Kudos to the Quanta Magazine writer and staff, such a well written article for layman understanding.
This make me wonder on the connection between 2, 3 and 4 dimensions, and Hamilton found out the hard that you need 4 numbering system or quaternion, in order to properly represent 3 dimensions [1].
This article hinting a direct connection between two and four dimensions, and the interplays between the two but not 3 dimensions and it seems to me that the 3 dimensions exist only as a curious transition.
Recently someone come up with the equivalent of complex number analytic signal (an indispensable tool in modern engineering) in the quartenion space and called it quaternion embedding, or probably the better name should be quaternion analytic signal [2].
It is great to see the synergy between Topology and Group Theory as the article mentioned in solving some of the the former's list of problems and their new found solutions. It looks like the topology group is rapidly cleaning their house as the article aptly put it, and at this rate (after 30 years of winter hiatus) we will probably see the results spilled over to the applied math fields for examples physics and engineering applications.
> This make me wonder on the connection between 2, 3 and 4 dimensions, and Hamilton found out the hard that you need 4 numbering system or quaternion, in order to properly represent 3 dimensions [1].
Sort of. Quarternions have some nice properties as representations of _rotations_ in 3d space. 3d space works just fine with 3 dimensions.
For rotations in 3D space, another nice representation is based on antisymmetric matrices [1]. All rotations can be represented as exp(A), where A has the form:
0 -z y
z 0 -x
-y x 0
Moreover, this generalizes to other dimensions. This allows to see that in 2D, rotations have one degree of freedom, in 3D they have three, and in 4D they have six.
Apparently (I am no expert) there are sequences of patterns in different dimensions based on: even-odd, powers of two, prime numbers and surely others. In some cases ±1 to get some effect.
There is the uniqueness of the 3D cross-product, even though various generalization do occur in higher dimensions.
One of the most obvious is the 1,2,4,8 sequence for normed division algebras: real, complex, quaternion and octonion. At each step you lose one mathematical property (commutative, associative, etc.). There cannot be any more, but there is a certain repetition, with factoring (doubling) known as Bott Periodicity:
String Theories occur in 10, 24, 8+8, 32, ... dimensions, based on various symmetries and dualities. And some solutions need the extra increment, say 10+1, for the actual theory in spacetime.
There are various theorems that seem to be trivial in 0D, 1D and 2D, then difficult in 3D, but perhaps eventually solved. In parallel, the upper limit existence proofs descend from infinite dimensions down to finite, then some impossibly (and apparently random) large finite number. Then a reasonable number less than 1,000, brought down to 100, and perhaps to less than 8. After a long struggle, and a Ph.D. for each number 7..6..5, it comes down to just 4, and 4D remains unsolved.
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...
> 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.
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)
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.
I wish this article talked more about how this plays out in the 3-dimensional analogue (knot theory) for comparison. I have very little picture of how I would expect this to work, so it would be helpful to start with a case I can picture, and then say how this is similar and how it's different.
Well, it's discussing having some larger ambient manifold and then embedding other manifolds in it of codimension 2 (and then also taking the complement and studying the fundamental group of said complement). If the ambient manifold has dimension 3 instead of 4, that's knot theory. Perhaps there's some reason why that's not the appropriate analogue, but it's certainly the naive analogue; if there's some reason it's not actually the right analogue, you're going to have to explain it.
And, again, the point isn't just to have an analogue that one can visualize, but a simpler case where the results are known so that one can compare the results and see how the dimension affects things.
I was trying to read the paper and it talked about putting shapes together, and I was sad to see that there were no graphical representations of what it was talking about. This is my only hope is that someone would build an accurate graphical representation of these 3D or even 4D shapes (using movement lines I imagine!).
This make me wonder on the connection between 2, 3 and 4 dimensions, and Hamilton found out the hard that you need 4 numbering system or quaternion, in order to properly represent 3 dimensions [1].
This article hinting a direct connection between two and four dimensions, and the interplays between the two but not 3 dimensions and it seems to me that the 3 dimensions exist only as a curious transition.
Recently someone come up with the equivalent of complex number analytic signal (an indispensable tool in modern engineering) in the quartenion space and called it quaternion embedding, or probably the better name should be quaternion analytic signal [2].
It is great to see the synergy between Topology and Group Theory as the article mentioned in solving some of the the former's list of problems and their new found solutions. It looks like the topology group is rapidly cleaning their house as the article aptly put it, and at this rate (after 30 years of winter hiatus) we will probably see the results spilled over to the applied math fields for examples physics and engineering applications.
[1] Quaternion:
https://en.wikipedia.org/wiki/Quaternion
[2] Polarization spectrogram of bivariate signals:
https://ieeexplore.ieee.org/abstract/document/7952905
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