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Geometric algebra (Clifford Algebra) unfortunately came late historically. It's a shame, because the whole theory is a very useful (eg for engineering / applie

Geometric algebra (Clifford Algebra) unfortunately came late historically. It's a shame, because the whole theory is a very useful (eg for engineering / applied math) superset of linear algebra. I really wish I had learned this first in my undergrad years, would have made a whole bunch of things way clearer from the get go: differential forms tensor calculus linear algebra etc From zero to geo is a very good video introduction to the topic: https://www.youtube.com/watch?v=2hBWCCAiCzQ&list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy&index=2 https://www.youtube.com/watch?v=2hBWCCAiCzQ&list=PLVuwZXwFua...

an1sotropy 5y ago
Yes, this. Though Zero to Geo is one of the links at the bottom of the article. It is really a shame that article does not clarify that, btw, what we've just derived is a re-derivation of a thing that has already been expressed and named, by Clifford, and well-characterized: https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra Such a bummer to see very slick but very ahistorical articles.
- mferraro89 5y ago
  Hi. I wrote whole section on the history of GA and what happened and why it isn't already the norm, but I chose to remove it because the article is already far too long, and I don't think that my intended audience (engineers, compsci people, university undergrads) would care about the history. Apologies that wasn't what you would have preferred.
  - an1sotropy 5y ago
    I think the article is great, and I think the interactive illustrations are sweet. Thanks for the taking the time to write it. As an educator, though, when I see presentations of existing ideas that present them as if they were new, I die a little. You're standing on the shoulders of giants whether or not you think so, and whether or not you say so. It's best to figure out who the giants are, and how you're standing on them. When you are up front about the connections to past scholars, you are giving the credit to those scholars that they deserve, and you are strengthening the storyline, and you are setting a good example for the people that look up to you. You can add a few sentences at the bottom saying, if you've gotten this far, congrats, you understand some basics of GA, and then link to other resources about the history and current applications of it.
adrian_b 5y ago
Its rediscovery was very late. Its initial discovery was not too late, but both Clifford and Maxwell died too young in 1879 and after their too early death there was no one left who could finalize the applications of this theory to physics. In their absence, the geometric algebra theory was ignored and both the theory of vectors and the theory of electromagnetic field were simplified to forms which are good enough for restricted contexts, but which are nonetheless inconsistent and fail in more general cases (the so-called Maxwell differential equations are valid only in much more restricted conditions than the original integral equations of Maxwell). As a child I have also learned the theory of vectors in the incorrect way, e.g. including the so-called "vectorial product", so I lost time later until understanding that it is not a vector, and then I lost more time until understanding that the so-called pseudovectors a.k.a. axial vectors and the so-called pseudoscalars are not independent entities that come from nowhere, but their existence is just the natural consequence of the properties of the vectors. It would have been much more efficient if the complete theory of geometric algebra would have been taught from the beginning.

an1sotropy 5y ago

Yes, this. Though Zero to Geo is one of the links at the bottom of the article. It is really a shame that article does not clarify that, btw, what we've just derived is a re-derivation of a thing that has already been expressed and named, by Clifford, and well-characterized: https://en.wikipedia.org/wiki/Geometric_algebra https://en.wikipedia.org/wiki/Geometric_algebra Such a bummer to see very slick but very ahistorical articles.

mferraro89 5y ago
Hi. I wrote whole section on the history of GA and what happened and why it isn't already the norm, but I chose to remove it because the article is already far too long, and I don't think that my intended audience (engineers, compsci people, university undergrads) would care about the history. Apologies that wasn't what you would have preferred.
- an1sotropy 5y ago
  I think the article is great, and I think the interactive illustrations are sweet. Thanks for the taking the time to write it. As an educator, though, when I see presentations of existing ideas that present them as if they were new, I die a little. You're standing on the shoulders of giants whether or not you think so, and whether or not you say so. It's best to figure out who the giants are, and how you're standing on them. When you are up front about the connections to past scholars, you are giving the credit to those scholars that they deserve, and you are strengthening the storyline, and you are setting a good example for the people that look up to you. You can add a few sentences at the bottom saying, if you've gotten this far, congrats, you understand some basics of GA, and then link to other resources about the history and current applications of it.

adrian_b 5y ago

Its rediscovery was very late. Its initial discovery was not too late, but both Clifford and Maxwell died too young in 1879 and after their too early death there was no one left who could finalize the applications of this theory to physics. In their absence, the geometric algebra theory was ignored and both the theory of vectors and the theory of electromagnetic field were simplified to forms which are good enough for restricted contexts, but which are nonetheless inconsistent and fail in more general cases (the so-called Maxwell differential equations are valid only in much more restricted conditions than the original integral equations of Maxwell). As a child I have also learned the theory of vectors in the incorrect way, e.g. including the so-called "vectorial product", so I lost time later until understanding that it is not a vector, and then I lost more time until understanding that the so-called pseudovectors a.k.a. axial vectors and the so-called pseudoscalars are not independent entities that come from nowhere, but their existence is just the natural consequence of the properties of the vectors. It would have been much more efficient if the complete theory of geometric algebra would have been taught from the beginning.