Trying to picture the warmup is already hard enough for me, so I’ll start with asking questions about that and revisit the rest later:
(Exercise: what are the lines of latitude? What about longitude? Can you picture the north and south pole in different placetimes, and the corresponding equators?)
I expect that the longitude are the lines taken by choosing one in the middle sphere, and tracing the line following it on both side? As for latitude, if I use the analogy of the 2-sphere, each circle in the film is one line of latitude; so maybe each 2-sphere in this film is a latitude?
Also, I don’t understand what you mean by your last question. In the 2-sphere version, the poles are only visible at time 0 and 2 and the equator is only visible at time 1.
Yeah, in my interpretation of “latitude line” it’s not really a line, but rather “those set of points that are exactly distance d from the south pole”. Which as you say are the 2-spheres that are visible at each instant of time.
>and tracing the line following it on both side
Sorry, I don’t know what you mean by this. I think you’re saying that we pick a longitude line in the equatorial S², and then draw the S² that passes through the south pole and that longitude line. That’s also what I had in mind for longitude lines. We could also interpret longitude lines as great circles (rather than “great S^(n-1)” where we’re considering S^n). Then it’s actually a line (I mean, a geodesic, i.e. a great circle), and it looks like two antipodal points on the S² at each instant, never seeming to move (except that they ride along with the expansion and contraction of S²). These describe a great circle passing through the north and south pole. There’s an S²’s worth of these lines, since the equator is an S² (analogous to there being a circle’s worth of longitude lines on S²).
>Also, I don’t understand what you mean by your last question
Edited the post. I mean to say, what if you put the south pole somewhere else. One point of the exercise is to get you to picture other great circles and “great S²s”.
(I misused “longitude line”; I guess that normally refers to half a great circle. So, we’d sweep out half of a great S². Which… bleh. I’d rather think about great circles.)
Trying to picture the warmup is already hard enough for me, so I’ll start with asking questions about that and revisit the rest later:
I expect that the longitude are the lines taken by choosing one in the middle sphere, and tracing the line following it on both side? As for latitude, if I use the analogy of the 2-sphere, each circle in the film is one line of latitude; so maybe each 2-sphere in this film is a latitude?
Also, I don’t understand what you mean by your last question. In the 2-sphere version, the poles are only visible at time 0 and 2 and the equator is only visible at time 1.
Yeah, in my interpretation of “latitude line” it’s not really a line, but rather “those set of points that are exactly distance d from the south pole”. Which as you say are the 2-spheres that are visible at each instant of time.
>and tracing the line following it on both side
Sorry, I don’t know what you mean by this. I think you’re saying that we pick a longitude line in the equatorial S², and then draw the S² that passes through the south pole and that longitude line. That’s also what I had in mind for longitude lines. We could also interpret longitude lines as great circles (rather than “great S^(n-1)” where we’re considering S^n). Then it’s actually a line (I mean, a geodesic, i.e. a great circle), and it looks like two antipodal points on the S² at each instant, never seeming to move (except that they ride along with the expansion and contraction of S²). These describe a great circle passing through the north and south pole. There’s an S²’s worth of these lines, since the equator is an S² (analogous to there being a circle’s worth of longitude lines on S²).
>Also, I don’t understand what you mean by your last question
Edited the post. I mean to say, what if you put the south pole somewhere else. One point of the exercise is to get you to picture other great circles and “great S²s”.
(I misused “longitude line”; I guess that normally refers to half a great circle. So, we’d sweep out half of a great S². Which… bleh. I’d rather think about great circles.)