Lectures on the Icosahedron and the Solution of the Fifth Degree, Felix Klein (Trübner & Co., 1884). Raise your hand if you spent your childhood wondering why the world has three spatial dimensions. I hope that’s all of you, because it’s really very odd once you stop to think about it. Isn’t three an awfully arbitrary number? Why not two? Or four? Or Seven thousand two hundred and sixty eight? What’s so special about three?
This was fun. thanks. The group theory stuff, reminds me of quaternions (and octonions (which now I've checked the spelling is my sum knowledge of octonions.)) As complex numbers are an extension of real numbers into another dimension, quaternions extend numbers into four dimensions. And for a math reason I don't know/understand you can't make a number system in three dimensions. And the only other system you can make a useful number in is eight. There's a nice three blue one brown video on quaternions. (I guess most of this comment was for your other readers... A good biography of William Rowan Hamilton?)
Yes! The fact that there are only four division algebras (reals, complex, quaternions, octonions) is a great example of the kind of thing I'm talking about at the start of the review.
Octonions indeed have all kinds of interesting connections to mathematical physics. Check out this video which I might have linked already in the review (http://www.youtube.com/watch?v=Tw8w4YPp4zM) or this paper (https://arxiv.org/abs/0711.2699). The paper is about quaternions, but the authors have a follow-up coming that connects octonions to quantum gravity.
That was great, but it makes me wish I were smarter. How accessible is the book? And is there a book that talks about the stuff you said about music and gravity and suchlike? I think you said that wasn’t in Klein’s book.
Klein's book isn't the most accessible (if it were published today it would basically be a research paper), but a great entry point to these ideas is Hermann Weyl's "Symmetry" (https://www.amazon.com/Symmetry-Princeton-Science-Library-Hermann/dp/0691173257). Weyl was one of the inventors of 20th century mathematical physics so he knows what he's talking about, and this is his book that tries to explain these ideas to laypeople -- starting with the popular conception of symmetry and gradually working up to the abstract mathematical notion.
This was fun. thanks. The group theory stuff, reminds me of quaternions (and octonions (which now I've checked the spelling is my sum knowledge of octonions.)) As complex numbers are an extension of real numbers into another dimension, quaternions extend numbers into four dimensions. And for a math reason I don't know/understand you can't make a number system in three dimensions. And the only other system you can make a useful number in is eight. There's a nice three blue one brown video on quaternions. (I guess most of this comment was for your other readers... A good biography of William Rowan Hamilton?)
Yes! The fact that there are only four division algebras (reals, complex, quaternions, octonions) is a great example of the kind of thing I'm talking about at the start of the review.
Octonions indeed have all kinds of interesting connections to mathematical physics. Check out this video which I might have linked already in the review (http://www.youtube.com/watch?v=Tw8w4YPp4zM) or this paper (https://arxiv.org/abs/0711.2699). The paper is about quaternions, but the authors have a follow-up coming that connects octonions to quantum gravity.
That was great, but it makes me wish I were smarter. How accessible is the book? And is there a book that talks about the stuff you said about music and gravity and suchlike? I think you said that wasn’t in Klein’s book.
Klein's book isn't the most accessible (if it were published today it would basically be a research paper), but a great entry point to these ideas is Hermann Weyl's "Symmetry" (https://www.amazon.com/Symmetry-Princeton-Science-Library-Hermann/dp/0691173257). Weyl was one of the inventors of 20th century mathematical physics so he knows what he's talking about, and this is his book that tries to explain these ideas to laypeople -- starting with the popular conception of symmetry and gradually working up to the abstract mathematical notion.