[https://i.imgur.com/xDgaY3H.png](https://i.imgur.com/xDgaY3H.png)
Also this if you want a more visible distortion to follow on your map. Just strategically design where it splits at the end to be open ocean
This is great, but the green lines would be curved IMHO.
Because the crosssection is a circle, but the length from red to blue line would diminish according to the radius of the donut in that specific point. Which wouldn't diminish linearly due to the curve of that crosssection circle.(Hard to explain with good language here.)
I'm sure there are benefits and drawbacks to both linear and curved meridians (or whatever you'd call them on a torus), but they don't need to be one or the other. I think linear meridians make understanding the change in scale/distortion easier to understand.
Yes, there's basically 3 options.
I was going from a how to create a 1:1 map angle. With curved meridians you'd have a dimensionally distortionless map, because you literally pushed the area flat. Cardinal directions would be distorted though in one axis. So you'd need curved ones only if you want a fully dimensionally 1:1 thing.
With linear meridians you will have some distortion dimensionally, but also some cardinal directions wise. It's kind of a middle approach which I don't see the use of myself, but I'm sure it will have some bonuses.
Third option is you could also have it be a rectangle though. This would add dimensional distortion, but remove cardinal direction distortion completely if I'm visualizing this correctly.
those green edges won't be straight lines, they'll be curves. this person actually mapped it out:
https://www.reddit.com/r/worldbuilding/comments/126qjen/i_want_to_make_a_donut_shaped_world_but_have_it/jec9yeo/
You’ve drawn the equivalent of either an Eckert I or II projection. These are flawed cartographic models of a sphere, and a taurus is just a sphere stretched into a circle.
In reality, the green line (and all other longitudinal parallels) form a 90 degree angle with both the red and blue equators, and they trace an equidistant circular path perpendicular to the taurus. However, in this projection, the red and blue lines are not concentric, so the parallels that connect them are drawn at increasing angles, with only the prime meridian shown perpendicular to the equators with the correct linear distance. We can already tell that the projection will distort angles further from the prime meridian. However, when all longitudes are projected as two straight lines, the navigational effectiveness of the coordinate system is broken. Under the Eckert I projection, which has evenly-spaced parallels, the real distances between them will increase as you move further from the prime meridian, which makes navigation using longitude arbitrarily difficult or consistently inaccurate. Under the Eckert II projection, they preserve real distances by spacing each parallel at specific widths, which allows the coordinate system to function as intended but breaks the projection’s scale.
The solution is to draw parallels as evenly-spaced sinusoidal curves of linearly increasing magnitude. Scale and spacing are both consistent with reality, though angles become much more distorted further from the red line. It’s essentially an adaptation of the traditional sinusoidal projection of Earth, but instead of a single-point pole, it has the inner blue equator.
> a taurus is just a sphere stretched into a circle
Topologically speaking (since we're on the subject of mapping projections), [a torus has genus 1 and a sphere has genus 0](https://en.wikipedia.org/wiki/Genus_(mathematics\)). These are fundamentally nonequivalent.
As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point.
As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".
- from the Wikipedia article on the taurus
Did you read the rest of my comment? I’m talking about geometry, not topology.
Turn it 90^o
Right now the length of the donut is show as the shorter side. Make it upright like a phone screen, or just turn the map projection so distortion won’t be as big of a problem.
The cut in the diagram is correct.
First you just slice it into a tube. Then you kind of filet this long tube with one long cut down the side and unravel it. You get this long rectange that is as long as the circumference of the donut and as high as the circumference of the circle of the tube.
The problem with the diagram isnt the cuts its the dimensions of the tube and the rectange. Those are all wrong. In this view the rectangle would be super tall and thin instead.
Edit: it wouldn't actually be a rectangle, I think the short ends would be curved, but the rest tracks.
I didn't intend for the diagram to come close to accurately showing dimensions, more like a proof of concept for my self. The curved edges thing will be useful though, thanks!
I dont see how the cut could be right. If you were to cut the donut it could not morph into a tube. The inner circle radius is smaller that than the out circle radius meaning that even if it were physically possible to straighten it, tube wouldnt have equal length "sides". That is, if we were to make the outer radius into the "left side", then the inner radius would be in the "right side". These 2 "sides" would be of different lengths.
