🧵 Is there any geometric property that is uniquely 3 dimensional ?
Anonymous at Thu, 10 Oct 2024 14:55:05 UTC No. 16419227
I just want to know if our realm is special in any way ?
(Also I know we actually live i a four dimensional space-time world, but I was just wondering, purely look at the space part, is there anything special to find ?)
Anonymous at Thu, 10 Oct 2024 15:24:16 UTC No. 16419280
the Cross Product
Inverse Square laws.
Anonymous at Thu, 10 Oct 2024 15:44:57 UTC No. 16419324
>>16419227
The is an exceptional Lie algebra isomorphism [math]\mathfrak{sl}_2 \cong \mathfrak{so}_3 \cong \mathfrak{sp}_2 [/math]. It roughly means that three-dimensional rotations in real space, two-dimensional rotations in complex space and two-dimensional rotations in symplectic space all act the same way.
>>16419280
The cross product can be represented in any arbitrary dimension using the Levi-Civita symbol. The three-dimensional cross-product is only unique in that it maps two vectors to a (pseudo)vector. In higher dimensions, two vectors map to an antisymmetric tensor. However, say 6 dimensions are just as unique, because they uniquely map two rank-2 antisymmetric tensors to a rank-2 antisymmetric tensor. And so on.
Anonymous at Thu, 10 Oct 2024 20:40:29 UTC No. 16419759
>>16419324
What about 14, 30 and 62 dimensions?
Anonymous at Thu, 10 Oct 2024 21:06:06 UTC No. 16419806
>>16419280
Yes inverse square laws apply to light and gravity any physical process that adheres to the geometric argument of continuously increasing radii of spheres. The fact that these processes dominate our experience leads us to insist on 3-dimensional solutions in general.
Of course there are times when this doesn't work so it's not like all of physical reality is perfectly 3-dimensional.
Anonymous at Thu, 10 Oct 2024 23:48:07 UTC No. 16420019
>>16419759
What about them? As a rule of thumb, the higher the dimension, the less interesting it is. You end up with so much structure that you won’t have those accidental equivalences, yet at the same time the structure is so “standardized” that the properties of a 101 dimensional space aren’t expected to be that different from a 100 dimensional one. There are, of course, exceptions as always. See exceptional Lie groups and exceptional finite groups. But they aren’t found by randomly choosing a dimension; they’re found through rigorous systematisation and classification of the whole mathematical framework they’re in.
Anonymous at Thu, 10 Oct 2024 23:49:27 UTC No. 16420020
>>16419227
4-dimensions is actually extremely special.
It is the only dimension where a manifold can have an infinite amount of non-equivalent smooth structures.
Anonymous at Fri, 11 Oct 2024 00:48:34 UTC No. 16420073
>>16419227
Nothing ever happens, therefore nothing changes, meaning there is no 4th dimension.
Anonymous at Fri, 11 Oct 2024 01:41:44 UTC No. 16420129
Anonymous at Fri, 11 Oct 2024 04:29:46 UTC No. 16420520
>>16420073
Anon cracks it kek
>>16419227
Well I'm not even a mathematician and I can think of one
3 dimensions is the minimum number that a point is able to move around at random in without a guarantee of ever returning to its starting location
But idk, that and even the others in this thread arent very convincing. It's always bothered me a bit. Hope we get an afterlife where we dont have to deal with this Saturn prison cube granuloma decordicate reality.
Anonymous at Fri, 11 Oct 2024 04:57:59 UTC No. 16420547
>>16419227
3D Ising model is unsolvable(?) and complicated
2D is solvable
4+D is boring mean field
In other words, 3D is the only real D from the point of view of mathematical statistical mechanics autism.
Anonymous at Fri, 11 Oct 2024 07:05:28 UTC No. 16420634
>>16419324
Thank you
Anonymous at Sun, 13 Oct 2024 02:18:40 UTC No. 16423991
Not really special
Anonymous at Sun, 13 Oct 2024 16:26:06 UTC No. 16428567
>>16419324
>cross product with LC symbol
Then it's the wedge product. but only in three dimensions does it take on the form it does, so it is special.
Anonymous at Sun, 13 Oct 2024 16:27:10 UTC No. 16428573
>>16420019
>the higher the dimension, the less interesting it is
Akshually infinite dimensional spaces are super interesting, proving you RONG.
Anonymous at Sun, 13 Oct 2024 16:29:21 UTC No. 16428588
>all good things come in threes
There you go.
Anonymous at Sun, 13 Oct 2024 16:30:12 UTC No. 16428594
>>16419324
>if you change the formula it's totally the same bro
Anonymous at Sun, 13 Oct 2024 16:36:00 UTC No. 16428619
>>16428594
I didn’t change it. The cross product in three dimensions can be written down using the Levi-Civita symbol. Try it. Makes deriving vector identities a lot easier.
Anonymous at Sun, 13 Oct 2024 17:50:29 UTC No. 16428921
>>16428619
>I didn't change it, I just changed the symbols and notations
Anonymous at Sun, 13 Oct 2024 18:27:05 UTC No. 16429022
>>16419227
It's the only space where the dimension is 3.
Anonymous at Sun, 13 Oct 2024 18:30:42 UTC No. 16429039
>>16428921
How fucking dumb do you have to be to think that the definition is just the notation? If I write 3@8=11, do you really fail to understand what algebraic operation I have in mind?
Anonymous at Sun, 13 Oct 2024 18:41:21 UTC No. 16429069
>>16429039
You are an[sic!] retard. they're saying you've changed the meaning. How fucking dense are you?
Anonymous at Sun, 13 Oct 2024 21:38:11 UTC No. 16429518
PĂłlya proved that a person almost surely would return to the origin in a 2-dimensional random walk, but for 3 dimensions or higher, the probability decreases to roughly 34%