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I'm developing a fantasy setting where people can go to other planes (universes) that are not made of mostly empty space, like our universe is. Instead, you have universes made mostly (but not entirely) of earth, water, or gas. (Gravity works differently on these planes, so you don't have all this matter collapsing in on itself. They're also not as big as our universe, though they're suspected to be larger than our solar system, at the very least.) In order to facilitate navigation in settings without stars or easily visible landmarks, I've decided that there's some kind of magical field that is detectable with a special compass. This magical field, when measured correctly, indicates a center point to the plane, four horizontal points (north-south-east-west) and two vertical points (up-down.)

But this is where I'm running into a problem: I need a simple, easy-to-understand naming convention for the up-to-down directions on this three-dimensional compass system. If a character says, "We need to go north," the reader automatically understands what he means. However, if the character is indicating a direction that is also on the Z axis, using terms that sound like "at 45 degrees" might be confusing and, more importantly, disorienting. I need my characters to be able to indicate they want to go not only north, but north and up or down. I believe I may have a naming convention figured out, one which I think might sound natural for my characters to speak, but while it makes sense to me, I need feedback from other people to make sure that it works.

So, here's how the system works. The center of the compass is the origin point (O) on the Cartesian coordinate system. (Remember, the magical field indicates a central point.) On the X,Y axes, the cardinal directions are the same as a conventional compass rose. (North: 0° = 360°, East: 90°, South: 180°, West: 270°) The Z axis has two points of simply called "Up" (0° = 360°) and "Down" (180°.) When characters describe specific degrees on the Z axis, they don't use terms like "north" or "south." Instead, 45° is "Upward" while 135° is "Downward." Thus, going north on a 45° angle is going "Upward North" while going north and down at a 135° angle is "Downward North." The same applies to all the other cardinal and intercardinal points on the compass, i.e. "Upward South," "Downward East," "Upward Northwest," "Downward Southeast," etc.

As for the points of 22.5° and 67.5° on a vertical compass, the equivalents of North-Northeast and East-Northeast on a horizontal compass rose, the terms "High" and "Low" are used. So, going north at a 22.5° is going "High Upward North." Going north at a 67.5° angle is going "Low Upward North." The same applies to the points of 112.5° and 157.5° from "Up" on the Z axis, which would be the equivalents of East-Southeast and South-Southeast on a horizontal compass rose. Thus, going North on a 112.5° angle is "High Downward North" and going North on a 157.4° angle is "Low Downward North."

Does this system make sense or is it too confusing to understand?

UPDATE

All right, so, after going over everyone’s feedback and giving things some thought, I’ve settled on a few ideas I have for how this compass system works, but I want to make sure that they’re sound before I commit to them. There’s no point in developing the terminology for this compass system if it wouldn't actually enable navigation within a sphere, is there?.

Before I get to my related questions, I need to give some specifics about how these planes, the magical field, and other things work. I will put the most important information in bold and give short descriptions of stuff if that's all the information you want, but I'll include more details in case you need them.

I’m calling this magical field that reacts with the magical compasses the “Cosmic Compass,” because calling in the “cosmic field,” like “magnetic field,” just didn’t sound right to me.

The Cosmic Compass

The Cosmic Compass is a magical field which affects certain materials in a similar manner to a magnetic field. This is how magical compasses can use it to determine one’s orientation on the planes in question. However, there is only one Cosmic Compass. Each plane does not have its own separate magical field. The magical field overlaps all of the planes where it is detectable. Thus, you only need one kind of magical compass to navigate on these different planes. The Cosmic Compass is not present on all planes, however, just a particular set of them. It is not present on the primary plane where my stories will be taking place.

The Wayfinder’s Compass

The name for the type of magical compass I’d like my characters to use would be a “Wayfinder’s Compass.” Usually, a Wayfinder’s Compass can function as a normal compass outside of these different planes. I’m thinking they also have other magical functions, like detecting the presence of any portals in your vicinity, functioning like a magical Geiger counter to alert people of the presence of dangerous magical energies or substances, and maybe even storing maps that can be projected like holograms. The point is that a Wayfinder’s Compass has applications on all sorts of planes, not just the ones we’re focusing on in this discussion.

What Are These Planes?

Short Answer: They’re called the Transitory Planes and they’re used to take shortcuts between different planets on the Celestial Plane, which is basically like our universe.

Long Answer: The worlds my characters inhabit are planets that exist in solar systems. Each solar system is in a separate galaxy. These galaxies are part of the same cluster found on what I am now calling the Celestial Plane. (I previously called it the Cosmic Plane in a few of my earlier responses.) However, this is just one cluster of galaxies on the Celestial Plane. There are probably about a trillion galaxies total. Because my characters don’t have spaceships, let alone spaceships with FTL capabilities, they can only travel to other planets by going through the Transitory Planes.

