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4D is really interesting and not because it's higher than 3D. There are many mathematical aspects that are unique to it! Check out the second answer to the MATH

by Jun8 2y ago

4D is really interesting and not because it's higher than 3D. There are many mathematical aspects that are unique to it! Check out the second answer to the MATH SE question to see one: https://math.stackexchange.com/questions/3344266/are-there-mathematical-concepts-that-exist-in-dimension-4-but-not-in-dimensio https://math.stackexchange.com/questions/3344266/are-there-m... Also the number of regular polytopes is highest (6) in 4D (https://oeis.org/A060296 https://oeis.org/A060296).

rstuart4133 2y ago
From that stackexchange link: > Another thing that can happen in 4 dimensional space but not 3 is that you can have two planes which only intersect at the origin (and nowhere else.) In 3 dimensions you'd get at least a line in the intersection. I tried to envisage how that could be, but failed miserably.
- rigmarole 2y ago
  Try shifting it down: Another thing that can happen in 3 dimensional space but not 2 is that you can have two infinitely long rectangles which only intersect as a line at the origin (and nowhere else.) In 2 dimensions you'd get at least an overlapping quadrilateral in the intersection. (That doesn't imply I grasp it in 4.. ha)
  - rstuart4133 2y ago
    I don't understand your comparison. It's true the intersection of two overlapping rectangles in a 2D space form another rectangle. But that's always true for all dimensions, isn't it? For example the intersection two overlapping lines in 1D space is another line, the intersection of two overlapping boxes in 3D space is another box. The stackoverflow quote is the intersection of two 2D objects (planes) in a 4D space. If you take it down one dimension, you are looking at the intersection of two 1D objects (lines) in a 3D space. That intersection is always 0D object (a point). It's also a point if the lines intersect in 2D space. I would humbly conjecture two intersecting lines in any space with more than 1 dimensions would be a point. My naive theory would be that the intersection of two N dimension objects in a space with more dimensions than N would always be a N-1 dimension object. Apparently not. The intersection of two infinite planes in 4D can be a point. Not any point mind you, only the origin. They must be very special planes.
- thawaya 2y ago
  Isn’t this just due to (2+2)-3=1? Why is it special to 4 dimensions?
jessekv 2y ago
Also, all knots are trivial in 4D: https://math.stackexchange.com/questions/1426501/why-are-all-knots-trivial-in-4d https://math.stackexchange.com/questions/1426501/why-are-all...
ithinkso 2y ago
The most unique thing about 4-dimensional space imho is that some simple manifolds like e.g R4 have uncountably many differential structures possible, in lower dimensions there is only one, in higher - finitelly many. When push comes to shove I'm not sure we have enough experimental physicists...

by Jun8 2y ago

rstuart4133 2y ago

From that stackexchange link: > Another thing that can happen in 4 dimensional space but not 3 is that you can have two planes which only intersect at the origin (and nowhere else.) In 3 dimensions you'd get at least a line in the intersection. I tried to envisage how that could be, but failed miserably.

rigmarole 2y ago
Try shifting it down: Another thing that can happen in 3 dimensional space but not 2 is that you can have two infinitely long rectangles which only intersect as a line at the origin (and nowhere else.) In 2 dimensions you'd get at least an overlapping quadrilateral in the intersection. (That doesn't imply I grasp it in 4.. ha)
- rstuart4133 2y ago
  I don't understand your comparison. It's true the intersection of two overlapping rectangles in a 2D space form another rectangle. But that's always true for all dimensions, isn't it? For example the intersection two overlapping lines in 1D space is another line, the intersection of two overlapping boxes in 3D space is another box. The stackoverflow quote is the intersection of two 2D objects (planes) in a 4D space. If you take it down one dimension, you are looking at the intersection of two 1D objects (lines) in a 3D space. That intersection is always 0D object (a point). It's also a point if the lines intersect in 2D space. I would humbly conjecture two intersecting lines in any space with more than 1 dimensions would be a point. My naive theory would be that the intersection of two N dimension objects in a space with more dimensions than N would always be a N-1 dimension object. Apparently not. The intersection of two infinite planes in 4D can be a point. Not any point mind you, only the origin. They must be very special planes.
thawaya 2y ago
Isn’t this just due to (2+2)-3=1? Why is it special to 4 dimensions?

jessekv 2y ago

Also, all knots are trivial in 4D: https://math.stackexchange.com/questions/1426501/why-are-all-knots-trivial-in-4d https://math.stackexchange.com/questions/1426501/why-are-all...

ithinkso 2y ago

The most unique thing about 4-dimensional space imho is that some simple manifolds like e.g R4 have uncountably many differential structures possible, in lower dimensions there is only one, in higher - finitelly many. When push comes to shove I'm not sure we have enough experimental physicists...