Have you ever wondered what lies upon the other side of a mirror? And another question: If there is another world beyond the mirror, how does it reflect on our world? Lewis Carroll dealt with these questions in his book “Through the Looking-Glass, and What Alice Found There”, and the result is… spectacular! Caroll fills his fantastical beyond-the-mirror world with nonsense, humor, creativity, and lots and lots of imagination. And of course, as it’s always with Caroll, the book is full of mathematics – you just have to look carefully to see it, which is exactly what we are going to do in this article! We did our first dive into Caroll’s math in our article “Alice in Mathland”, and now it’s time to reverse our logic, enter the mirror-land and see what surprises wait for us there.
The world behind Alice’s mirror is a game, a chess game. If Asimov included a chess game in his novel, then Caroll wrote a whole novel about one, incredibly big, chess game. Alice is a Pawn, moving only forward over the chessboard to become a Queen, and during the entire book, she can only talk with the pieces that surround her in adjacent squares. All the pieces behave as they should, according to the game’s rules: the queens always run, the knights frequently fall off their horses due to their weird moving shape (forming the shape of an L), and the kings are slow, helpless, or even asleep. Although, one must admit that the game is a bit weird, and does not fully follow the chess rules.
You see, the sequence of moves (of white and red pieces) is not always followed, and each color may perform more than one move in a row. Less significant but still worth mentioning: Some checkmate opportunities are ignored, the white queen runs away from the red knight instead of attacking him, and in general, the reasoning behind most of the moves is doubtful. Luckily, most of it can be explained by the quirkiness of the characters and artistic choices.
Many tried to “fix” Carroll’s game, but one such attempt deserves special mention: In 1910, Donald M. Liddell constructed a genuine 66-moves chess game that fully follows the story events and opens with the Bird (an opening that alludes to a great British player and chess historian of the nineteenth century). It was published in the British Chess Magazine, Vol 30, p181-184, however, in the same issue we found this: A response by the chess editor of the Bradford Observer Budget newspaper that wrote –
“Mr. Donald M. Liddell submits a game to illustrate the advances of Alice in ‘Through the Looking-glass.’ Mr. Liddell undoubtedly means well, but we may tell him quite frankly that his attempt to compete with the author of ‘Alice’ in this way is predestined to ignominious failure. We played over the whole of Mr. Liddell’s game for the benefit of a young lady of about Alice’s own age who wanted a story to be read to her, and she wasn’t a bit pleased. Indeed, not to mince matters, she was very cross.”
Nevertheless, we believe that what Mr. Liddell did is an admirable achievement. Regarding Carroll’s game, it’s worth noting that it was one of the first attempts to combine a plot with a chess game, and as such, it does a good job.
Mirror, mirror on the wall
“Tweedledum and Tweedledee
Agreed to have a battle;
For Tweedledum said Tweedledee
Had spoiled his nice new rattle.
Just then flew down a monstrous crow,
As black as a tar-barrel;
Which frightened both the heroes so,
They quite forgot their quarrel.”
They were standing under a tree, each with an arm round the other’s neck… “The first thing in a visit is to say ‘How d’ye do?’ and shake hands!” And here the two brothers gave each other a hug, and then they held out the two hands that were free, to shake hands with her.
Chapter 4: Tweedledum and Tweedledee
Tweedledee and Tweedledum look and act in identical ways as if each one is the reflection of the other. “Nohow” would Tweedledum say, “Contrariwise” would Tweedledee continue. Tweedledee would use his right hand, while Tweedledum his left. Symmetry and asymmetry stimulated a great deal of research in mathematics, which leads to defining these two characters as enantiomorphs, 3-dimensional objects that are like mirrors of each other, sharing one property: Chirality. An object is chiral if it cannot be made to look like its mirror-image by rotating it or displacing it, i.e. it’s distinguishable from its mirror image – just like our hands.
Chirality is very important in our world, and Alice wonders about that as well: Just a moment before entering the mirror-land Alice shares her thoughts with her cat: “How would you like to live in Looking-glass House, Kitty? I wonder if they’d give you milk in there? Perhaps Looking-glass milk isn’t good to drink”. Surprise surprise, but nowadays we know that this seemingly innocent question actually reveals an interesting scientific finding: The looking-glass milk would almost certainly not be good to drink, or to the least, Kitty won’t feel its taste!
It appears that the lack of symmetry in molecules is crucial to our existence. Natural chiral molecules almost always occur only in the left-handed or the right-handed form, not both. For example, we may enjoy a cup of tea with D-glucose, but we won’t feel the taste of L-glucose at all. Chiral molecules need to be handled by chiral enzymes with the same handedness, while different handedness may cause the molecules to react differently in the human body – the consequences can be dangerous. As for the milk, its mirror-image molecules would be with the wrong handedness for Kitty’s enzymes to digest. Please, don’t try mirror-milk at home!
Carroll could not know about the milk, but his thoughts about symmetry and chirality were spot-on!
“I know what you’re thinking about,” said Tweedledum: “but it isn’t so, nohow.”
“Contrariwise,” continued Tweedledee, “if it was so, it might be; and if it were so, it would be: but as it isn’t, it ain’t. That’s logic.”
In the name of
Ah… Logic – Carroll’s favorite. On her journey, Alice meets the White Knight, which is one of the kindest characters in the book. The White Knight decides to sing her a song:
“Everybody that hears me sing it
——either it brings the tears into their eyes, or else——““Or else what?” said Alice, for the Knight had made a sudden pause.
“Or else it doesn’t, you know. The name of the song is called ‘Haddocks’ Eyes.'”
