3 ms·

They're really not stable in two dimensions? Hmm [1], and another answer [2] (mentioned by Hawking apparently) but that's less convincing since there could be a

They're really not stable in two dimensions? Hmm [1], and another answer [2] (mentioned by Hawking apparently) but that's less convincing since there could be a 2D universe with intelligent organisms that just have their digestive systems on the outside... 1. https://www.reddit.com/r/askscience/comments/q8fmo/what_would_orbits_look_like_in_a_2d_universe/ https://www.reddit.com/r/askscience/comments/q8fmo/what_woul... 2. http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/2-D_Digestion.gif http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/2-D_Di...

chaxor 5y ago
Right hand rule only works in 3d and 7d
- lordnacho 5y ago
  That is rather fascinating, where can I read about this?
  - octoberfranklin 5y ago
    In the early chapters of the Feynman Lectures on Physics (volume II) there is a really good explanation of why cross product and curl make absolutely no sense in more than three dimensions. The fact that they can be defined unambiguously is a complete accident which occurs only in three dimensions. Not sure if this is what GP was talking about. But the explanation there is excellent. It's in the chapter where curl is introduced for the first time. After you read it, you'll get annoyed whenever you see curls or cross products. Also: https://en.wikipedia.org/wiki/Seven-dimensional_cross_product https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc...
  - ogogmad 5y ago
    Given a cross product in n dimensions, it's possible to construct a normed division algebra in n+1 dimensions. Hurwitz's theorem says that the only normed division algebras are the reals, complexes, quaternions and octonions. So there can be no cross products outside of 0, 1, 3 and 7 dimensions. In computing applications, if you want to find a vector perpendicular to two or more vectors outside of 3 and 7 dimensions, you would use the QR decomposition to do it. If on the other hand, you want to generalise the "directed area" properties of the cross product, you would use Exterior Algebra, or its elaborations like Clifford Algebra or the theory of differential forms.
lisper 5y ago
> They're really not stable in two dimensions? Stable is not really the right word. "Well behaved" is better. In 2D all orbits are stable in the sense that escape velocity is infinite, but a consequence of that is that you don't naturally get nice (nearly) circular orbits that keep planets in habitable zones. That only happens in 3D.

chaxor 5y ago

Right hand rule only works in 3d and 7d

lordnacho 5y ago
That is rather fascinating, where can I read about this?
- octoberfranklin 5y ago
  In the early chapters of the Feynman Lectures on Physics (volume II) there is a really good explanation of why cross product and curl make absolutely no sense in more than three dimensions. The fact that they can be defined unambiguously is a complete accident which occurs only in three dimensions. Not sure if this is what GP was talking about. But the explanation there is excellent. It's in the chapter where curl is introduced for the first time. After you read it, you'll get annoyed whenever you see curls or cross products. Also: https://en.wikipedia.org/wiki/Seven-dimensional_cross_product https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc...
- ogogmad 5y ago
  Given a cross product in n dimensions, it's possible to construct a normed division algebra in n+1 dimensions. Hurwitz's theorem says that the only normed division algebras are the reals, complexes, quaternions and octonions. So there can be no cross products outside of 0, 1, 3 and 7 dimensions. In computing applications, if you want to find a vector perpendicular to two or more vectors outside of 3 and 7 dimensions, you would use the QR decomposition to do it. If on the other hand, you want to generalise the "directed area" properties of the cross product, you would use Exterior Algebra, or its elaborations like Clifford Algebra or the theory of differential forms.

lisper 5y ago

> They're really not stable in two dimensions? Stable is not really the right word. "Well behaved" is better. In 2D all orbits are stable in the sense that escape velocity is infinite, but a consequence of that is that you don't naturally get nice (nearly) circular orbits that keep planets in habitable zones. That only happens in 3D.