Connected components of the orthogonal group by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
The orthogonal group has 2 connected components:
- one with determinant +1, which is itself a subgroup known as the special orthogonal group. These are pure rotations without a reflection.
- the other with determinant -1. This is not a subgroup as it does not contain the origin. It represents rotations with a reflection.
It is instructive to visualize how the looks like in :
- you take the first basis vector and move it to any other. You have therefore two angular parameters.
- you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
- at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing
As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2:
A low dimensional example:because you can only do two things: to flip or not to flip the line around zero.
Note that having the determinant plus or minus 1 is not a definition: there are non-orthogonal groups with determinant plus or minus 1. This is just a property. E.g.:has determinant 1, but:so is not orthogonal.
Maiden Voyage by Herbie Hancock (1965) by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Terminal multiplexers are CLI desktop environments by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Predecessor to the synchrotron.
Matrix representation of a symmetric bilinear form by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Training, validation, and test data sets by
Ciro Santilli 35 Updated 2025-04-24 +Created 1970-01-01
Ciro Santilli likes this.
He doesn't have the patience to actually watch full episodes with rare exceptions, rather just watching selected scenes from: www.youtube.com/channel/UCdeIGY2DIjpGf0A2m9GSE3g, but still, it is interesting.
What appeals to Ciro in this series is how almost nothing is solved by violence, almost everything is decided in the bridge, at the "cerebral" level of the command structure. This reminds Ciro of a courtroom of law sometimes.
Maybe there's also a bit of 90's nostalgia involved too though, as this is something that would show on some random cable channels a bored young Ciro would have browsed during weekends, never really watching full episodes.
One crime of many episodes is being completely based on some stupid new scientific concept, which any character to back it up.
Another thing that hurt is that due to their obsession with the senior members of the crew, sometimes those senior members are sent in ridiculously risky operations, which is very unrealistic.
Episodes that Ciro watched fully and didn't regret:
- s02e09 Measure of a man en.wikipedia.org/wiki/The_Measure_of_a_Man_(Star_Trek:_The_Next_Generation) see also physics and the illusion of life
- s04e14 Clues en.wikipedia.org/wiki/Clues_(Star_Trek:_The_Next_Generation)
- s04e15 First contact en.wikipedia.org/wiki/First_Contact_(Star_Trek:_The_Next_Generation). Although the premise that the aliens look so much like humans, and worse, that Decker could speak their language to the point of passing as one of their race is preposterous, the idea of inversion of first contact is just too cute.
- s07e15 The Lower Decks en.wikipedia.org/wiki/Lower_Decks_(Star_Trek:_The_Next_Generation), sliding scale of idealism vs. cynicism near cynincism, yes please
Semi worth it:
- s03e11 The Hunted memory-alpha.fandom.com/wiki/The_Hunted_(episode) if it weren't for the ending, maybe this would have been decent
Not worth it:
- Cause and effect
TODO
- s06e11 Chain of command
Group theory, abstraction, and the 196,883-dimensional monster by 3Blue1Brown (2020)
Source. Too basic, starts motivating groups themselves, therefore does not give anything new or rare.We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis:then we derive and evaluate at 0: therefore represents the infinitesimal rotation.
Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator :
Repeating the same process for the other directions gives:We have now found 3 linearly independent elements of the Lie algebra, and since has dimension 3, we are done.
Answer: it can give some qualitative intuition on what is larger/smaller happens before/after based only on arguably more intuitive geometric considerations, without requiring you to do any calculations, see e.g. Figure "Spacetime diagram illustrating how faster-than-light travel implies time travel".
There are unlisted articles, also show them or only show them.