it’s a bit of a “spirit of the law vs letter of the law” kind of thing.
technically speaking, you can’t have a straight line on a sphere. but, a very important property of straight lines is that they serve as the shortest paths between two points. (i.e., the shortest path between A and B is given by the line from A to B.) while it doesn’t make sense to talk about “straight lines” on a sphere, it does make sense to talk about “shortest paths” on a sphere, and that’s the “spirit of the law” approach.
the “shortest paths” are called geodesics, and on the sphere, these correspond to the largest circles that can be drawn on the surface of the sphere. (e.g., the equator is a geodesic.)
i’m not really sure if the line in question is a geodesic, though
You are absolutely correct, but to add on to that even more:
When we talk about space, we usually think about 3D euclidean space. That means that straight lines are the shortest way between two points, parallel lines stay the same distance forever, and a whole bunch of other nice features.
Another way of thinking about objects like the earth is to think of them as 2D spherical manifolds. That means we concern ourself only to the surface of the earth, with no concept of going below the surface or flying up into the sky. In S2 (that’s what you call a 2D spherical manifold), and in spherical geometry in general, parallel straight lines will eventually cross, and further on loop back and form a closed loop. Sounds weird, right? Well, we do it all the time. Look at lines of Longitude, for example.
We call the shortest line connecting two points in curved manifolds geodesics, as you said, and for all intents and purposes, they are straight. Remember, there is no concept of leaving the sphere, these two coordinates is all there is.
What one can do, if one wants to, is embed any manifold into a higher-dimensional euclidean one. Geodesics in the embedded manifold are usually not straight in higher-dimensional euclidean space. Geodesics on a sphere, for example, look like great circles in 3D.
i think it depends on what you mean by “accurately”.
from the perspective of someone living on the sphere, a geodesic looks like a straight line, in the sense that if you walk along a geodesic you’ll always be facing the “same direction”. (e.g., if you walk across the equator you’ll end up where you started, facing the exact same direction.)
but you’re right that from the perspective of euclidean geometry, (i.e. if you’re looking at the earth from a satellite), then it’s not a straight line.
one other thing to note is that you can make the “perspective of someone living on the sphere” thing into a rigorous argument. it’s possible to use some advanced tricks to cook up a definition of something that’s basically like “what someone living on the sphere thinks the derivative is”. and from the perspective of someone on the sphere, the “derivative” of a geodesic is 0. so in this sense, the geodesics do have “constant slope”. but there is a ton of hand waving here since the details are super complicated and messy.
this definition of the “derivative” that i mentioned is something that turns out to be very important in things like the theory of general relativity, so it’s not entirely just an arbitrary construction. the relevant concepts are “affine connection” and “parallel transport”, and they’re discussed a little bit on the wikipedia page for geodesics.
This whole post is a good illustration to how math is much more creative and flexible than we are lead to believe in school.
The whole concept of “manifolds” is basically that you can take something like a globe, and make atlases out of it. You could look at each map of your town and say that it’s wrong since it shouldn’t be flat. Maps are really useful, though, so why not use math on maps, even if they are “wrong”? Traveling 3 km east and 4 km north will put you 5 km from where you started, even if those aren’t straight lines in a 3d sense.
One way to think about a line being “straight” is if it never has a “turn”. If you are walking in a field, and you don’t ever turn, you’d say you walked in a straight line. A ship following this path would never turn, and if you traced it’s path on an atlas, you would be drawing a straight line on map after map.
I think those last 2 paragraphs are due to people approximating math that would otherwise be quite complex to calculate, or making models that are approximations due to widespread available technology. Just because I don’t turn if I cross over Mt Everest, does not mean that is the fastest route by foot.
I’m not saying to not use these approximations.
I really recommend the book “Where Mathematics Comes From,” to really think deeply about what math is to us as an animal. Even other animals can do some rudimentary math, and arguably athletes are doing math innately as they perform their sports. Birds and dolphins do physics and calculus. Sort of. In this view, what we teach as math to each other as humans is essentially a language describing these phenomenon and how they work together. Calling this approximation a “straight line” in this language sense isn’t very accurate and it’s what’s causing the debate.
To clarify, as youve not understood the joke, nor read the comments. As far as I understand it, were you to start sailing at the first point, you never have to turn to arrive at the second. That’s why it’s “straight”. On the 2d plane you are completely correct however.