How Many Transitory Planes Are There?

Short Answer: The are eight, but the five I have developed so far are:

  1. The Plane of Earth
  2. The Plane of Water
  3. The Plane of Air
  4. The Plane of Fire
  5. The Plane of Mirrors

Long Answer: There are a total of eight Transitory Planes, four that are elemental themed and four that are non-elemental theme. I’ve only come up with one of the non-elemental themed planes, the Plane of Mirrors. My thinking is that, in keeping with the theme of compasses, the four elemental planes are like the cardinal points of a compass while the four non-elemental planes are like the intercardinal points of a compass. That doesn’t mean that these planes are arranged in such a formation, however. It’s more like they all occupy the same space but, being separate universes, don’t actually interact with each other. This is how the Cosmic Compass is able to overlap all of them. It’s like a bubble around all eight Transitory Planes.

How Big Are These Planes?

Short Answer: 20 billion km in diameter.

How I came to that size: I wanted them to be at least the same size as our solar system. However, there’s more than one way to decide where we put the edge of our solar system. One possible boundary is the orbit of Neptune, the other is the heliopause. The former is a radius of around 30 AU while the latter is around 90 AU. Or, to put those into really, really big numbers, the former is 4,487,936,120.73‬ km while the latter is 13,463,808,362.19‬ km. And that’s the radius. We have to double those for the possible diameters of the solar system. Those are mind-boggling numbers.

So, I decided to simplify things a bit.

Since 1 AU = 149,597,870.691 km, I rounded up to 150 million km. Now, since our choices of AU were 30 and 90, I went halfway between those and chose 60 as my multiplier for my new AU. 150 million times 60 = 9 billion. From there, I decided to round up to an even 10 billion km for the radius, meaning each Transitory Plane has a diameter of 20 billion kilometers.

Why my characters aren’t usually traveling all that far: The planets my characters are traveling between on the Celestial Plane are in different galaxies, but those galaxies are part of the same cluster. Furthermore, while this cluster of galaxies probably isn’t at the center of the Celestial Plane, the portals connecting those galaxies to the Transitory Planes are mostly (but not exclusively) found within the central region of those planes. These areas are called the Core Region, which surrounds the Core of each Transitory Plane. I’ll describe the scale of the Core Regions next because it does relate to the Cosmic Compass and I want to make sure that the Wayfinder’s Compasses would actually work effectively within this area.

The Core Regions of the Transitory Planes

Okay, so the portals connecting these galaxies to each other are found within the Core Region, but there is a pattern to the way the portals are arranged. The home worlds of the nine original races are each in their own galaxy, with eight of those galaxies forming a ring around one center galaxy. That is, the nine galaxies form a horizontal disk in their arrangement. If viewed from above, this disk would look a lot like a compass rose, with one galaxy in the middle, four galaxies located at the cardinal points of the compass, and four galaxies located at the intercardinal points.

How does this relate to the Core Regions of the Transitory Planes and the Cosmic Compass?

Short Answer: Because the directions on the Cosmic Compass are always the same, so if you want to go to the north galaxy, you travel to the northern part of the Core Regions on the Transitory Planes.

Long Answer: In terms of their placement on the Transitory Planes, the portals that connect to planets in those galaxies are located in the same general areas of the Core Region as the point of the compass the galaxy occupies on the Celestial Plane. That is, the “north” galaxy has most of its portals opening in the north area of the Core Regions, the “south” galaxy has most of its portals opening in the south area of the Core Regions, and the center galaxy has most of its portals opening in the area immediately surrounding the Core of each Transitory Plane. This is why I want people to be able to use magical compasses to navigate the Transitory Planes.

Exceptions to the portal placement rule: There are some portals that don’t follow the pattern, of course. That is, some portals to the north, south, east, west, etc., galaxies are found in the region surrounding the Core of a given Transitory Plane. Likewise, some portals from the nine galaxies open into very remote parts of the Transitory Planes. They may connect to regions that are millions or even a billion kilometers from the Core Region. The reverse is occasionally the case as well, with a portal to a galaxy on the fringes of the Celestial Plane connecting to the Core Region.

This is another reason I need to know if the Cosmic Compass is feasible. I may have a story or two where my characters go through an Earth Portal expecting to find themselves in the Core Region of the Plane of Earth only to check their magical compasses and realize they are five billion kilometers away from the Core. (And then they’ll probably run into something of the Lovecraftian variety, because why limit the shock and horror of the moment to a mere measure of distance?)

