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In my setting, the inhabited world is basically a huge cylinder sticking out of the pole of a gas giant with a breathable atmosphere:

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Everyone lives on a side of a giant wall, basically - the top is in a vacuum and thus is beyond reach. And I stuck with a conundrum, how do you determine longitude to navigate this thing (latitude is easy since you can track your elevation by atmospheric pressure or visually by the clouds surrounding the pillar)? I imagine compasses would be useless - the magnetosphere of the gas giant produces strong and chaotic magnetic fields, and even if it would be calm, the compass would always point towards the center of the cylinder, or up or down. So this leaves the navigation by stars as the only other option I know besides short-range navigation via landmarks, but how reliable would it be, considering that the planet is spinning?

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    $\begingroup$ Can people see the floor or the roof of the cylinder? Can they see through the middle, to the other side? And what is the source of light? $\endgroup$
    – KeizerHarm
    Apr 24, 2021 at 10:20
  • $\begingroup$ They can't reach the roof since it's in a literal vacuum of space, and there's no floor, the thing goes all the way down to the core of the gas giant. They live on the outside of it. $\endgroup$ Apr 24, 2021 at 10:21
  • $\begingroup$ How do they stick to it? Is the rotation of the Gas Giant likely to produce a periodic day/night cycle (ie. can there be "time-zones")? $\endgroup$ Apr 24, 2021 at 10:36
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    $\begingroup$ There's ledges and gorges of various sizes, up to those allowing to build cities in. That's where everybody sticks. And yes, there's day-night cycle based on gas giant rotation (~30 hours long, if that matters). $\endgroup$ Apr 24, 2021 at 10:53
  • $\begingroup$ Since there is a predictable day-night cycle, you can determine longitude using the same method as 17th-century mariners: A sextant and a reliable clock. An ephemeris is useful so you're not always using the sun for reference. $\endgroup$
    – user535733
    Apr 24, 2021 at 13:40

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Three possibilities, in ascending order of usefulness.

  1. Guesswork: works very badly, extremely limiting, but was used for millennia.

  2. Clocks: If the day-night cycle varies from one part of the cylinder to the other (you don't mention whether it does), determine when dawn occurred according to your very precise clock set to a particular location. However, if the day's based on the gas-giant rotation, it will probably not help, because all the cylinder sees day at the same time.

  3. Radio signals: Your fixed locations broadcast radio signals with their location. This will let your navigators triangulate between them.

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Same way it was done on Earth: timekeeping.

Assuming the planet rotates, the tower, being at the pole, will rotate with the same angular velocity. Arbitrarily pick one place that will be the designated Prime Meridian. We'll call it Grenich. That will be 0 longitude. Now use a reliable clock set so that when the sun is at the highest point overhead at Grenich, it is the midpoint of the day, or start of the day, or however you arrange your timekeeping.

Have a portable second clock that keeps the same time, and travel spinward or anti-spinward from Grenich. If you want to know what longitude you are at, take a sighting on the sun when it's at the highest point in the sky (local noon) and compare it to what the time reads on your clock. The difference between noon where you are and noon in Grenich (which you know, because the clock is set based on it) can be used to determine calculate your latitude.

Simple example using Earth: Earth rotates 360 degrees in 24 hours, so 15 degrees an hour. When you determine local noon, it's when the clock, set at Greenwich Mean Time, reads 11:30am. So you've reached local noon a half hour before, so you know you are about 7.5 degrees spinward (East) of Greenwich, thus your longitude is 7.5 East. If your clock reads 2:45 pm, you are 2.75 hours behind Greenwich, thus around 41.25 (or 41 deg 15') West.

Note that this method works on Earth even in polar regions in 24 hour daylight because the sun still reaches the highest point in the sky at noon local time. In your setup, it's even easier because there will never be 24 hour sunlight except on the top of the cylinder.

Now, that suffices for low-tech. There are a couple of options for more high-tech solutions. One that comes to mind is that the civilization sets up radio transmitters at regular intervals around the cylinder, one high up, one low down, with paired transmitters at each interval. Let's say they set up 36, so one pair every 10 degrees, and each broadcasts a simple signal, say its longitude (from 0 to 350) plus N or S to indicate if it's in the north or south.

Your "compass" would consist of a radio direction finder that can measure the angle between two or more radio signals. You would first look for the signals the closest to directly north and south of you. Let's say it's the 30N and 30S signals. You then measure the angle between the transmitters. If they are 180 degrees apart, you are directly on 30 longitude. If they are closer than 180 degrees, basic trigonometry will tell you how far east or west of 30 degrees longitude you are. If you wanted to get fancy, it might lock on the 20N/20S or 40N/40S transmitters to make a second reading to verify.

If you can't get evenly space paired intervals, the system still works, the math is just slightly more advanced: you might be reading off the 33N signal and the 38S one, for instance.

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I don't think you need much tech at all. A barometer can tell you how far up the cylinder you are, and at least at night, looking out the top of the cylinder at the stars can tell you which side of the cylinder you're on.

Directions will be given in terms of up or down, clockwise or counterclockwise. "Steve, the rendezvous point is 3km up, 2km clockwise." Absolute positions will be given in altitudes and bearings; "Cloudtown is located at 15km, 287 degrees." What these measurements are relative to is mostly arbitrary, much like deciding the prime meridian here on Earth.

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/There's ledges and gorges of various sizes, up to those allowing to build cities in. That's where everybody sticks./

These ledges and gorges are built things. They are part of the cylinder and they are symmetric. They are marked to be distinctive. They can be seen from a distance. Atop one, a person can see the neighboring ledges in the distance. Travelling between them one can see the two ledges you are between.

I could imagine coordinates corresponding to the relative apparent size of the two visible ledges. Right between them it would be A50B50 as the the viewer they are the same size. As i approach A the A value gets larger and the B value smaller. One could calculate this with a piece of paper held to the eye as one looked at the ledges.

Or perhaps an icon or figurine worn as a necklace and used for this purpose of determining relative size. I envision the protagonist uses a little animal made from a striped stone. When things are slow the protagonist and her companion debate over what the animal is supposed to be.

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