3 ms·

My understanding is that all of this (wedge/alternating products, etc.) is considered a part of linear algebra, and has been established that way for quite a lo

by j2kun 11y ago

My understanding is that all of this (wedge/alternating products, etc.) is considered a part of linear algebra, and has been established that way for quite a long time. It's hardly the first idea of translating between geometry and algebra (there's a whole field called "algebraic geometry" going back to antiquity, come on!), and it's not the "universal language of mathematics" by a long shot. > This entire analysis of Clifford Algebra was based on my own foundational assumption that mathematics is not a human invention, but more of a discovery of the essential principles of computation in the brain. And now I'm suspicious. I prefer an objective mathematical treatment and not one with some self-described "radical" agenda.

defen 11y ago
The equations of geometric algebra can be represented using matrices (perhaps with a non-euclidean metric for the inner product depending on application), so in that sense you don't gain the ability to do anything that you couldn't do with regular linear algebra. The "big wins" are that the notation is simpler and easier to reason about, and everything is coordinate-free. As an example, there is a model of 3 dimensional euclidean space that can be represented using a 5-dimensional GA (with 4,1 metric signature). Within this model, you have lines, planes, circles, spheres, etc; but you also have rotations, reflections, translations; and all of these things are just standard elements of the mathematics with no special cases. So you can multiply a translation and a rotation and you get back the composition of those two things. Or you can multiply a sphere by a translation and you get back a sphere that has been translated by the specified amount. Or if you want to smoothly interpolate a rigid body motion (translation+rotation) between two points, you can take the logarithm of that motion. And again, this can all be done symbolically without resorting to coordinates - except at the very end, perhaps, if you want to map it onto an actual coordinate system :) In terms of people applying it to more "advanced" subjects like Minkowski space or differential geometry - I'm not so sure. I haven't spent too much time looking into that.
reagency 11y ago
http://www.av8n.com/physics/clifford-intro.htm http://www.av8n.com/physics/clifford-intro.htm is a much better source. It looks like OP read av8n and that translated it into some spiritual interpretation.
- j2kun 11y ago
  After reading this I am thoroughly convinced that this is standard material for introductory graduate algebra. And they do it in far less space and with far less philosophy.
deleted 11y ago
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by j2kun 11y ago

defen 11y ago

The equations of geometric algebra can be represented using matrices (perhaps with a non-euclidean metric for the inner product depending on application), so in that sense you don't gain the ability to do anything that you couldn't do with regular linear algebra. The "big wins" are that the notation is simpler and easier to reason about, and everything is coordinate-free. As an example, there is a model of 3 dimensional euclidean space that can be represented using a 5-dimensional GA (with 4,1 metric signature). Within this model, you have lines, planes, circles, spheres, etc; but you also have rotations, reflections, translations; and all of these things are just standard elements of the mathematics with no special cases. So you can multiply a translation and a rotation and you get back the composition of those two things. Or you can multiply a sphere by a translation and you get back a sphere that has been translated by the specified amount. Or if you want to smoothly interpolate a rigid body motion (translation+rotation) between two points, you can take the logarithm of that motion. And again, this can all be done symbolically without resorting to coordinates - except at the very end, perhaps, if you want to map it onto an actual coordinate system :) In terms of people applying it to more "advanced" subjects like Minkowski space or differential geometry - I'm not so sure. I haven't spent too much time looking into that.

reagency 11y ago

http://www.av8n.com/physics/clifford-intro.htm http://www.av8n.com/physics/clifford-intro.htm is a much better source. It looks like OP read av8n and that translated it into some spiritual interpretation.

j2kun 11y ago
After reading this I am thoroughly convinced that this is standard material for introductory graduate algebra. And they do it in far less space and with far less philosophy.

deleted 11y ago

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