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Sorry to spoil the party, but this is the old 19th century way of teaching quaternions (and also complex numbers). It is much easier to start with some https://

by amai 8y ago

Sorry to spoil the party, but this is the old 19th century way of teaching quaternions (and also complex numbers). It is much easier to start with some https://en.wikipedia.org/wiki/Group_theory https://en.wikipedia.org/wiki/Group_theory and then you understand that quaternions are simply matrices of a specific form: https://en.wikipedia.org/wiki/Quaternion#Matrix_representations https://en.wikipedia.org/wiki/Quaternion#Matrix_representati... . Quaternion multiplication is simply matrix multiplication of these matrices. And that's it. No mysteries, this is just simple linear algebra (you do't even need complex numbers, the real representation is enough and makes the connection to 4d rotations manifest).

gmadsen 8y ago
it shouldn't be an either or with geometry and algebra. both are valid and it helps to know both. representing H as a matrix is certainly valid and makes computations easier, but I dont really think that is building intuition
- earthicus 8y ago
  The modern approach is not lacking all geometry, and the 19th century presentation is not lacking all algebra. Any respectable treatment will include both, the question is which makes the relationship clearer? The abstract algebraic approach talks of 'the rotation group' which maps on to geometric concepts as cleanly and directly as possible. Then we talk of different parameterizations or representations of this group, with the unit quaternions being the 'double cover' of the rotation group (which gives rise to the primary difficulty in understanding them - that a rotation by 360 degrees reverses orientation, and 720 degrees returns us to where we started). I think the modern approach is much clearer - the geometric ideas appear more directly, and the algebra is far less messy.
  - edflsafoiewq 8y ago
    > I think the modern approach is much clearer - the geometric ideas appear more directly, and the algebra is far less messy. Can you show us an example?
twtw 8y ago
Do you have any books you could recommend that present this approach?
- earthicus 8y ago
  Stillwell[1] discusses quaternions in chapters 1 & 2, although the meat and potatoes of the book is more generally about lie groups and lie algebras. It's pitched at a lower level than most books on this topic by restricting attention to the classical matrix groups. [1]https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387782141/ref=sr_1_1?ie=UTF8&qid=1540144950&sr=8-1 https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math...
  - twtw 8y ago
    Thank you.
kwoff 8y ago
If you want to go the opposite direction, Hamilton's "Elements of Quaternions" is long out of copyright: https://archive.org/details/elementsquaterni00hamirich/page/n0 https://archive.org/details/elementsquaterni00hamirich/page/... (I doubt most people get past the impressively long table of contents. :)
joppy 8y ago
A lot of open problems in mathematics boil down to trying to analyse "matrices of a specific form" - I think while this viewpoint is important to recognise (essentially: quaternions are a finite-dimensional real associative algebra), it might not give as much intuition as other interpretations. It's kind of like how in computer science, many many hard problems can be restated as problems about some graph constructed from the input, but now you have two problems...

by amai 8y ago

gmadsen 8y ago

it shouldn't be an either or with geometry and algebra. both are valid and it helps to know both. representing H as a matrix is certainly valid and makes computations easier, but I dont really think that is building intuition

earthicus 8y ago
The modern approach is not lacking all geometry, and the 19th century presentation is not lacking all algebra. Any respectable treatment will include both, the question is which makes the relationship clearer? The abstract algebraic approach talks of 'the rotation group' which maps on to geometric concepts as cleanly and directly as possible. Then we talk of different parameterizations or representations of this group, with the unit quaternions being the 'double cover' of the rotation group (which gives rise to the primary difficulty in understanding them - that a rotation by 360 degrees reverses orientation, and 720 degrees returns us to where we started). I think the modern approach is much clearer - the geometric ideas appear more directly, and the algebra is far less messy.
- edflsafoiewq 8y ago
  > I think the modern approach is much clearer - the geometric ideas appear more directly, and the algebra is far less messy. Can you show us an example?

twtw 8y ago

Do you have any books you could recommend that present this approach?

earthicus 8y ago
Stillwell[1] discusses quaternions in chapters 1 & 2, although the meat and potatoes of the book is more generally about lie groups and lie algebras. It's pitched at a lower level than most books on this topic by restricting attention to the classical matrix groups. [1]https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Mathematics/dp/0387782141/ref=sr_1_1?ie=UTF8&qid=1540144950&sr=8-1 https://www.amazon.com/Naive-Theory-Undergraduate-Texts-Math...
- twtw 8y ago
  Thank you.

kwoff 8y ago

If you want to go the opposite direction, Hamilton's "Elements of Quaternions" is long out of copyright: https://archive.org/details/elementsquaterni00hamirich/page/n0 https://archive.org/details/elementsquaterni00hamirich/page/... (I doubt most people get past the impressively long table of contents. :)

joppy 8y ago

A lot of open problems in mathematics boil down to trying to analyse "matrices of a specific form" - I think while this viewpoint is important to recognise (essentially: quaternions are a finite-dimensional real associative algebra), it might not give as much intuition as other interpretations. It's kind of like how in computer science, many many hard problems can be restated as problems about some graph constructed from the input, but now you have two problems...