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While I get what you're saying, and clearly there is "more stuff", I think we shouldn't discount how progress can make things simpler on this front. Proofs we l

by rtpg 4y ago

While I get what you're saying, and clearly there is "more stuff", I think we shouldn't discount how progress can make things simpler on this front. Proofs we learn are usually much more refined than the original versions. Notational improvements and more interesting abstractions mean we can cover more ground more quickly. Another angle here, I remember seeing a study where some children were just... taught algebra. Like given high school algebra classes in 3rd grade, and kids were able to absorb all that abstract reasoning "just fine" (according to the study). Of course there's only so much abstraction that can be done, but I think we shouldn't assume we are at the end of history on much of anything (except for parsing algorithms)

rnvannatta 4y ago
To add to your comment, quaternions predate rotational matrix operators by a considerable amount (1843 vs not exactly clear, ~1900 with Peano or ~1920 with Weyl), despite quaternions being much more challenging to manipulate. There are definitely simpler ways to view the same things. There was a cottage industry of exotic hypercomplex numbers that disappeared when linear algebra matured to eclipse them. In fact, Maxwell's Equations were originally derived with quaternions.
- BlueTemplar 4y ago
  I am somewhat unconvinced that quaternions are more challenging, or at least a worse way to think about the issue : https://eater.net/quaternions https://eater.net/quaternions And speaking of Maxwell's Equation"s" : http://www.av8n.com/physics/maxwell-ga.htm#sec-preview http://www.av8n.com/physics/maxwell-ga.htm#sec-preview
  - rnvannatta 4y ago
    Well, a rotation matrix doesn't require doing 2 half rotations, and doesn't require reaching into the 4th dimension in such a way that it gets perfectly cancelled out. It doesn't require abstract analogies about cubes with strings glued to them or people holding coffee cups. With some familiarity with linear algebra, it's easy to derive the formula for constructing a rotation matrix. You just have to think about what the operation does to the axes. The derivation for quaternion rotation is far more abstract, by virtue of the operation we actually care about involving a sandwich of multiplications with unclear 4 dimensional meaning. There's no hyperspheres with a rotation matrix. Augmenting your space to handle not just rotations & scaling, but translations is easy for matrices, just requires a homogeneous coordinate and you get 4x4 matrices with intuitive columns. Augmenting quaternions to handle translations requires the 8 dimensional dual-quaternions. I definitely like geometric algebra, it's a very nice continuation of topics in linear algebra and makes it clear why things like normals behave differently from standard vectors. But I don't use it every day. I use standard linear algebra every day.

by rtpg 4y ago

rnvannatta 4y ago

To add to your comment, quaternions predate rotational matrix operators by a considerable amount (1843 vs not exactly clear, ~1900 with Peano or ~1920 with Weyl), despite quaternions being much more challenging to manipulate. There are definitely simpler ways to view the same things. There was a cottage industry of exotic hypercomplex numbers that disappeared when linear algebra matured to eclipse them. In fact, Maxwell's Equations were originally derived with quaternions.

BlueTemplar 4y ago
I am somewhat unconvinced that quaternions are more challenging, or at least a worse way to think about the issue : https://eater.net/quaternions https://eater.net/quaternions And speaking of Maxwell's Equation"s" : http://www.av8n.com/physics/maxwell-ga.htm#sec-preview http://www.av8n.com/physics/maxwell-ga.htm#sec-preview
- rnvannatta 4y ago
  Well, a rotation matrix doesn't require doing 2 half rotations, and doesn't require reaching into the 4th dimension in such a way that it gets perfectly cancelled out. It doesn't require abstract analogies about cubes with strings glued to them or people holding coffee cups. With some familiarity with linear algebra, it's easy to derive the formula for constructing a rotation matrix. You just have to think about what the operation does to the axes. The derivation for quaternion rotation is far more abstract, by virtue of the operation we actually care about involving a sandwich of multiplications with unclear 4 dimensional meaning. There's no hyperspheres with a rotation matrix. Augmenting your space to handle not just rotations & scaling, but translations is easy for matrices, just requires a homogeneous coordinate and you get 4x4 matrices with intuitive columns. Augmenting quaternions to handle translations requires the 8 dimensional dual-quaternions. I definitely like geometric algebra, it's a very nice continuation of topics in linear algebra and makes it clear why things like normals behave differently from standard vectors. But I don't use it every day. I use standard linear algebra every day.