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Most readers are aware of the rotational movement around the table in A Mad Tea-Party, yet in the episode prior to this, the one with the Cheshire Cat on the branch, the feline had directed Alice towards the Hatter’s and Hare’s house, by using these strange words:

“In that direction,” the Cat said waving its right paw round, lives a Hatter, and in that direction,” waving the other paw, “lives a March Hare.” (italics around the words “that” are Carroll’s).

The reader is soon told that:

Alice waited a little...and after a minute or two she walked on in the direction in which the March Hare was said to live.

After this particular “walk” Alice finds herself looking at exactly the same tree branch and Cat, from what appears to be exactly the same perspective and distance! (For this refer to Tenniel’s two pictures of the event.) Therefore Alice must have walked around her own axis, just as the earth was said to “turn round on its axis,” leading to the Duchess’s axis/axes pun.

Alice then tells the Cat, who now vanishes from tail to head:

...I wish you wouldn’t keep appearing and vanishing so suddenly: you make one quite giddy.

In this case, as in similar ones in Wonderland and Looking-Glass, Carroll uses the word “giddy” overwhelmingly when there is some rotational movement occurring and someone is feeling giddy or dizzy. For Carroll use of the word “giddy,” see A Kwic Concordance to Lewis Carroll’s Alice’s adventures in Wonderland and Through the Looking-Glass, p. 163)

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  • Also, gyreing and gimbleing in the wabe. Mar 7, 2021 at 19:20
  • Yes, the geometer Carroll would also be into not only rotating around one’s own axis but also around another axis, as well as corkscrew types of rotations. Mar 7, 2021 at 19:28
  • Probably you're going to tell us that Alice was going around and around the tree, quoting The Nursery Alice.
    – Valorum
    Mar 7, 2021 at 19:45
  • Not at all, as there doesn’t seem to be any evidence in the text for that. Do you understand what walking round one’s axis entails? A geometer and logician such as Carroll certainly would understand these complex motions and what they may mean in terms of time and time keeping. Mar 7, 2021 at 19:53
  • By considering that in ‘Looking-Glass,’ Alice, the Hatter (now Hatta) and the March Hare (now Haigha) seem to have travelled back in time, it may be possible to go the other way as well. These T-Partiers (“t” being the algebraic letter for equations involving time) may have found a way to upset Time enough to have gone back hundreds of years. So I guess, that it may be possible for them to make amends for their “crimes,” or find a way to please Time ,and so travel forwards in time, to a time after the publication of ‘The Nursery Alice.’ Mar 7, 2021 at 20:22

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This article https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved/?ignored=irrelevant advances the theory that Lewis Carroll's works for children contain parodies of concepts from (what was then) modern mathematics that he didn't like, in particular, concepts involving matrices and rotations.

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

Alice’s ensuing attempt to solve the riddle pokes fun at another aspect of quaternions: their multiplication is non-commutative, meaning that x × y is not the same as y × x. 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’!”

It’s an idea that must have grated on a conservative mathematician like Dodgson, since non-commutative algebras contradicted the basic laws of arithmetic and opened up a strange new world of mathematics, even more abstract than that of the symbolic algebraists.

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