Alice underground: the door, the quaternion and the relativity

Alice underground is the first version of Aline in the wonderland by Lewis Carroll. The original manuscript, illustrated by Carroll himself, was given to the little Alice Liddell for Christmas in 1864 and picked up the story that he had told to Alice and her sisters Lorina and Edith during a summer's afternoon, precisely on July the 4th, 1862. This first version of the carrollian fantasy novel is, ultimately, a restricted version of Alice, where various characters and episodes completely absent in Underground are added, such as the Duchess or the team composed by the Mad Hatter, the March Hare and the Dormouse.
The intial, interesting considerations about underground is about the importance of the trees and the doors: following the suggestion by Adele Cammarata(3), we can assume that the tree and the door that Alice cross to enter the garden of the Queen of Hearts, completely absent in Wonderland, is linked with the Celtic tradition. Indeed the oak is one of the sacred trees of the druids, symbolizing a link between heaven and earth(1). In this way the oak, which in Celtic was called duir, is a real door that connects people with the gods, but also ourselves with our inner part. So, from an etymological point of view, a carved door in a tree trunk is a Celtic symbol used to identify the Alice's passage towards a more stable phase after the size's changes of the previous scenes.
These changes in size, alluding both to the transition to adulthood, in perfect connection with the Druidic symbolism, and with the more classic homothetic transformations, i.e. the transformations which, without changing the proportions of a geometric figure, change its size. All these changes remain unchanged in the transition to the second version, including the meeting with the Caterpillar, who continues to ask Alice:
Who are you?
One piece, absent in Underground, is the Mad Hatter's tea party, that it is not only a particular scene in which mention the time (in this case stopped for the protagonists of the party), and even a way to mention the cyclical partitions, but it could be a way to introduce in a fairy tale quaternions, discovered by William Rowan Hamilton.
Hamilton began to work on an extension of complex numbers, constituted by two parts, a real and an imaginary (where for imaginary number is defined as a multiple of the square root of $-1 = i^2$). Earlier this extension had only three numbers, and Hamilton could only get rotations in the plane; then he added the fourth number, allowing him to get also the spatial rotations. In order to give an interpretation to his results, Hamilton wrote in 1853 on Lectures on Quaternions:
It seemed (and still seems) to me natural to connect this extra-spatial unit with the conception of time.
Starting from these elements, we can intend the party in this way: the Hatter, the March Hare and the Dormouse are three elements of a quaternion orphans of the fourth element, and this forces them to constantly go around the table, always drenching the same biscuits in the same cups without ever being able to change their condition. The fact that Alice is not able to change their condition suggests that she is like a space coordinate.
Another passage that seems to fit with the mathematics of quaternions is the Hatter's response to Alice at the end of this exchange:
'You should learn not to make personal remarks,' Alice said with some severity; 'it's very rude.'
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. 'You might just as well say that "I see what I eat" is the same thing as "I eat what I see"!'
that we can intend as an allusion to the non-commutativity of quaternions.
Shortly, a quaternion is a kind of 4-vector written (or defined) as follows: \[q = a + bi + cj + dk\] where $a$, $b$, $c$, $d$ are real numbers, while $i$, $j$, $k$ are the coordinates into a 3d imaginary space (in some sense the equivalent of $x$, $y$, $z$ in the real space). It's not a case if $a$ is called the scalar part, while $bi + cj + dk$ is the vectoral part of the quaternion. The three imaginary vector directions are then related by the following equation: \[i^2 = j^2 = k^2 = ijk = -1\] But when you start playing a little with quaternions, some interesting properties check out: for example, you can build your own group of rotations starting from quaternionic units (to every quaternion we can associate a rotation in space) or even the so-called quaternion group, non-commutative, and which we can provide a representation using both matrices $2 \times 2$ with complex values, both $4 \times 4$ with real values(2).
Th last observation: the group theory comes from the studies by Evariste Galois and Niels Abel about the solutions of the 5th grade, and higher polynomials; in the same way, as shown in 1981 by Richard Dean, quaternion group comes from the following polynomial: \[x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36\]
Alice at theatre
Carroll's novel has had in its history several theater transpositions. Of all, I saw one of the Milan theater "Elfo Puccini", in 2013, based on Alice Underground: absurd and unusual scenes in carrollian style with some scientific ideas. For example the presence of Mr. Time and Mr. Space that opened the piece talking to each other while Alice spleeping her adventures in the Wonderland. In the meanwhile on the white wall behind the actors images and drawings are projected: clocks, numbers, math symbols and equations like the famous Einstein's relativity field equation: \[R_{\mu \nu} - {1 \over 2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}\] Despite the absence of the Queen of Hearts (only as a projected image and voiceover), appreciable the attempt to make the logic as one of the subject of the piece: there is not only the Mad Hatter's tea party, but also the scene with the Duchess, the encounter with the twins Tweedledee and Tweedledum and, above all, the famous race of the Red Queen. In particular this last scene is beautiful thanks to the actress: as Alice runs to enter into the garden of the Queen of Hearts, or to go into the world beyond the mirror (through a metaphorical hall of mirrors), the actress runs on site, making so a run which leaves the runner at the starting point!
Interesting the closing considerations about dream and dreamer: what could happen to the dream if the dreamer wake up? This theme was explored by a lot of science fiction, for example in Tonight the Sky Will Fall by Daniel F. Galouye or in the Joe Lansdale's Drive-In saga. And, in one of the various interpretation of carrollian works, it could be also a Lewis Carroll's quest.
(1) Many Celtic gods are represented with a face set in a tree trunk. I also remember the tradition of the tree of life, apparently symbolic in the Jewish tradition, but, if we think of its Celtic equivalent, present also in The Silmarillion and in the Tolkien's epic.
(2) Read also Alice adventures in Algebra: Wonderland solved (3rd part)
(3) Adele Cammarata (2002). Introduction to Alice underground, Stampa Alternativa, Vietrbo

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