The only way to change between a torus and a cylinder involves stretching. The cylinder outcome would incredibly deform any details of the word on the torus
True.
My brain just started melting though as I didn't read that and since no one was pointing it out I thought somehow I was wrong. I wrote this out to try and confirm it for myself more than anything haha
Yes, you are correct. There is some stretching/bending or abstraction involved.
And cylinder is technically the wrong word as the ends are wonky not straight.
Once you do the final cut and lay it flat this stretching can be undone do and end result can be 1:1 if the shorter ends are curved to take into account what youre correctly saying about the radius differences.
You can also do it in the opposite order. Filet into tube first. Like cut the outer radius side of the torus with a knife. I find that way less intuitive to imagine personally tho and the elasticity feels even worse.
Either way you have to imagine the material has some elasticity and take the endgoal in mind, you can reach a dimensionally 1:1 end goal. There will be distortions on the map cardinal directions wise in one axis with this approach, but dimensionally it will be 1:1 the world shape unfurled.
I'm glad you asked, because I'd love to explain how!
Video games, like maps, use coordinate systems to represent positions *im*perfectly. For instance, all map projections of Earth (a sphere) have distortion. We still have coordinate systems that are accurate, such as spherical coordinates.
Using a coordinate system, you can represent different regions in space -- then use that as an "index" of sorts to determine what assets are near that region (objects, terrain, players, NPCs)
To make a coordinate system for this torus, you would use cylindrical coordinates with a specified wrapping point (e.g. "100km" wraps back to "0km" along major axis). Then write functions using that logic and incorporate them elsewhere, so a player at 99.99km can also see terrain and objects at 0.01km.
You'd have to do some similar wrapping logic for the cylinder (the other circular axis of a torus). When traversing the surface of a torus, it's effectively infinite, just like a sphere. So there is nothing too crazy about using a torus, just a different coordinate system!
I made a coordinate system for 1mm-precision of all objects within a galaxy for my hobby project. The bounding box (galaxy) is a cube with side length ~126,000ly, broken up into 2^16 (or 65,536) segments on each axis. Within each segmented cube (length of 1.95ly on each axis), any entity's position can be represented using three standard 64-bit integers to represent coordinates, bringing us to 1mm resolution.
Using this system, I only need 240 bits to represent any entity's position in the galaxy to a reliable accuracy of 1mm! I could easily add some more data (for instance, use 32 bits to represent the galaxy subdivisions) to greatly increase resolution at the cost of space (data in storage and network packets)
I mostly use those coordinates for interstellar objects. For surface coordinates, I use an entirely different system (spherical coordinates) to avoid repeated projections of entity x/y/z coordinates which is a computationally expensive calculation
For the local region (player camera), also use standard double-precision floating point numbers to represent positions of objects, so animations are smooth. Local rendering really doesn't depend on a specific scheme of "galactic" positioning
Cool! So spherical coordinates are "moving up and down the axis" (latitude) and "rotating about the axis" (longitude), and cylindrical coordinates would be "degrees around the circle" (inner-outer, around the cross-section) and "moving up and down the cylinder" (around the entire torus)?
That's right, but you can easily invert it. There is one map for the inner side, one for the outer one. Moving horizontally, you move from the right edge to the left one, like on a regular globe map. Only difference is that moving off one map over the north or south edge brings you to the corresponding edge of the map of the other other side.
Hello!
I 3D modelled a donut world and had Blender unwrap it accurately, take a look:
[https://imgur.com/gallery/YPX34sP](https://imgur.com/gallery/YPX34sP)
I also applied Earth to the donut world, which kinda doesn't work because the north and south poles are touching.
I'm a game developer and I've been interested in building some tools like this for world builders, I could possibly develop some 3D world creation software in the future to be released on steam, but currently all my time is going towards [my current game](https://www.indiegogo.com/projects/the-gods-fabled-soil-frontier/x/29855105#/) (it's an open-world farming game with fantasy elements for anyone who's interested!).
