck its way into the story.
After falling down the rabbit hole Alice is in a bit of distress. She says
"I'll try if I know all the things I used to know. Let me see: 4 times 5 is 12,
and 4 times 6 is 13, and 4 times 7 is -oh dear! I shall never get to 20 at that rate!"
According to Martin Gardner who has written an annotated version of Alice in Wonderland
the most simple explanation here is that Alice is adding one every time she does
a multiplication by four and if you keep going this way you'll get to four times 12
being equal to 19 and children especially only know multiplication up to the 12s
so she won't be able to go any further and won't be able to get to 20.
This is a nonsense form of multiplication but it sets us up for the nonsense to come.
Later in the story we meet the Cheshire Cat
"'All right', said the Cat; and this time it vanished quite slowly, beginning with the
end of the tail, and ending with the grin, which remained some time after the rest of it had gone.
'Well I've often seen a cat without a grin,' thought Alice 'but a grin without a cat,
it's the most curious thing I ever saw in all my life.'"
This grin might be referring to abstract or pure mathematics. That it doesn't necessarily have
a cat to go along with it which would be the real life applications or even
obvious uses of what the research mathematicians are working on, they're doing it just for the joy
of the mathematics itself and are living in the abstract world of the equations.
Further into wonderland Alice is talking to Gryphon
and Mock Turtle, she asked them about lessons
"'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,
because they lesson 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 11th day must have been a holiday.'
'Of course it was' said the mock turtle 'and how did you manage on the 12th?'
Alice went on eagerly. 'That's enough about lessons.'"
This might be a fun way to introduce the idea of negative numbers,
something that also puzzled early mathematicians.
In this case what would happen on the 12th day, would the students start to teach the teacher?
Then we go to the famous tea party scene, there's the Mad Hatter,
a hare and a mouse and Alice is also informed of a fourth character, time,
who is absent from the party due to a falling out with the hatter. As a result the three at the tea
party are forever stuck at tea time they're forever going around and around the table.
"The hare takes out a watch and Alice remarks 'what a funny watch! It tells the day of the month
and doesn't tell what o'clock it is' "Why should it?' muttered the hatter
'Does your watch tell you what year it is?' 'Of course not,' Alice replied very readily
'but that's because it stays the same year for such a long time.'
'Which is just the case with mine' said the hatter."
Later he says that "'It's always six o'clock'
'Is that the reason so many tea things are put out here?' Alice asked 'Yes that's it' said the
hatter 'it's always tea time and we've no time to wash the things between whiles.'
'Then you keep moving around I suppose' said Alice 'Exactly so' said the hatter."
Now it might be nothing but the idea of three characters stuck going around and around
and not making much sense without time shares a few similarities with the idea of quaternions,
something that was introduced not too long before the story
was written. Quaternions consist of three spatial dimensions i, j and k
which are imaginary and a fourth dimension which is real and that is time. Perhaps at the tea party
the three characters are the imaginary units and without time they don't really make sense.
A property of quaternions is that when you multiply them they are non-commutative
which means the order actually matters. Normal multiplication such as 3 times 4 is the same as
4 times 3 it is commutative but with quaternions i times j is not the same as j times i
and there's a scene at the tea party where this comes up in the language.
They are discussing a riddle "'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 '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 in the March hare 'that I like what I get is the same thing as I get what I like.'
So if the characters at the tea party are used to a non-commutative form of multiplication then
that's maybe why they're so unsatisfied with Alice claiming that the order of words doesn't matter.
Something that happens a lot to Alice during the story is that she's often changing size
and having to eat or drink something to return to the right proportions. A caterpillar even
tells Alice to keep her temper and Alice interprets this to mean to not get angry
however with our mathematical eyes in place we could see that Carroll is making a play on words
and that there is a more mathematical definition of temper that would have
been more familiar to victorians at the time and has to do with scale and proportions.
We see examples of both enlargements and stretches which affect Alice's body differently. We tend
to use the word enlargement even if we're making something smaller we would just use a scale factor
between 0 and 1 in that case and that mathematical use of the word enlargement
might make sentences like "you will grow shorter" make a little bit more sense.
Stretches would act about a point so if Alice is stretched about her middle that would make every
point twice as far away from the middle as it used to be which might explain why
her neck grows so long and in this case Alice has lost her mathematical temper.
The caterpillar might be telling Alice to keep her body in proportion no matter what size she
is. Carroll did love Euclidean geometry and wrote some books based on Euclid's elements,
and in Euclidean geometry it doesn't matter what the
magnitude of shapes are all that matters is the ratio between sides or the proportions.
I have seen many other interpretations of the Alice in Wonderland story,
people claim to have found commentary on social and political issues or references
to victorian literature. Given the author's background it does make sense for there to
be a little bit of mathematics in there too but maybe the nature of the story just lends itself
to these interpretations. Regardless I hope you've enjoyed taking a little look down the rabbit hole
and let me know if you find any other mathematical references yourself.
If you enjoy playing with math or finding creative ways to share science then take a
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I made a flamingo, something that Alice later uses to play croquet.
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A big thank you to KiwiCo and as always thanks to my Patreon supporters for helping me to make
these videos. A special shout out to today's Patreon cat of the day Xena Warrior Princess.