3 ms·

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 d

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. [1] Quaternion: https://en.wikipedia.org/wiki/Quaternion https://en.wikipedia.org/wiki/Quaternion [2] Polarization spectrogram of bivariate signals: https://ieeexplore.ieee.org/abstract/document/7952905 https://ieeexplore.ieee.org/abstract/document/7952905

_kb 3y ago
> such a well written article for layman understanding As an undereducated idiot who’s interested in mathematics, I couldn’t agree more. Quanta’s articles are a continuous source of approachable knowledge.
mikhailfranco 3y ago
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: https://math.ucr.edu/home/baez/week105.html https://math.ucr.edu/home/baez/week105.html https://math.ucr.edu/home/baez/octonions/conway_smith/conway_smith.pdf https://math.ucr.edu/home/baez/octonions/conway_smith/conway... https://math.ucr.edu/home/baez/octonions/ https://math.ucr.edu/home/baez/octonions/ 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.
empath-nirvana 3y ago
> 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.
- guyomes 3y ago
  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. [1]: https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_map https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_ma...
  - jasomill 3y ago
    The two dimensional case is my favorite: let I be the identity matrix and J the "imaginary unit matrix" 0 -1 1 0, then JJ = -I = exp(Jπ) and exp(Jθ) = cos(θ) -sin(θ) sin(θ) cos(θ).

_kb 3y ago

> such a well written article for layman understanding As an undereducated idiot who’s interested in mathematics, I couldn’t agree more. Quanta’s articles are a continuous source of approachable knowledge.

mikhailfranco 3y ago

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: https://math.ucr.edu/home/baez/week105.html https://math.ucr.edu/home/baez/week105.html https://math.ucr.edu/home/baez/octonions/conway_smith/conway_smith.pdf https://math.ucr.edu/home/baez/octonions/conway_smith/conway... https://math.ucr.edu/home/baez/octonions/ https://math.ucr.edu/home/baez/octonions/ 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.

empath-nirvana 3y ago

> 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.

guyomes 3y ago
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. [1]: https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_map https://en.wikipedia.org/wiki/Rotation_matrix#Exponential_ma...
- jasomill 3y ago
  The two dimensional case is my favorite: let I be the identity matrix and J the "imaginary unit matrix" 0 -1 1 0, then JJ = -I = exp(Jπ) and exp(Jθ) = cos(θ) -sin(θ) sin(θ) cos(θ).