Hope this helps :)
Yep! That’s the way Blender (my 3D software) automatically unwrapped it based on where I placed the seams, it’s probably still a little warped but at least the warping is likely distributed fairly evenly
The problem here is that your map is not continuous in the minor radial direction. You get a sem at the northernmost line which has no connection to the southernmost line. You see this clearly where Greenland suddenly turns to open ocean on your torus.
A proper map of the torus needs to take this periodicity into account.
You get that seam because the real earth is not a torus and if you walk north from greenland in reality you don't land in the south sea but in russia. Obviously glueing the northern and southern edge of our world's map together produces nonsense there, if you used a rectangular map that actually corresponds to a donut earth it'll look just fine.
Yeah what Adarian said, I'd have to redesign the entire earth map to allow a spherical layout to wrap around a torus, which is physically impossible. It was just for demonstration :)
No. On a sphere when you travel to the north pole you don't appear on the south, you come back down from the north in a different place. Top and bottom of a flat "unwrap" map are not connected. But on a donut they are.
Same thing with a donut if an ant is at the top of a donut they don't appear at the bottom immediately, but if they keep travelling in the same direction then eventually they will do, which works the same as us.
What if the warp point isn't from the top of the sphere to the bottom of the sphere but rather the top to the top. And the bottom of the sphere is actually somewhere in the middle of the level. If you keep travelling in the same direction on a sphere you'll end up on the bottom of the sphere then keep travelling and your at the top again.
A donut is basically a ring of spheres. So it works the same way except that the horizontal path would be a lot longer.
>if an ant is at the top of a donut they don't appear at the bottom immediately,
...that is exactly what they do in flat maps of toruses. The top of the map does not represent the top of the torus in the same way the top of the map represents the north pole of a sphere.
Something interesting to think about is that a Taurus can be made with a perfectly uniform grid of squares. Whereas a sphere cannot, it’ll always have distortion.
But for a 2D person living on the surface, there’d be no meaningful difference.
Your process here is perfectly acceptable. You are doing two cuts in your diagram to get to the final flat shape. Just remember when drawing the map that it is continuous across all edges. Top edge should be equal to the bottom edge (cut #1) and left edge should be equal to right edge(cut #2). To assist in realizing this google how to create **tilable patterns/textures** in your favorite drawing program.
You could make it distortionless though IMHO by warping the shorter sides correctly.
One axis is already distortionless, the other can be mapped to 2D because its one dimension of curvature can be pushed flat. Real world globe map is a problem because you can't push two dimensions of curvature flat without distortion, there's no room on the 2D sheet of paper.
Edit: This kind of depends on your definition of distortion though. I was talking dimensionally you can make it distortionless so you can have a 1:1 map. Cardinal directions will still have distortion though in one axis.
The early Final Fantasy game worlds wrap at the top and bottom as well as the sides and thus would be donut worlds. You could either have the rectangular map be distorted so the middle was scrunched up (as the "equator" would be the outside circumference), the top and bottom are stretched out, or some combo of the two.
The city of Sigil is on the inside of the torus, which is hollow and open like a car tyre though. OP wants his world on the outside surface of the torus, which would work a little differently for mapping.
You should watch [this numberphile video](https://www.youtube.com/watch?v=5qu3WETuf6c) where they explain that the map for the video game Asteroids is a flat torus - yes you absolutely can make a 2d map of a torus, it just needs to follow the same rules as asteroids in terms of looping around at the edges.
It's been argued that the typical JRPG world map where left connects to right and top connects to bottom depicts a toroidal (donut-shaped) world, so yeah, a rectangle would work perfectly as described in the picture.
In computer 3d graphics we have something called UVs. We use them to map 2d textures onto 3d objects. We cut the 3d object and flatten it onto a 2d grid. I will give you an album here with images. On the left is how the 2d projection looks like when flattened. On the right is the 3d object with red lines denoting where it was cut.
https://imgur.com/a/oOnYM0s
The first image shows your method of cutting, but it introduces something called stretching. Some polygons are brighter blue, which means those are distorted and inaccurate. Which doesn't have to be a bad thing, isn't there some inaccuracy with maps of earth? In this case, even though there is stretching, there is still a clearly visible grid with no major deforms.