How Big Are the Core Regions?

Short Answer: 200,000 km in diameter.

How I came to this size: I didn’t think the Core Regions should be quite as large as the sun, since the sun has a diameter of about 1.39 million kilometers. That is much too large for people to traverse, by aircraft, boat, and especially by mount or by foot. So, I scaled things down so that 1) the Core Regions could hypothetically be traversed from one side to the other by foot, within around a decade.

Though it’s not entirely realistic, I set the amount of distance my hypothetical traveler could walk at 50 km a day. A bit high, I know, but multiplying 50 by the number of days in a year seemed more “tidy.” (And, no, before any of you ask, I’m NOT factoring in leap years.) The result for 1 year was 18,250, which I discovered was 5,494 km more than the equatorial diameter of the earth. Multiplying my result by 10 years, I got 182,500 km, or 39,516 km more than the equatorial diameter of Jupiter. I then subtracted 182,500 from 200,000, just to see how close the result was to 18,250. It was 17,500, which means it was only 750 km less than another year’s worth of travel for my hypothetical traveler.

Well, one more year of walking isn’t so bad to hit the 200,000 km mark, is it? So, I just rounded up to 200,000.

How Big Are the Cores of the Transitory Planes?

Short Answer: 2 km in diameter.

How I came to this size: With the diameter of the Transitory Planes being 20 billion km and the Core Regions being 200,000 km in diameter, I determined the Core Regions’ diameters are 0.001% of the diameters of the Transitory Planes. So, I checked to see what 0.001% of 200,000 was and got 2. (If I got that wrong, please correct me. I’m always second-guessing myself when it comes to math.)

To further facilitate the characters being able to determine their position in terms of distance from the Core without them needing to stop and do some calculations, I've decided to add another aspect to the Cosmic Compass - the Cosmic Pulse.

What Is the Cosmic Pulse?

Description: The Cores of the Transitory Planes emit a magical frequency which changes in oscillation depending on where you are on the Transitory Planes. The closer you are to the Core, the faster the Cosmic Pulse, the further away you are, the slower it is. This means the Cosmic Pulse would enable characters to gauge their exact distance from the Core of each Transitory Plane.

Frequency Range of the Cosmic Pulse: The Cosmic Pulse is 1 Hz at the edges of the Transitory Planes and increases by 1 Hz every kilometer. Thus, the range is 1 Hz to 10 GHz.

Story Purpose of the Cosmic Pulse: In addition to helping characters determine their location in the Core Regions, it will also show them when they have gone through a portal to a remote part of the Transitory Planes. The further away they are, the more likely they are to run into something of the Lovecraftian variety, so finding out they're 5 billion km from the Core will be a very, very frightening thing.

How Is the Cosmic Pulse Measured?

Currently, I’m thinking that the Cosmic Pulse only affects one type of material. Whatever this material is, it resonates at the same frequency as the Cosmic Pulse. Thus, if installed into a magical compass, it can give a reading that indicates how far the user is from the Core of a Transitory Plane. The question is, what material would work best? I want this material to be utterly mundane on the Celestial Plane and only be of use to anyone on a Transitory Plane for measuring the Cosmic Pulse. I’m thinking a crystal of some kind would be good for a fantasy setting, but I’m not sure if a regular crystal material can oscillate at 10 Ghz without having issues.

Regardless, I’m going to probably use the Cosmic Pulse in other ways, such as it affects the biological clocks of certain creatures so the Transitory Planes have an artificial day/night cycle.

So, here are the updated questions regarding the Cosmic Compass:

1. How accurate would the magical compasses be at determining not only direction but also location on the Transitory Plane, particularly within the Core Regions?

2. Is the Cosmic Pulse necessary for determining one's distance from the Cores of the Transitory Planes or can it be done solely with the directions of the six points of the Cosmic Compass and a bit (or a lot) of math?

3. Would a regular crystal material be able to oscillate at frequencies between 1 Hz and 10 GHz without complications or would a different material be necessary? If so, what kind of material?

