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Apart from the exterior algebra part, there is little added value to using geometric algebra compared to other representations. My guess is that most applicati

Apart from the exterior algebra part, there is little added value to using geometric algebra compared to other representations. My guess is that most applications don't really need exterior algebra, or at least they would not benefit much from the nice unified representation it provides.

DreamScatter 7y ago
Hi, I'm the developer of Grassmann.jl, Actually, it has a lot more value to provide than what is taught in school. It never caught on because it was historically difficult to understand without computer algebra to help build intuition.. so the only people who understood it are the ones bold to explore mathematics deeply enough to arrive at this system and utilize it.
jacobolus 7y ago
> there is little added value to using geometric algebra compared to other representations This is not right in my experience. When trying to solve geometry problems I find that working out the details in terms of the GA formalism typically saves big piles of work compared to other formalisms (such as matrices, differential forms, trigonometry, analytic geometry, ...). The biggest features are (a) access to a concept of “multivectors”, and (b) access to a concept of vector division. Mathematicians use these concepts all the time in the planar setting by pretending that points and vectors are just funny kinds of complex numbers, and eliding the semantic differences between these different types of objects. Similarly for the use of quaternions in 3D (though this is less common nowadays than complex numbers). But multivectors and vector division are very powerful tools in more general settings. What sometimes ends up happening is I try to solve a problem using my (more extensively trained) understanding from standard undergraduate math courses and other standard textbooks, flail around with a bunch of horrible fiddly algebra for a while, then decide to redo my work in terms of GA and end up with like 4–5 lines of simple manipulation replacing a page of messy work. And I am by no means an expert. There are many extremely convenient (multi)vector identities which I have not properly learned and end up laboriously working out for myself while solving geometric problems. If I had spent more time doing guided exercises I am sure I could be still much more efficient.
- chombier 7y ago
  I completely agree: for pen/paper computations it is probably hard to beat. I was more referring to computer applications, where once you've coded data structures for point/lines/planes/etc you don't really care how they're implemented.

DreamScatter 7y ago

Hi, I'm the developer of Grassmann.jl, Actually, it has a lot more value to provide than what is taught in school. It never caught on because it was historically difficult to understand without computer algebra to help build intuition.. so the only people who understood it are the ones bold to explore mathematics deeply enough to arrive at this system and utilize it.

jacobolus 7y ago

> there is little added value to using geometric algebra compared to other representations This is not right in my experience. When trying to solve geometry problems I find that working out the details in terms of the GA formalism typically saves big piles of work compared to other formalisms (such as matrices, differential forms, trigonometry, analytic geometry, ...). The biggest features are (a) access to a concept of “multivectors”, and (b) access to a concept of vector division. Mathematicians use these concepts all the time in the planar setting by pretending that points and vectors are just funny kinds of complex numbers, and eliding the semantic differences between these different types of objects. Similarly for the use of quaternions in 3D (though this is less common nowadays than complex numbers). But multivectors and vector division are very powerful tools in more general settings. What sometimes ends up happening is I try to solve a problem using my (more extensively trained) understanding from standard undergraduate math courses and other standard textbooks, flail around with a bunch of horrible fiddly algebra for a while, then decide to redo my work in terms of GA and end up with like 4–5 lines of simple manipulation replacing a page of messy work. And I am by no means an expert. There are many extremely convenient (multi)vector identities which I have not properly learned and end up laboriously working out for myself while solving geometric problems. If I had spent more time doing guided exercises I am sure I could be still much more efficient.

chombier 7y ago
I completely agree: for pen/paper computations it is probably hard to beat. I was more referring to computer applications, where once you've coded data structures for point/lines/planes/etc you don't really care how they're implemented.