loppy

It does indeed scale with r^(n-1), but your factors are not close at all. It involves the gamma function, which in this case can be expanded into various factorials and also a factor of sqrt(pi) when n is odd. According to Wikipedia, the expression is 2pi^(n/2)r^(n-1)/Gamma(n/2).

Planes can be thought of as spheres of infinite radius, and have infinite surface area. This point of view is very natural in conformal geometry.

I understand that for many people loneliness isn't a blessing

I think there is a misunderstanding about the word "loneliness" here (which I used to share). I was skimming an academic article about this some time ago, and was very surprised when they gave a definition of loneliness that is apparently standard in academia: something like "your emotional needs are not met by the people around you/in your life". It has nothing at all to do with whether or not or how many people are present in your life, and whether or not you like the company of other people. By this definition, you are not lonely, and furthermore loneliness is definitionally bad and cannot be a "blessing".

Maybe I just personally never had a proper definition of "loneliness" in my head to begin with, but I think this rather technical usage of the word "lonely" is an extreme disconnect between researchers/academic writing and how the general populace interprets the word.

日本語は得意じゃないけど、ちょっと分かるから頑張ってみようと思います。

なるほど、DeepLでしたね。いまさっきRemedultさんの他のコメントを読んでいるうちに、これはAIかなと思っていました。

もう一つのアニメをすすめたいけど、「京騒戯画」と呼ばれています。どれよりも僕の気に入ったアニメなんで何回みても、色がめっちゃすぶらしい〜！と思えざるをえないんです。全くフリーレンと違っているけど、ほんの少しでもお気になったらぜひ見てください。

None of these are the same genre, but for "beautiful depictions of light and color" I don't think KyoAni can be beat, first one of theirs that comes to mind is Hibike Euphonium. Violet Evergarden would probably be the closest to Frieren.

Some off the top of my head other than KyoAni: Shoshimin Series, Yofukashi no Uta. These are decidedly darker color palettes though.

This one is "muddier", but if you like Frieren, maybe Mushishi is worth a shot.

Below is just one possible aspect of this, the other answers you've received are also valid. Writing systems are complicated!

Your making the mistake that writing systems are supposed to represent speech sounds. They do not (or at least they don't have to). As an example, in my accent (midwestern American English) there are at least three different sounds I make for "t":

• "touch": (aspirated) voiceless alveolar plosive
• "matter": voiced alveolar tap
• "mat": glottal stop

These are the technical names linguists use for these sounds; you can find them on Wikipedia if you want to know more. English speakers can agree though that they are all "the same thing"; the technical terminology is that they are all allophones of the same phoneme. Different accents will have different allophones, for example some English accents may pronounce this "t" phoneme in "matter" and "mat" the same way as my "touch". If you think this is splitting hairs, that's just false; the way languages divide sounds into phonemes varies greatly. For example, Japanese speakers consider my "touch" "t" and my "matter" "t" to be two completely different sounds, i.e. two different phonemes which are not interchangeable.

(Very) roughly speaking, writing systems tend to map better onto phonemes than onto actual sounds. Part of your frustration with Vietnamese writing could partly be from this: Vietnamese possibly has some sounds as allophones which in English are not allophones and belong to different phonemes. In other words, to a Vietnamese speaker they are the same sound. On the flipside, it could be that Vietnamese uses different letters for different phonemes, but those sounds are part of the same phoneme in English and you perceive them as "the same sound" when they are in fact distinct.

One more example is the Cot-Caught merger present in some varieties of English. In my accent, the vowels in these words are two separate sounds for two separate phonemes. In English accents which have the merger, they have become the same phoneme and in fact are pronounced identically, with the exact sound depending on the particular variety of English.

This shows one way you can end up with different spellings for identically-pronounced words.

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.

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.

Preprint: https://arxiv.org/abs/2502.14367