3 ms·

> I'd like to see a rewrite of EM in Geometric Algebra focused not on pages and pages of abstract equations, but practical numerical applications. What do you

> I'd like to see a rewrite of EM in Geometric Algebra focused not on pages and pages of abstract equations, but practical numerical applications. What do you mean by practical? Do you mean something you can use to calculate the electric and the magnetic fields as if we didn't understand they're the same thing? That exists, they're described by their own fields of science which used to be called "Electricity" and "Magnetism" respectively. Of course, what's the point when you have Maxwell's equations?

jiggawatts 6y ago
Geometric Algebra is merely a variant of linear algebra that is purpose-designed for describing geometric concepts such as signed areas. For comparison, vector algebra is mostly a direct extension of linear algebra. Linear algebra was originally largely designed for solving large systems of simultaneous equations. Its native objects, vectors and matrices, are not ideal for geometric concepts. Trying to pretend everything is a vector ends up causing issues with the equations that are hideous to solve. Some of these issues are not obvious when manipulating vector algebra purely symbolically. They manifest when it comes time to create a numerical simulation. You suddenly realise that there are practical issues such as high numerical error, conditional statements all over the place, and corner-cases like gimbal lock. For comparison, something like simulating the full electromagnetic interaction in a 4D spacetime ought to be straightforward with a geometric algebra! It's just that I haven't seen anyone try. Every EM textbook is the same. They go through the same equations. The same algebra. The same symbolic solutions. With the same corner cases. The same numerical stability issues. The same result: "Oh we can talk about it, but rendering it is too hard!" I want an "acid test" of EM, a... wavepool to play with. Not a toy model. The full thing, capable of simulating even obscure but experimentally-verified results such such as the Aharonov–Bohm effect. See: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect That is, I want something where instead of plugging numbers into a simplified equation to derive such effects, I want to be able to set up a bunch of charges, press "simulate", and watch the test particle get deflected in the area with zero field.

jiggawatts 6y ago

Geometric Algebra is merely a variant of linear algebra that is purpose-designed for describing geometric concepts such as signed areas. For comparison, vector algebra is mostly a direct extension of linear algebra. Linear algebra was originally largely designed for solving large systems of simultaneous equations. Its native objects, vectors and matrices, are not ideal for geometric concepts. Trying to pretend everything is a vector ends up causing issues with the equations that are hideous to solve. Some of these issues are not obvious when manipulating vector algebra purely symbolically. They manifest when it comes time to create a numerical simulation. You suddenly realise that there are practical issues such as high numerical error, conditional statements all over the place, and corner-cases like gimbal lock. For comparison, something like simulating the full electromagnetic interaction in a 4D spacetime ought to be straightforward with a geometric algebra! It's just that I haven't seen anyone try. Every EM textbook is the same. They go through the same equations. The same algebra. The same symbolic solutions. With the same corner cases. The same numerical stability issues. The same result: "Oh we can talk about it, but rendering it is too hard!" I want an "acid test" of EM, a... wavepool to play with. Not a toy model. The full thing, capable of simulating even obscure but experimentally-verified results such such as the Aharonov–Bohm effect. See: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect That is, I want something where instead of plugging numbers into a simplified equation to derive such effects, I want to be able to set up a bunch of charges, press "simulate", and watch the test particle get deflected in the area with zero field.