“Oh, that’s the name of the song, is it?” Alice said, trying to feel interested.
“No, you don’t understand,” the Knight said, looking a little vexed. “That’s what the name is called. The name really is ‘The Aged Aged Man.'”
“Then I ought to have said ‘That’s what the song is called’?” Alice corrected herself.
“No, you oughtn’t: that’s quite another thing! The song is called ‘Ways And Means‘: but that’s only what it’s called, you know!”
“Well, what is the song, then? ” said Alice, who was by this time completely bewildered.
“I was coming to that,” the Knight said. “The song really is ‘A-sitting On A Gate‘: and the tune’s my own invention.”
Chapter 8: “It’s My Own Invention”
That may sound confusing, but to a student of logic that sounds perfectly fine. And not only to a student of logic, but to every computer programmer as well. In computers, information is stored in memory units. Each such unit has an address on the computer. We can store the speed of light 299792458 m/s in a memory unit and call it SpeedOfLight, the address of that value we can store in another memory unit: ReferenceToSpeedOfLight, and of course, we can store the address of that information in another memory unit ReferenceToReferenceToSpeedOfLight and so on and so forth. In math and especially in logic, it’s the same and Carroll is distinguishing here among things, the name of things, and the names of names of things. Yes, logic can be confusing.
Nobody
“Just look along the road, and tell me if you can see either of them.”
“I see nobody on the road,” said Alice.
“I only wish I had such eyes,” the King remarked in a fretful tone. “To be able to see Nobody! And at that distance too! Why, it’s as much as I can do to see real people, by this light!”
All this was lost on Alice, who was still looking intently along the road, shading her eyes with one hand. “I see somebody now!” she exclaimed at last. “But he’s coming very slowly
——and what curious attitudes he goes into!” (For the Messenger kept skipping up and down, and wriggling like an eel, as he came along, with his great hands spread out like fans on each side.)At this moment the Messenger arrived: he was far too much out of breath to say a word, and could only wave his hands about, and make the most fearful faces at the poor King.
“Who did you pass on the road?” the King went on, holding out his hand to the Messenger for some more hay.
“Nobody,” said the Messenger.
“Quite right,” said the King: “this young lady saw him too. So of course Nobody walks slower than you.”
“I do my best,” the Messenger said in a sullen tone. “I’m sure nobody walks much faster than I do!”
“He can’t do that,” said the King, “or else he’d have been here first. However, now you’ve got your breath, you may tell us what’s happened in the town.”
Chapter 7: The Lion and the Unicorn
Nonsense, right? It’s funny because of how silly it is, unless… while normally for us nobody is nothing, in mathematics it is a real thing! And it all has to do with “sets”.
A set is a well-defined collection of distinct objects, like numbers. For example, a set of 2, 4, and 8 is denoted by {2, 4, 8}, and it’s the same as {4, 2, 8} because it has the same elements. If every element of one set is included in another set, then the first set is a subset of the second set. Therefore, the set {4} is a subset of {2, 4, 8}, which is the subset of the set that includes all even numbers, and the set {2, 6} is not a subset of {2, 4, 8}, because {2, 6} has the element 6 that is not included in {2, 4, 8}.
There is a unique set with no members that is called the “Empty Set”, and it is commonly denoted by the symbols Ø or {}. The empty set is a subset of every set – you cannot “see” it, because we don’t write {2, 4, 8, Ø}, but it’s still there. So is it a thing? Yes! There is even an approach, that won’t be discussed here, that defines all natural numbers (0, 1, 2, 3, 4, …) from a combination of many empty sets. It’s “bizarre”, but exists – you can read about it by searching for “Von Neumann definition of ordinals”.
Since Alice experiences many different mathematical weirdnesses, seeing “nobody” shouldn’t come as a surprise. As to making sense, it’s as good as it gets. The Red Queen phrased it better:
“I only wanted to see what the garden was like, your Majesty
——““That’s right,” said the Queen, patting her on the head, which Alice didn’t like at all, “though, when you say ‘garden.’—I’ve seen gardens, compared with which this would be a wilderness.”
Alice didn’t dare to argue the point, but went on: “
——and I thought I’d try and find my way to the top of that hill——““When you say ‘hill,'” the Queen interrupted, “I could show you hills, in comparison with which you’d call that a valley.”
“No, I shouldn’t,” said Alice, surprised into contradicting her at last: “a hill can’t be a valley, you know. That would be nonsense
——“The Red Queen shook her head. “You may call it ‘nonsense’ if you like,” she said, “but I’ve heard nonsense, compared with which that would be as sensible as a dictionary!”
Chapter 2: The Garden of Live Flowers
Sense and Nonsense
We didn’t cover gems like Red’s Queen “Now, here, you see, it takes all the running you can do, to keep in the same place” or The Red King that dreams about Alice that dreams about the Red King, in an endless loop, and many others.
We began our journey with the question of how the mirror-land reflects on our world – That world is full of nonsensical situations, many of which are based on various mathematical aspects. Things that seem to be impossible are based on fundamental mathematical ideas, which are true not only in mirror-land, but in our world as well, and are just exaggerated in the mirror-land’s fantastical world.
“I’m just one hundred and one, five months and a day.”
Chapter 5: Wool and Water
“I can’t believe that!” said Alice.
“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.”
Alice laughed. “There’s no use trying,” she said: “one can’t believe impossible things.”
“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”
As Stephanie Pradier said in The Science Show:
“Nonsense opens our minds to the infinite possibilities of our world, and allows us to think of six impossible things before breakfast. I believe the human soul needs nonsense, just as it needs art, literature, music, and maths.”
Related Articles: