In February 2020, I entered a “#700STEMChallenge” writing competition, open to students from KS3-KS5 from over 14 schools. The task was to write a 700-word article on an area of STEM, and I was luckily awarded 1st place for the Under 18 category for Maths. At first, I couldn’t decide what to write on – Euclidean geometry? Topology? The golden ratio topic I’m researching for my EPQ, but with a different focus?
In the end, I found an area that combined several areas of maths, including its history, in a very clever, hidden and symbolic way. Everyone is familiar with this topic, but not many realise: there’s maths, even behind this.
Magic, Madness, Maths: A Curious Combination
The world of mathematics was in turmoil. Current and controversial, the new concepts of imaginary numbers and non-Euclidean geometries seemed incredibly far away from the physical reality of arithmetic and proofs. For a particular English mathematician named Charles Dodgson, such abstract contemporary ideas of the 19th century were “semi-logical”. Thus, he decided to combine fact with fiction and math with myth: by creating his own nonsensical fiction world, he could mock the new radical ideas by revealing their absurd conclusions. Perhaps better known as Lewis Carroll, Dodgson used this technique based on Euclid’s proofs, reductio ad absurdum, to create a mix of mad hatter maths. The result? Alice’s Adventures in Wonderland.
Having fallen down a rabbit hole into a wonderland of hidden maths, Alice eats a cake that reduces her height to three inches. Smoking a hookah pipe, the Caterpillar comes in with a mushroom to restore her size. However, Alice must eat exactly the right amount to return to the correct proportions, given that different parts of the mushroom either stretch or shrink various body parts. This scene arguably symbolises Dodgson’s ridicule of symbolic algebra, which allows “absurd” calculations with negative and impossible solutions. Taking a closer look at the Caterpillar’s pipe, “hookah” is, like “algebra”, of Arabic origin; the first Arabic translation of “algebra” was in fact “al jebr e al mokabala” (De Morgan, 1849) meaning “restoration and reduction” – which is exactly what Alice experiences. Alice seeks restoration of her size, yet encounters reduction. This is a striking parallel to De Morgan’s proposition: “reduce” algebra from universal arithmetic to symbolic operations, in order to “restore” the significance of symbolic algebra.
The madness of this mathematical wonderland only becomes “curiouser and curiouser”, as Alice notes. The next chapter, “Pig and Pepper”, describes the kitchen where underneath the action a geometry parody is at play. In this scene, the Duchess gives Alice her baby who suddenly transforms into a pig, which represents Dodgson’s derision of projective geometry. In particular, this concerns the “principle of continuity”, defined by its French inventor Jean-Victor Poncelet as the idea that for geometric figures undergoing a continuous transformation, as long as there are no sudden changes, “the same property will belong to all the successive states of the figure”. An example of this would be two intersecting circles. By solving their equations simultaneously, two distinct points of intersection are obtained. The principle claims that this property will be preserved with any continuous transformation, such as moving the circles’ centres apart. Even when the circles no longer touch, they will still intersect twice, but the solution will include imaginary numbers where i is √-1. Dodgson presents a geometric figure in the form of a baby, turning it into a pig through the continuity principle. Indeed, most of the baby’s original features stay the same, as should an object under a continuous transformation.
By describing the Hatter’s tea party, Dodgson explores the breakthrough discovery of quaternions in 1843 by William Rowan Hamilton. Quaternions are a number system based on four terms which enable mathematicians to calculate rotations. With three terms (one for each dimension of space), Hamilton could achieve rotation in a plane, but could only speculate that the fourth term was the concept of time, which would allow a three-dimensional rotation. This is mirrored at the tea party where Alice sits with three characters: the Hatter, the Dormouse and the March Hare. Dodgson uses these characters to represent three terms of a quaternion, where the fourth term – portrayed as the character Time – is missing. This only allows plane rotations, symbolised by the characters continuously circling the table. With quaternion multiplication being non-commutative, x x y does not equal y x x which is reflected in the characters’ exchange. The Hare tells Alice to “say what she means”, to which she replies, “at least I mean what I say”, which the Hatter says is “not the same thing”. A contradiction to basic arithmetic, non-commutative algebras arguably seemed even more absurd to Dodgson than symbolic algebra.
If we look through the looking glass carefully enough, we discover a whimsical wonderland of magic, madness and maths. What would remain of Alice’s Adventures in Wonderland without this curious combination?
Reference list:
Anon., 2010. The Mad Hatter's Secret Ingredient: Math. [Online] Available at: https://www.npr.org/templates/story/story.php?storyId=124632317&t=1581441141860 [Accessed 10 February 2020].
Anon., n.d. Mathematics in the 19th Century. [Online] Available at: https://www.britannica.com/science/mathematics/Mathematics-in-the-19th-century [Accessed 6 February 2020].
Asthana, D., 2018. The Hidden Math Behind Alice in Wonderland. [Online] [Accessed 7 February 2020].
Bayley, M., 2009. NewScientist. [Online] Available at: https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/ [Accessed 4 February 2020].
Devlin, K., 2010. The Hidden Math Behind Alice in Wonderland. [Online] Available at: https://www.maa.org/external_archive/devlin/devlin_03_10.html [Accessed 6 February 2020].
Morgan, A. D., 1849. Trigonometry and Double Algebra. s.l.:s.n.