For proper and better informed explanations read the other comments :D
The Eustachian tube goes between the middle ear and back of the throat and is there to equalize the air pressure in your middle ear. When you yawn or swallow it opens. That’s why yawning pops your ears when you’re in an airplane. Often children’s Eustachian tubes so not develope and they have chronic issues with fluid in the middle ear that can’t drain, so they will have “tubes” installed in their eardrum which allows fluid to drain out of the middle ear into the ear canal.
I currently haven’t been able to hear for 3 weeks because of trapped fluid due to some kind of sinus issue. The problem is that the snot gets thicker as it sits in there so it won’t drain out. Hopefully it resolves on it’s own, but I may have to have my eardrum sliced open and have it sucked out.
Lots of people can pop their ears on demand by flexing the muscles that open the tube. Those muscles are the same ones that flex when yawning and sometimes when chewing.
If your ears are always popping, I don’t think that’s necessarily bad, but maybe you’re more susceptible to ear infections? If it’s not painful and has always been like that it’s probably fine, but mentioning it to your doctor next time you see them won’t hurt.
It can be bad. I also have this superpower, I can pop my ears (open my eustachian tubes) by flexing my throat muscles. The bad part comes when this happens on its own over and over and makes me dizzy. Thankfully that’s only happened twice, but once was almost a whole day. I pop my ears much less now to try and avoid overdoing it.
Oh, I see! For many people it’s just not something they notice, and it often doesn’t happen unless there’s a pressure difference. For other people, ears pop every time they yawn, or even eat. And lots of people can pop their ears manually in the same way people can flare their nostrils or wiggle their ears.
You might be able to clear (or at least relieve) some of the pressure by doing a Eustachian Tube Massage
Apply pressure to the area below the ear and just behind the jaw bone, then follow the jawline down to the neck (I’m not sure if you go under the jaw or straight down the neck. I do both)
I’ve been trying all the tricks I can find including that one. Been breathing in steam from a bowl with a towel other my head, using a various forms of heat on my ears and face. Taking the good cold medicine that they keep behind the counter at the pharmacy plus Flonase, and antihistamines. Went through a week of antibiotics just in case. I started wearing a mask at work and outside in case it is allergies. I have one side that is getting unblocked finally, which was the first side to get blocked.
One thing I will say about tubes in the ear is that when done right, the procedure for getting them in is pretty painless. Had tubes throughout most of my early childhood, so I don’t really remember them, but I did have to get a tube put in within the last 5 or so years. Worst part was the slight burning sensation of the stuff they put in to ensure it wouldn’t hurt. Otherwise, no pain whatsoever.
If you mean hearing, then probably as a kid considering I used to get a lot of ear infections. Pretty sure it might have helped last time as well because of the inside being cleared of things blocking sound.
As for healing, it must have been doing something helpful as a kid considering I think I had something like 2-3 sets put in to help with fluid/whatever drainage. And I assume it helped last time considering they found out I needed another one when I went in for an ear infection.
Would clarifying words have helped? “If you only sailed with forward force…” or “Following along the surface of the earth…” or… what?
Obviously they mean that you don’t need to make any turns and that straight means an arc around the earth and not through the Earth, unless someone has a very different idea what sailing means…
Yes I think they mean it’s a continuous line, not a “straight” line. As in the line is uninterrupted (continuous). It’s also possible they mean the line qualifies as a nonlinear function since it also doesn’t double back over itself (A function is a relationship where each input value (X) will create only one output value (Y)).
Math is hard. Describing lines like this is math - calculus actually due to the curve, and actually not just basic calculus but vector calculus because it involves an x,y, and z axis. Most laypeople will struggle to describe a line with the correct jargon.
Depends on what you mean by help. Yes, it would communicate the point better, but it’s engagement bait, so the ambiguity is a feature rather than a bug.
I assume you mean “both capes.” While this line does come within a few thousand miles of the Horn of Africa, that’s not known as an especially hard sailing area but maybe for pirates.
Sailing this line in the other direction would be considerably harder.
Lolololol. Bro I’ve been around Africa in a 30 foot sailboat with an 8 foot draft. ‘Not hard sailing’ ? You have obviously never been on a boat at sea, let alone around either horns, capes, or whatever. Look up Shipbreakers, it’s a type of wave, then come tell me its not a hard sailing area.
Cmon man. Yes I’ve been a few places in sailboats. North sea in the winter for one. You clearly were trying to refer to Cape Horn and The Cape of Good Hope (or Cape Agulhas). Just take the L and don’t be a twat.
memes
Oldest
This magazine is from a federated server and may be incomplete. Browse more on the original instance.