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    $\begingroup$ Is this fantasy setting for a role playing game or for a work of fiction? I ask because for fiction you can get into the weeds a little with your direction system which you readers might enjoy, and then have your characters use it fluently because you are writing them and they rock. If you have players that have to learn and use the system they may chafe because they want loot, not figuring out whether it is high upward north or just plain high north or upward north. $\endgroup$ – Willk Apr 14 at 13:58
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    $\begingroup$ It’s a story setting. Basically, each race (humans, elves, dwarves, etc.,) lives on a different planet in a different galaxy. They don’t have spaceships, but they can reach other planets by taking shortcuts through other planes, like the Plane of Earth. However, I needed a way for people to keep their sense of direction on those planes, since they don’t work the same way as the Cosmic Plane, hence a three dimensional compass. $\endgroup$ – Patrick-Leigh Apr 14 at 16:39
  • $\begingroup$ Isn't there the potential for your readers to get confused between up/north and down/south when reading papercharts etc? Maybe inward/outward might be an alternate naming for the z dimension. $\endgroup$ – EveryBitHelps Apr 14 at 17:56
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    $\begingroup$ Been done long ago by aviators. "Twelve O'Clock High" "4 O'Clock Low" and so on. $\endgroup$ – Carl Witthoft Apr 14 at 17:58
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    $\begingroup$ "sideways" is a term in "The Long Earth" series, I think. $\endgroup$ – Tom Apr 14 at 18:56
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Keep it simple. Use the cardinal directions as normal, then vertical direction based on what fraction of a right angle your directional vector is. Note that this is completely independent of what you use for units, whether your culture has divided a circle into 360 degrees or 100 units or some other interval.

  • "Northeast up half" is heading northeast going upward at half a right angle (45°).
  • "South-southwest down two-thirds" is heading south-southwest going downward at 60°.
  • "East flat" (or "East up/down none", or simple "East") means heading straight east at the current altitude.
  • "Up full" and "Down full" mean straight up and down, obviously. A cardinal heading would be pointless, but could be included if, for instance, you wanted to be facing a specific direction for whatever reason. If you can imagine a helicopter, then an order like "West down full 100 meters" would mean hovering and descending 100 meters straight down, facing west.

The advantage of this type of system is that it's easily adapted to numerical values if your culture and the technology permits, as should be obvious. "Three-one-five down three-zero" is (assuming a 360 unit circle) a heading of northwest (315°) descending at an angle of 30°. In the older system it would be "Northwest down one-third."

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  • $\begingroup$ Ah, this is excellent! It also has a nautical sound to it, which works perfectly with the story setting in other ways. (It’s a flintlock fantasy heavily influenced by the 18th century and the Golden Age of Piracy.) $\endgroup$ – Patrick-Leigh Apr 14 at 17:45
  • $\begingroup$ I was wondering if you'd be looking at ship-like movement, which means you probably need two other factors: speed and possibly distance, but that's easy. "Northeast up half ahead a third" is easily understood (head northeast going upward at 45° at 1/3 speed). If you need distance, just append it at the front or back "Ten leagues northeast up half ahead a third" or "Northeast up half ahead a third for 10 leagues". If you're not worried about speed but just distance, ""Ten leagues northeast up half". $\endgroup$ – Keith Morrison Apr 14 at 17:51
  • $\begingroup$ @Patrick-Leigh, if you're going for nautical flavor, you might try the points of the compass, eg "Northeast up four points" or "South by west down a point". $\endgroup$ – Mark Apr 16 at 0:18
  • $\begingroup$ @Mark, that system works, but it requires you predefine what a "point" is. Using a fraction is dimensionless and requires no prior knowledge other than what a right angle is. $\endgroup$ – Keith Morrison Apr 16 at 15:24
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Your ideas would work well. Here are some adjustments to be get a feel for what it would look like.

Three dimensions takes three coordinates to express a unique point relative to another point.

However, direction on a sphere can be expressed with only two coordinates. This is known as spherical coordinates and direction can be express with two angles; the azimuthal angle $\theta$ and the polar angle $\Phi$.

The image below shows how this works (The R vector can be ignored) 1:

Weisstein, Eric W. "Spherical Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalCoordinates.html

If you wanted to make analogy to the traditional compass then you could keep North (0$^o$), East (90$^o$), South (180$^o$) and West (270$^o$) about azimuthal angle (x-y plane in the picture), just like you described.

As you described the positive polar angles could be "Up" and negative polar angles are "Down". For polar angles:

(0) : The Up and Down could be replaced by naught: Naught North-East

(22.5) : Low

(45) : Mid

(67.5) : High

($\pm$90) : The direction might just be Up-Naught/Down-Naught

So in the image, the direction would be "Up North-East High".

Heading in the direction of the properly orientated needle would be "Naught-North"

I included the "Naught" to add flair. Its not strictly necessary.

Of course to work, the "compass" would need to always point towards a specific point in space. The compass might be a glass sphere which is has etchings marked corresponding to the angles. The needle would run nearly the diameter of the sphere, but have just enough room to move freely.

Note, in real life this wouldn't work in space for a variety of reasons. But if there were some force which would always point the needle towards a specific point within the context of a localized environment, this sort of compass might be useful for navigation, if not just cool. The further away the point is the better the bearing is going to work for moving in a straight lines.

1 [Weisstein, Eric W. "Spherical Coordinates." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalCoordinates.html ]

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  • $\begingroup$ I like this one. It allows for much more specific directions, more akin to modern navigation, and is much simpler $\endgroup$ – awsirkis Apr 14 at 19:51
  • $\begingroup$ I like your suggestion on the compass design and that the needle always points toward a specific point. Actually, perhaps multiple needles are what's needed. On second thought, forget the needles. This is a MAGIC compass, so perhaps there are tiny beams of light that radiate from the center of the glass sphere, indicating the directions of the various points. Heck, since these planes are all finite, they might be able to gauge the distance from those points with some degree of accuracy. $\endgroup$ – Patrick-Leigh Apr 15 at 1:27
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You could always take a page from Star Trek (which I believe comes from aeronautics); directions are "<yaw> by <pitch>" or "<yaw> mark <pitch>". (You could also substitute other words to connect the two.)

This allows great precision if you use decimal degrees (0-360 by -90-90 with as many decimals of precision as you want), or you can use clock hands (1-12 by 1-5, ignoring 6 and 12 for the latter because those make the yaw irrelevant). In either case, you can talk about relative (i.e. I am always facing 0 by 0) or absolute directions ("north" is always 0 by 0). You can add "flair" to this by having them divide their circles by some number other than 12 or 360, although this may be too confusing for readers.

Note that, if you use absolute directions, you also need a 3D compass; you need to know "up" as well as "north". (Depending on how gravity works, this may be easy. For example, you could use this system to navigate Earth's oceans with a regular compass if you also have a reliable way to determine 'down'... which, in water, can be less easy than you might expect.

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Just do what we do, but with an extra dimension.

We describe directions with four base words: North, South, East and West. We can also combine them: Northwest, Southeast, and so on. For even more precision, we have constructs like North-northwest and East-southeast.

We simply extend this system with two additional directions, "Up" and "Down" (though one may want separate words for them to make a distinction between absolute ("West") and relative ("Left") terms, but I digress).

You want to go up? That's "Up". You want to go up and north? "Upnorth". Down, west and south? "Downsouthwest". Just north? That's just "North".

We can continue with more detailed directions. You want to go north and slightly upwards? "North-upnorth". Northeast and slightly up? "Northeast-upnortheast".

Seeing as a third dimension adds more information to convey, I suspect that super-specific directions will be less commonly used due to how cumbersome they are to say.

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    $\begingroup$ This was my initial thought, but you have to watch out for ambiguous combinations. North up north north east can be read in a variety of ways that actually are very different directions, so the convention for how you combine them has to be very specific, because memorising a 420 point 3d compass (the equivalent of the 16 point rose) is not really a nice thing to do. $\endgroup$ – Joe Bloggs Apr 14 at 15:42
  • $\begingroup$ @JoeBloggs Indeed, and that's the major downside with this approach. I was actually planning on designing and providing an explicit example grammar for how to construct and interpret complex directions. However, I quickly realized that even if such an unambiguous grammar were to be made, the non-trivial cases would be so long, cumbersome and difficult to visualize that nobody, in or out of universe, would ever bother using such directions given such a scheme. Instead, I just hinted at the problem towards the end and left it at that. $\endgroup$ – BambooleanLogic Apr 14 at 16:32
  • $\begingroup$ That was exactly the problem I ran into. Best I could formulate was always giving 'flat' coordinates priority and then repeating the first cardinal given when trying to identify the vertical component, but even then you can end up with godawful constructs like 'north north east north up' (22.5 degrees up from north north east) and up north north east up (67.5 degrees up from north north east). $\endgroup$ – Joe Bloggs Apr 14 at 16:45
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Using a Solar Reference Point

There are many good answers here for a cartesian navigation system, but compasses don't follow the rules of cartesian space. They aim at a point and not an infinite direction. Since you mention that your universe is the size of a solar system, instead of hand waving away how the compass works, one option is to put a star at the middle of your universe, and make that the your reference point.

Your compass could be any variety of tools designed to measure the sun as your primary reference point, but with a star it does not actually have to be magic. You could just use a pinhole sunspot viewer, or a magnetic compass that orients to the star's magnetic field.

Stars have a spin which can be used to define an equatorial line as well as up vs down. If you perceive the spin as clockwise, you are right-side up, if your perceive it as counter-clockwise, you are upside down. From that perspective, any point in space above the sun's equator or moving in that direction is Up and below it is Down. Moving toward or away from the star would be In and Out, and moving in or against the direction of the star's rotation would be With and Counter.

