I have a question: how is a "geometric vector" defined? I get the intuitive picture, but mathematically, in my knowledge a (mathematician's) "vector" has always been defined as one that satisfies the vector axioms, while a physicist's vector is also defined similarly, with the additional constraint that it be invariant under a rotation.
math the beautiful uploaded a video named "3 dimensions ..." and you realize the video itself is 3 minutes long. and when you look closely it is 3 minutes 14 seconds. beautiful math indeed.
It would seem to me that basic geometry is possible in 4d. Certainly the 6 regular convex 4-topes can be described and analyzed in the same way as the Platonic solids. So I am getting from this that apparently vectors do not work to describe geometry in 4d, but this doesn't exactly mean we can't describe the geometry of 4d.
I don't understand why you say the geometric vector concept is limited to three dimensions. Do you say that just because you feel that you are unable to visualize it? Geometry still exists in higher dimensions. If you learn to think in 4D space, Euclidean geometry can be easily extended to 4D. For instance planes can now intersect at a point, a line can be skew to a plane, and 3D hyperplanes can intersect in a 2D plane. Maybe I'm missing something.
I understand that since vectors are a sort of line, that as dimensions increase, their spatial influence goes down (what is a point to a cube?). But does that necessarily mean directed segments have no role to play in 4D and beyond? If you just mean from a practical standpoint I 100% agree but as an impossibility to even have them exist in higher dimensions is not obvious to me
Doesn't this conceptualization of "geometric" rely on knowing the limitations of the entity that employs the term? Using this, a 2D entity would be forced to claim that 3D cannot be geometric.
I have a question: how is a "geometric vector" defined? I get the intuitive picture, but mathematically, in my knowledge a (mathematician's) "vector" has always been defined as one that satisfies the vector axioms, while a physicist's vector is also defined similarly, with the additional constraint that it be invariant under a rotation.
This is the exact right question! Here's the answer: czcams.com/video/N32KI6qoeRA/video.htmlsi=_i08XayIrc_kF0DD
math the beautiful uploaded a video named "3 dimensions ..." and you realize the video itself is 3 minutes long. and when you look closely it is 3 minutes 14 seconds. beautiful math indeed.
I didn't notice any of these things!
It would seem to me that basic geometry is possible in 4d. Certainly the 6 regular convex 4-topes can be described and analyzed in the same way as the Platonic solids. So I am getting from this that apparently vectors do not work to describe geometry in 4d, but this doesn't exactly mean we can't describe the geometry of 4d.
I don't understand why you say the geometric vector concept is limited to three dimensions. Do you say that just because you feel that you are unable to visualize it? Geometry still exists in higher dimensions. If you learn to think in 4D space, Euclidean geometry can be easily extended to 4D. For instance planes can now intersect at a point, a line can be skew to a plane, and 3D hyperplanes can intersect in a 2D plane. Maybe I'm missing something.
I understand that since vectors are a sort of line, that as dimensions increase, their spatial influence goes down (what is a point to a cube?). But does that necessarily mean directed segments have no role to play in 4D and beyond? If you just mean from a practical standpoint I 100% agree but as an impossibility to even have them exist in higher dimensions is not obvious to me
Agreed - he doesn't explain what the limitation is other than his inability to visualize it.
Great lectures on Generalized Geometry! Awesome!
Awesome
Glad you enjoyed it!
Doesn't this conceptualization of "geometric" rely on knowing the limitations of the entity that employs the term? Using this, a 2D entity would be forced to claim that 3D cannot be geometric.
I don't know
Thorold Gosset banished a second time to the intellectual netherworld, first by Burnside, now by Grinfeld ... tsk, tsk
I did not such thing
@@MathTheBeautiful You seem to say that 4-dimensional geometric objects do not exist, but Gosset and many others show otherwise.
that is why schools should teach philosophy