# Quaternions: Part of The Hidden Math Behind Alice in Wonderland

Quote: Alice, angry now at the strange turn of events, leaves the Duchess’s house and wanders into the Mad Hatter’s tea party. This, Bayley surmises, explores the work of the Irish mathematician William Rowan Hamilton, who died in 1865, just after Alice was published. Hamilton’s discovery of quaternions in 1843 was hailed as an important milestone in abstract algebra, since they allowed rotations to be calculated algebraically.

Just as complex numbers work with two terms, quaternions belong to a number system based on four terms. Hamilton spent years working with three terms – one for each dimension of space – but could only make them rotate in a plane. When he added the fourth, he got the three-dimensional rotation he was looking for, but he had trouble conceptualizing what this extra term meant. Like most Victorians, he assumed this term had to mean something, so in the preface to his Lectures on Quaternions of 1853 he added a footnote: “It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time.”

As Bayley points out, the parallels between Hamilton’s mathematics and the Mad Hatter’s tea party are uncanny. Alice is now at a table with three strange characters: the Hatter, the March Hare and the Dormouse. The character Time, who has fallen out with the Hatter, is absent, and out of pique he won’t let the Hatter move the clocks past six.

Reading this scene with Hamilton’s ideas in mind, the members of the Hatter’s tea party represent three terms of a quaternion, in which the all-important fourth term, time, is missing. Without Time, we are told, the characters are stuck at the tea table, constantly moving round to find clean cups and saucers.

Their movement around the table is reminiscent of Hamilton’s early attempts to calculate motion, which was limited to rotatations in a plane before he added time to the mix. Even when Alice joins the party, she can’t stop the Hatter, the Hare and the Dormouse shuffling round the table, because she’s not an extra-spatial unit like Time.

The Hatter’s nonsensical riddle in this scene – “Why is a raven like a writing desk?” – may more specifically target the theory of pure time. In the realm of pure time, Hamilton claimed, cause and effect are no longer linked, and the madness of the Hatter’s unanswerable question may reflect this.

Alice’s ensuing attempt to solve the riddle pokes fun at another aspect of quaternions that Dodgson would have found absurd: their multiplication is non-commutative. Alice’s answers are equally non-commutative. When the Hare tells her to “say what she means”, she replies that she does, “at least I mean what I say – that’s the same thing”. “Not the same thing a bit!” says 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’!”

When the scene ends, the Hatter and the Hare are trying to put the Dormouse into the teapot. This could be their route to freedom. If they could only lose him, they could exist independently, as a complex number with two terms. Still mad, according to Dodgson, but free from an endless rotation around the table.

Sources:
1) The Hidden Math Behind Alice in Wonderland, by Keith Devlin, http://www.maa.org/devlin/devlin_03_10.html, Mathematical Association of America
2) Topic: Quaternions in Alice in Wonderland, in http://groups.google.com/group/geometric_algebra/topics

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### 5 responses to “Quaternions: Part of The Hidden Math Behind Alice in Wonderland”

1. Peter L. Griffiths

Where Hamilton is wrong over Quaternions is that he fails to distinguish angles which are variables from abstract numbers such as i and -i which are constants. This arises because Hamilton fails to discuss the Cotes formula
which clearly distinguishes angles u from i and -i, particularly cosu + isinu.

2. Peter L. Griffiths

Why do the roots of +1,-1,+i and -i always add up to zero? The answer is that computing all these roots is a matter of going round the angles of a circle and finishing at the 360 degrees (=0 degrees) starting point. Incidentally if you have computed the roots of +i, then the roots of -i are easily obtained by substituting -i for +i.

3. Further to my comment of 27 February 2012, -1 cannot have more than two square roots, nevertheless -1 can have three cube roots which are cos60+isin60, cos180+isin180 which equals -1, and cos300+isin300. Hamilton had no understanding of these matters.

4. Peter L.Griffiths

Further to my previous comment of 27 August 2011, on 16 October 1843 Hamilton alleged that i^2=j^2=k^2=ijk=-1. This equation is incorrect because every real, imaginary and complex number has only two square roots, so k^2=-1 is wrong unless k can equal either i or j. It can likewise be demonstrated that every real,imaginary and complex number has only three cube roots, four fourth roots, five fifth roots etc.

5. Peter L. Griffiths

Hamilton in his letter of 17 October 1843 is very confused about the relationship between i, j, +1 and -1. He asks what are we to do with ij? In fact i and j are the unequal roots of a common square, and there is no law of arithmetic which makes ij equal to anything but +1. It is these doubts of Hamilton which are the source of his fallacious theory of the non-commutative properties of the multiplication of imaginary numbers. All multiplication whether of imaginary or real numbers is commutative.