Navigating a 3-d Solar Map

In the map below, point A and B are dots with lines that go up or down as far as they are from the equator to where they intersect with it. So, to get a heading from A to B you would travel "about Down-Down-In-Counter" in layman's terms or if using a degree system "315 mark 30".

For the degree system it would be 0 to 360 with 0 being In, 90 With, 180 Out, and 270 Counter, and 0 to 180 with 90 being parallel to the equatorial plane, 0 being Down and 180 being Up.

To define an absolute location (as you would in GPS), you would need an arbitrary marker you define as 0 degrees. For this you would want a very large planet that is on the equatorial plane that can be seen from most places (Point C on the map). It will server as sort of a Prime Meridian/North Star. From it you measure a planet's location by X-angle/Y-angle/Distance. So let's say "C" lies as "0/0/100", the "A" would be at about "283/30/120" and "B" would be at about "296/-20/80".

enter image description here

What does a star look like in each plane?

If you want to get scientific here, the 4 elements of fire, air, water, and earth don't all make sense on thier own. Fire without air to burn is not a thing which makes explaining your planes in this since impossible (I know you can just magic handwavium this, but you don't have to). In science, there is a better explanation of these 4 states that our ancestors first identified which are the 4 phases of matter: plasma (fire), gas (air), liquid (water), and solid (earth). In this since your can make your material planes places where matter only exists in a single state of matter while maintaining its other chemical and atomic properties.

Plasma: Our sun could best be described as a plasma star; so, no real modifications needed. In the plane of fire, all astral bodies, no matter how small would look like stars; and the sun is simply the biggest and brightest star in the sky.

Gas: In this plane, all asterial bodies would look like gas giants, and the sun would just be a really really big gas giant. Looking at the sun would be like looking at a giant jupiter. Fusion does not work so well without a plasma state; so, it would be too dark to see it, but it would still have a magnetic field from the pseudo magnetic gases at its core which you could detect with a compass.

Liquid: In this plane asterial bodies would all be endless oceans with no floors. Like the Gas sun, the Liquid sun would be lightless but generate a magnetic field

Solid: In this plane, planets would all be solid stone or ice and the sun is no exception. This may actually lead to a problem since the spin of a solid core against a liquid mantle is where we get the magnetic field on rocky planets. That said, You could dismiss by saying that the star is solidified in a polarized state which you could align to with a magnetic compass.

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    $\begingroup$ This could work for some of the planes, like the Plane of Air and maybe the Plane of Fire, but visibility on the Plane of Water is limited and on the Plane of Earth it is heavily obstructed by all the rock. But some kind of big, visible object at the center of the Plane of Air seems very appealing to me, so I think I'll find a way to use it. $\endgroup$ – Patrick-Leigh Apr 14 at 22:17
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Most of the answers I've seen here take the compass rose, and then add a yaw angle. This makes the handling of the Up-Down dimension fundamentally different from the handling of the two planar dimensions. My approach aims to be a true 3D approach that treats all six poles the same.


The Compass star

So, what is the working principle of the compass rose?
In 2D, the compass rose subdivides arcs on a circle.
How do we extend this to 3D?
In 3D, the compass star subdivides surface patches on a sphere.
What are the surface patches, that we need to subdivide?
The surface patches are all triangles.

Whenever we take one pole on each of the tree axes, we see that they form a spherical triangle. Like this:

     U

   /   \
  /     \

N  -----  E

Splitting a triangle into four smaller triangles is rather simple. We just need to take the center points on its sides:

          U

        /   \
       /     \

    UN  -----  UE

   /   \     /   \
  /     \   /     \

N  ----- NE  -----  E

This gives you a total of 18 directions: The 6 poles plus the 12 points half-way between the poles.

For more fine-grained orientation, we continue to subdivide the triangles, just like the compass rose continues to subdivide the arcs. The second iteration is this:

                    U

                  /   \
                 /     \

              UUN ----- UUE

             /   \     /   \
            /     \   /     \

         UN  -----UUNE -----  UE

        /   \     /   \     /   \
       /     \   /     \   /     \

    UNN -----UNNE -----UNEE ----- UEE

   /   \     /   \     /   \     /   \
  /     \   /     \   /     \   /     \

N  ----- NNE ----- NE  ----- NEE -----  E

Now we have the 6 poles, three intermediate directions on the 12 lines between the poles, and three directions in the center of each of the 8 celestial octants. That gives 6 + 3*12 + 3*8 = 66 directions.


