Strange

The sign of what? And if you think your question in the OP can be answered by quaternions (not octonions?) then what is the relationship of that to your equation? Why aren't you pursuing the answer using quaternions? That is interesting (for people who find that sort of thing interesting!) as is almost anything by Baez, but it just seems like another tangent from your equation. Can you explain how your equation realties to octonions and/or quaternions?

Has this actually been proved? Or is it just a guess? I don't see why you think there is any ad hominem involved. I am just curious why you brought this up. Presumably you think there is some connection between your idea and octonions? Can you expand on that? (I am actually going to make an exception to my usual practice and take a look at these videos; I have read about Furey's work and it is pretty interesting.)

Why do you think this is relevant?

How does it do that? The direction of up, for a particular location, does depend on the movement of the Earth. But eve if you ignore movement and time, there is still no consistent definition of "up". For fairly obvious reasons. You can call it what you like but you need to use it to support your claims.

Really? What equipment would that be? What could we build to detect the presence of gods or a soul or ...?

I just noticed that you described the circle, earlier, as being "inside" the sphere(*), whereas I was thinking of it being drawn on the surface of the sphere. (sorry for the rather vague description; I'm not sure how to formalise it to make it precise!) Perhaps epicentre would be a better term? (*) Or at least, I thought you did...

And that kind of makes my point! But if you heard them say it, then it isn't really the same situation, is it?

Plugging those into the equation gives a result of 0°. But I don't know what that means; 0° from what? That is the trouble with specifying directions: it always has to be relative to something. But maybe 0° means "up"? Isn't that obvious? (It is about the only thing that is!) If anyone else wants to test other values: https://www.wolframalpha.com/input/?i=cot^-1(cos(v)+tan(sin^-1(sin(L%2F2)%2Fsin(v)))),+v%3Dpi%2F4,+L%3Dpi%2F2 (replaced lambda with L for simplicity) Assuming I have got that right, this gives imaginary results for some inputs. I'm not sure what that is supposed to mean, in terms of direction.

While that seems a simple and "obvious" analysis, I can't imagine anyone in that situation saying "stupid man". The phrase would normally be "stupid <generic insult>" (eg. stupid idiot, stupid bugger, stupid fool). The only times I think the phrase would use a term that specifically identifies the person is when that identification already has negative or demeaning connotations (to the speaker); eg "wop", "fag", "woman", etc. "Stupid boy, Pike!"

The point I was making (about "stupid woman", the comic and many other examples) is that the use of language differently for men and women is so widespread that it is often not seen. And yet it can have a corrosive effect. So a specific use of "stupid woman" addressed to a woman who has just aid something stupid (which is entirely plausible as she was talking about Brexit and rarely has anything intelligent to say on the subject) may not matter too much. But it is symptomatic of a wider problem.

That thing that flew over your head just now? That was the point. I think you missed it.

I was trying to draw an analogy between the structure of the two sentences.

I think it is intimately related. The problem with "stupid woman" (if there is one, he adds, hedging his bets!) is just one example of the way gendered language is used more generally.

Yep, nothing wrong with that. I am often wrong. Being wrong is an opportunity to learn.

The problem may be that gendered terms often have different meanings, even when used in similar contexts. Saying "stupid man" sounds somewhat unnatural (to me). I think "stupid idiot" or similar is more likely to be used if talking about a man (perhaps because "male" is assumed in so much communication). And, as he was talking only to himself, the use of "woman" can't have been necessary to disambiguate who he was talking about so it must have been intended, as hypervalent says, to emphasise the insult. Calling a man "stupid idiot" for example says that he is stupid and, worse still, and idiot. So what does that tell us about "stupid woman"? You only have to look at studies of the words used to describe men and women in business to realise that there is inbuilt prejudice and stereotyping in the language we use. For example, "he's strong" but "she's bossy". "He is ambitious" but "she is pushy" and so on.

So you have no support for any of your claims. You are not willing to retract or apologise for any of your fundamental errors. You have no proof that time dilation due to relative velocity is the same as gravitational time dilation. In other words, it is all guesswork. And wrong. Sounds like a reasonable place to wind it up.

OK. That's an important detail that was missing! Go on then. But your three planes are not (in general) orthogonal. In the case that they are orthogonal, then the system appears identical to traditional Euclidean coordinates.

Don' worry. I reminded myself of the definition of geoid, so I am happy with that now.

What on Earth are you talking about? (excuse the pun) That has nothing to do with time dilation. By "technical" you mean correct, then. Nope. It is the potential not the "strength" of gravity. If you had a planet with twice the mass of Earth and twice the radius, then the time dilation at the surface would be the same as on Earth but g would be half the value. This can't possibly be correct because gravitational time dilation depends on the distance from the centre of mass (you know, potential) while time dilation due to relative velocity is independent of position (within that frame of reference).

So no comment about the fact your claim about time dilation on the moon appears to be incorrect? Not even to show that I have made a mistake?

Thats a good practical example. Although slightly ambiguous. I think you said that alpha, lamb and upsilon are planes. So by defining two of them as "the latitude" do you mean that the upsilon and lambda planes are parallel to one another. But that seems to contradict your requirement that the three directions are different. To clarify, there are an infinite number of planes that could go through the circle centre. So when you say "the latitude" I interpret to mean a slice though the Earth at that latitude. But that would make the two slices (lambda and upsilon) parallel slices. It is not clear how this relates to either definition of "up". I guess (trying to visualise this) that up would be defined by the intersection of lambda and alpha? But I don't see how it defines an "absolute" up. It is also not obvious how this system is useful. I don't understand why you can't tell which direction the person is moving. And if you can't tell, doesn't that cast even more doubt on the usefulness of the system? (Not knowing a value is nothing to do with entanglement.)

I was just assuming they were three arbitrarily (and differently) oriented planes. If there is more to it than that, you will need to explain.

Excellent post, @CharonY. Makes me realise how poorly I had expressed myself!

A good case has been made that it was the development of air conditioning that enabled large parts of the world to get involved in commerce and industry.

I don't see how math can resolve something that is a logical contradiction. But go ahead and show us. I thought you were defining them.