No, the original paper is saying something like: given the sequence say 12 -> 3 -> 7, there is a sequence 12 -> 3x -> 7x -> 7x + 3x -> 7x + 3x + 4x where 7x+3x+4x = 12. Obviously here x=6/7, but doing something like this with arbitrary rotations in 3D isn’t so simple.
If you’re referring to doing moves on a Rubik’s cube, no that’s irrelevant to the theorem. If you’re referring to applying a sequence of rotations to a whole solid cube, then yes.
The example given in the OP is incorrect. /u/gameryamen is implying something like: given a sequence of rotations W there is a scale factor a>0 such that W(a)W(a)W = 1, with W(a) the same sequence of rotations as W but with all rotation angles scaled by a.
This is not what the paper does. The paper finds an a such that W(a)W(a) = 1.
His whole post seems bunk, honestly. Example:
Having one more shot in your follow up acts as kind of a hinge, opening up more possibilities.
This seems completely irrelevant. It seems that maybe they’re referring to the probabilistic argument the authors give to justify why their theorem should be true (before giving a complete proof), but this argument involves repeating the same exact rotation two times, not two different rotations in sequence.
They use continuity and the fact that F(0) = 1 to conclude that, since it also takes on a negative value (what they put effort into arguing), it must also attain 0. So no, they don’t find it directly, but it is technically possible to express F explicitly and then numerically find a root.
Should be Steam/logs, with Steam your Steam folder where e.g. steamapps is.
I’ve had an issue with controllers before because Steam’s udev rules straight up give the wrong permissions to the device files they create. Check Steam logs, there’s one specifically for controllers if I remember correctly (or maybe it was a generic “console” log) and it should be very clear if this is the issue because there will be a permission error recorded.
Funnily enough, yaoi actually is an acronym, I think referring to stereotypical yaoi stories:
Yama nashi, Ochi nashi, Imi nashi
“No climax, no resolution, no meaning”
Yes, “quantum teleportation” just means transfering a quantum state from one location to another without knowing what that state is, in the same sense that you can copy a file from one place to another without knowing its contents. So it’s just a particular kind of information being transfered, not anything about how quickly it’s transfered. There is a lot of awful terminology in quantum mechanics.
i,j,k for basis vectors is an interesting one. Historically, Hamilton invented his quaternions before any notion of “vector” existed (as an algebraic object; I believe the geometric notion is older). (So, what, did people just write out everything componentwise? Yes, yes they did. For example, that’s how things like Maxwell’s equations were originally presented.) The reason he chose i,j,k for the unit quaternions is because i was already in use for complex numbers, and i was in use for complex numbers probably to stand for “imaginary”.
The notion of “vector” was invented specifically as a “de-algebraicization” of quaternions. People did not like working with quaternions because they thought it was weird, particurlary because they required 4 numbers but space only required 3, so the likes of Gibbs and Heaviside gutted them and gave us modern 3D vector calculus. The reason we work with the dot product and cross product in 3D is specifically because, given pure imaginary quaternions v, w the product (vw) has real part (-v.w) and imaginary part (v x w).
Also, your last paragraph is somewhat misinformed. Sequences of Greek letters are used all the time, and Hebrew letters are also used in set theory to denote cardinalities (though I can only think of aleph and beth, no sequences of such letters). It is also well-known that some people like to use Japanese よ (yo) for the Yoneda embedding in category theory. But beyond Latin and Greek, there is quite a dearth.
This sentence is shit, but I want to point out at least that “magic” is jargon for “well-behaved” or “particularly well-suited (for our application)”. It’s essentially “magic” like in the phrase “it works like magic”.
Have you read the authors current series, Dai Dark? I’ve only read a couple of chapters so I can’t say too much about how it compares to Dorohedoro, but…
You made me realize this is actually pretty common in math, e.g. “Let x, y be real numbers” instead of “Let x and y be real numbers”. I imagine this comes from the infuence of notation like “Let x, y ∈ ℝ”.
Read the actual actual article: https://arxiv.org/abs/2502.14367
The authors prove that given any sequence of rotations W “almost always” there is a sequence of rotations W’ formed by scaling every rotation angle in W by the same positive factor, such that the sequence W’W’ is the identity (that is, apply all the rotations in W’ and then apply all of them again).
The issue isn’t the result, it’s popsci writing.