The number system

You can continue this construction to finer subdivisions, but on the next iteration you will run into problems with the naming. Your names will become rather unwieldy and they become hard to define in an unambiguous way. However, we observe that every direction is nothing more or less than a weighted sum of up to three pole directions. And we can easily express these with integer numbers. Up to now, we had these directions:

Principal directions:

  U

N,  E

Mixing of two directions:

        U = 2U

    UN,     UE

N = 2N, NE,     E = 2E

Mixing of four directions:

                            U = 4U

                     UUN = 3UN,    UUE = 3UE

              UN = 2U2N,    UUNE = 2UNE,  UE = 2U2E

       UNN = U3N,    UNNE = U2NE,  UNEE = UN2E,  UEE = U3E

N = 4N,       NNE = 3NE,    NE = 2N2E,    NEE = N3E,    E = 4E

We can continue this principle, doubling the number of mixed directions in every step and denoting the weight of each principal direction with a single integer:

Mixing of eight directions:

                                8U

                            7UN,    7UE

                        6U2N,   6UNE,   6U2E

                    5U3N,   5U2NE,  5UN2E,  5U3E

                4U4N,   4U3NE,  4U2N2E, 4UN3E,  4U4E

            3U5N,   3U4NE,  3U3N2E, 3U2N3E, 3UN4E,  3U5E

        2U6N,   2U5NE,  2U4N2E, 2U3N3E, 2U2N4E, 2UN5E,  2U6E

    U7N,    U6NE,   U5N2E,  U4N3E,  U3N4E,  U2N5E,  UN6E,   U7E

8N,     7NE,    6N2E,   5N3E,   4N4E,   3N5E,   2N6E,   N7E,    8E 

I each direction, the sum of the integers is exactly 8. Or 16 in the next subdivision step. Or 32 in the following one. I'm not going to write all those directions down... That is, the expression of a direction in the numerical system is fully unambiguous: It gives the count of directions that were mixed, and it provides the weights of all the constituents. And, because it only ever uses three numbers and three letters/names, it remains concise even when denoting a direction with high precision.

Of course, this number system would only ever be used by navigation professionals. Normal people will stop at the second subdivision and stick to the non-numerical names that I outlined above. 66 directions should be plenty for a layman's use.


Further thoughts

The above construction is based on the octahedron: The 6 poles correspond to the 6 vertices of the octahedron, and the 8 triangles that form the octahedron are the basis for the triangle subdivision above. However, there are two other platonic solids that are composed of triangles: The tetrahedron and the icosahedron. Both of these allow the exact same triangle subdivision process to derive more precise directions.

The tetrahedron would only use 4 poles and divide the sphere into 4 patches. That's rather coarse and of little use.

The icosahedron would use 12 poles, twice as much as the octahedron, and provide a whopping 20 triangular patches (2.5 times as much as the octahedron). It would make for a good basis for the construction above, and it would add a significant magical flair if the compass is an icosahedron that's suspended within a sphere made of glass (crystal?). You'd need to invent names for 12 poles, though.

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  • $\begingroup$ I love the icosahedron in a crystal sphere idea. My Dwarves are obsessed with mathematics and geometry, and they regard Platonic solids as almost sacred, so having on inside a magical compass fits perfectly into my story setting. I’m going to give your other suggestions a lot of thought, because I think there’s a ton of potential in them. Thanks for the feedback! $\endgroup$ – Patrick-Leigh Apr 16 at 19:15
  • $\begingroup$ @Patrick-Leigh Glad to hear that. Actually, the icosahedron had been my first thought because I have once contributed to a piece of scientific software that actually uses the icosahedron to structure its data. They use it because it provides the smoothest mapping of a sphere to a discrete grid of cells, solving their pole problem. However, most people are not really that accustomed to the icosahedron, and I figured that the octahedron would be more intuitive to them, especially since we already got names for its poles. But if that's not the case for your dwarves, all the better :-) $\endgroup$ – cmaster - reinstate monica Apr 16 at 20:18
  • $\begingroup$ I’m thinking the magic compasses come in different models, depending on what you can afford and how precise you want to be. The octahedron would be the type most people would use, since they’re mostly interested in keeping their bearings, while Dwarves, being obsessed with precision in all things, would have the models with the icosahedrons in them. Like I said, they revere Platonic solids. I think they consider them the “divine polygons” or something to that effect. $\endgroup$ – Patrick-Leigh Apr 16 at 21:32
  • $\begingroup$ Darn autocorrect! I meant “divine polyHEDRONS.” $\endgroup$ – Patrick-Leigh Apr 16 at 21:38
  • $\begingroup$ I'm actually having a bit of difficulty visualizing how an octahedron in a crystal sphere would indicate the direction of the six poles. Could you clarify that point for me? $\endgroup$ – Patrick-Leigh Apr 18 at 22:52
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Just use the word "and".

