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They were motivating examples of both the Algebraic Geometry and Differential Geometry courses that I took. They were used in multiple homework/exam problems in

They were motivating examples of both the Algebraic Geometry and Differential Geometry courses that I took. They were used in multiple homework/exam problems in those courses. And I didn't go to some fancy private school, this was senior level math classes at a flagship public state school in the south.

jacobolus 6y ago
Algebraic geometry / differential geometry are both treated as “sophisticated”, “advanced” subjects. They typically require high-level preparation (several years of pure math courses as prerequisites) and are taught in an extremely abstract way, often are deferred to grad school, and are seldom if ever taught to non-math-major undergraduates. But geometric algebra, a.k.a. real Clifford algebra (“geometric algebra” was Clifford’s own name for it), can easily be taught to high school students in a very simple concrete way, and then used throughout science/engineering/computing courses aimed at non-math-majors. It can unify and simplify the wide variety of other mathematical formalisms that are used throughout undergraduate technical curricula. It is straight-forward to implement in a computer, and it makes calculations much easier to write down.
- WoahNoun 6y ago
  The way GA is promoted by the non-math world does not unify mathematical formalisms. I've yet to see a GA implementation that does more than reimplement rotations/quaternions.
  - adamnemecek 6y ago
    How many differential geometry implementations do you know?
  - jacobolus 6y ago
    > I've yet to see [...] Most implementations I have seen do much more than quaternions. How hard did you look? Here’s an example you can play with in a browser, https://enkimute.github.io/ganja.js/ https://enkimute.github.io/ganja.js/ But what is implemented in computers is not 1:1 with the algebraic tools that can be profitably employed to solve geometry problems symbolically. GA is a rich language full of useful tools which can describe/model many kinds of problems. Coming from a math background, let me recommend you start with https://arxiv.org/abs/1205.5935 https://arxiv.org/abs/1205.5935
adamnemecek 6y ago
Ok but how much time did you spend on them?

jacobolus 6y ago

Algebraic geometry / differential geometry are both treated as “sophisticated”, “advanced” subjects. They typically require high-level preparation (several years of pure math courses as prerequisites) and are taught in an extremely abstract way, often are deferred to grad school, and are seldom if ever taught to non-math-major undergraduates. But geometric algebra, a.k.a. real Clifford algebra (“geometric algebra” was Clifford’s own name for it), can easily be taught to high school students in a very simple concrete way, and then used throughout science/engineering/computing courses aimed at non-math-majors. It can unify and simplify the wide variety of other mathematical formalisms that are used throughout undergraduate technical curricula. It is straight-forward to implement in a computer, and it makes calculations much easier to write down.

WoahNoun 6y ago
The way GA is promoted by the non-math world does not unify mathematical formalisms. I've yet to see a GA implementation that does more than reimplement rotations/quaternions.
- adamnemecek 6y ago
  How many differential geometry implementations do you know?
- jacobolus 6y ago
  > I've yet to see [...] Most implementations I have seen do much more than quaternions. How hard did you look? Here’s an example you can play with in a browser, https://enkimute.github.io/ganja.js/ https://enkimute.github.io/ganja.js/ But what is implemented in computers is not 1:1 with the algebraic tools that can be profitably employed to solve geometry problems symbolically. GA is a rich language full of useful tools which can describe/model many kinds of problems. Coming from a math background, let me recommend you start with https://arxiv.org/abs/1205.5935 https://arxiv.org/abs/1205.5935

adamnemecek 6y ago

Ok but how much time did you spend on them?