There is nothing wrong with your coordinate system. But focusing too much on the directions and angles makes it more confusing that needs be, and complicates the language as well.

You can fix this just by saying how far things are apart along each axis.

Rather than "The fortress is 6 miles high upward northwest from here" use "The fortress is six miles to the northwest and 1 mile up".

Rather than "We need to go 6 miles high upwards northwest" say "we need to go 6 miles northwest and 1 mile up".

After all, we don't need to go at any particular angle do we? The angle/direction is not important. It's the start and end that matters.

If I HAD to describe directions. . .

If for some reason I wanted someone to start moving in a certain direction and just keep going forever. The simplest terminology is "Go 1 North by 1 upwards". That means go North at the angle so that every time you go 1 unit north you also go 1 unit up. In other words go north at a $45^\circ$ angle from the floor. This is the same as your "Upwards North".

We also have "Go 1 North by 2 upwards" that means go North at the angle so that every time you go 1 unit north you also go 2 units up. This is the same as your "High Upwards North".

Also "Go 2 North by 3 upwards" means every time you go 2 units north you go 3 units up. So a $60^\circ$ angle to the floor.

You can plug whatever ratios you want in to get all the directions.

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I love your system (noting Carl Witthoft’s endorsing comment that it lines up with real-world praxis); I think it is indeed very intelligible.
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There is also a lot of interesting and good material among the answers here. A couple of comments.

I would suggest that you have (at least) two verbal expression systems — e.g. a military one and a civilian one, or an adventurer one and a casual one. Each system would be more about precision, or brevity, or everyday understandability, or what-have-you. (It might add a bit of spice, in some situations, to have a character switch from one to the other (for the sake of speed, or precision, or intelligibility).) This ties in with Matthew’s and Keith Morrison’s comments about relative directions. (By the same token, there might be different compassOrbs with markings for one or all of the main systems, and perhaps other ones with various etiologies; see also below.)

For precision, it looks like it would be hard to beat “cmaster - reinstate monica” ’s system — just add more characters (although it would be good to have a trailing 0’s element).

(For one system) I like the idea of using a polar system, with either 9 or 12 (or 4) main divisions for each quarter-circle… or just 0 to whatever (36/48/16) for the whole circle… in the NWSE plane. [You could have, correspondingly, a people with 9 or 12 fingers.] The NWSE plain works like (our) everyday clock system, basically. It goes either by quadrants — “2-8” — or simple — “35” (anchored presumably with 0 for north). The former sounds more plausible to me. (Actually, I would expect the quadrants to be labelled 1 to 4, rather than 0 to 3… but I shall ignore that herebelow.)

Up/down is a suffix from 1 to 9/12/4, preceded by a signal for down — e.g. “3-5-neg2” — or just 0 to 23. (You could have 0-8 [viz 9] instead of 11 [viz 12] for the elevation, for interest.) [Again, you could have 1 to 12 instead of 0 to 11.]

“Oh-oh-oh” is north/level. “3-6-6” is south-west, 45° upwards. “1-oh-neg10” or “1-oh-22” is east and steeply down.

For precision you could add a qualifier — e.g. “3-6dash85-3dash17”. [Or “dot”… or “boo” or whatever.]

Again, you could use the system you describe (which I very much like) in one case, and the above in another.

(I would like an adaptation of this with only one-syllable words, but you are probably better sticking with English. [Possibly “sven” and “ ’lven ” for 7 and 11 respectively, or what-have-you.])
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The justification of this is that you do not always need the reader to understand the direction that the story characters are discussing. You could have a (hopefully not too contrived) conversation in which someone is explaining the more esoteric system to someone else, in terms of the casual and easy-to-understand one, and then carry on with it. Given that, the keen reader can indeed follow the other system. By the same token, when you are explaining directions to the reader, you can use the intelligible one.
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[Only some of the following is consistent with the system you have described.] I thought it might also be worth mentioning the concept that the magic system was not quite so anthropocentric. That is (perhaps {for instance}?)… it can tell you which way is up/north/east, but not where the centre is. That is… you know that the centre is in “that” direction, but not how far away it is. The extreme version of this idea is that it does not tell you where the centre is; only which way is up/east. You could also consider variations such as having it tell you the direction to the centre, and perhaps up and down, but nothing else… or the direction and distance of the centre but nothing else… or it tells you EWNS but you get up/down from “gravity” (meaning that you do not know your height). (Conversely, of course, you can have it so that it does tell you exactly where you are.) (You could also have compassOrbs that work only in some domains (and expensive ones that are universal), or that malfunction in some domains.)

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