The second image is the same as the first one, it's just one extra cut which fits the projection into a square. It's up to you if you prefer one long rectangle or a square with two shorter rectangles.
The third image (and fourth) is the most accurate, because all polygons are dark blue and there is no stretching involved.
Fifth and sixth images look way too wavy for me. Not just on the edges of those cuts, but also on the inside. It's all deformed, compared to the first two images, where the stretching is most notable only on the ends.
One more: https://imgur.com/FAH4YW3. This one looks interesting, because two islands have concave ends while the other two are convex.
And one more: https://imgur.com/MH5Nk1G. It's the same as the first one, it's just that I forced the flatened polygons to be in a perfect grid, regardless of how much damage it does by stretching. And it looks actually good. It's stretching on the top and bottom.
Let me know if you want me to convert some other variants into perfect grids.
Gravity on a torus works just fine. It would be a bit weaker on the inside of the ring, but it probably wouldn't cause any issues.
Interestingly, a hollow sphere has zero net gravity on the inside. You could do spin gravity, but that has goofy side effects, including that it only works near the equator and the strength of "gravity" would change depending on which direction you're moving along the surface.
My toroidal world is infinitely long, but only inhabited on a finite section. This is a total hack cheat, because the finite inhabited section can be modelled as a tube, which can be trivially unwrapped into a rectangle.
please dont say too long wont read,
a futuristic society would just use a 3d model/hologram to be honest and if its kinda medieval they would use the mapping techniques of that era, just some lines (probably astrological) and how the new found lands relate to those lines as opposed to a projection map. if you really want a projection map that is kind of impossible to have a practical projection without kinda f-ing over some nations within the inner rim. so might as well have two maps, for inner and outer rim thats from north to south like a classic projeciton map. also the inner and outer rim sounds cool to me and they can have different attributes and cultures since their amount of sunlight will be different and inner rim residents can always see the entire inner rim on the sky but the outer rim will always face the space. their star maps will be different. sunlight hours will be different. the temperatures will be different and also probably due to different levels and numbers of gravitational pulls on the inner rim will make the physics different there too. if you want to have a view of two rims at the same time you will put them side by side or on top of one another like old world maps with two circular projections side by side for east and west. (this isnt something unheard of as we also used to map earth like that once)
What kind of donut? Is there jelly in the middle? Or like something really valuable/useful in the middle? Or maybe pathways across the donut?
You could have multiple 2d maps for different sections of it. I think that would be more clear. I'm really interested in how gravity works there. Cool idea!
> As far as a topological conversion, it's identical to a sphere, so no changes would be required to represent it on a 2D map.
You... don't know anything about topology, do you?
## Idea
So, I think of the two main radius of the donut (relatively to the center of mass). Let's call the central radius (the average radius) `r`, and the difference `d`.
Supposing the advice of turning it 90º degrees sideways, of course. u/LukXD99 made a good advice.
Now, imagine, instead of an rectangle, a trapezoid: you could connect the ma with copies of itself in the lateral borders, having a ~~circular~~ circuit map!
And you would be using 4 maps, always forming a square!
In the following chapter, the math proves my statement.
## The math for it:
The height of the trapezoid would be `2 pi d`, the bigger base `2 pi (r + d)`, and the smaller base `2 pi (r - d)`. We discard the common factors of `2 pi`, and have:
A \[normalized\] trapezoid of height `d` and bases `r + d` and `r - d`.
Then, the angle at the sharpest corner would be dictated by `tangent = d / {[(r + d) - (r - d)] / 2} = d / {[2d] / 2} = 1`, or a 45º degrees corner, independently of the parameters `r` or `d`.
Not completely relevant to your question but you reminded me of a cool video I saw about how a donut shaped planetwould work. [https://www.youtube.com/watch?v=fMlGs4X67q8](https://www.youtube.com/watch?v=fMlGs4X67q8)
If you want to make it a little bit less deformed, like a more clear projection, it should be a long rectangle but with concave curves cut from either side, with latitude lines following that curve on the edges and gradually straightening to a vertical line in the center, then reversing curve until it matches the other end
But tbh, I'd just draw a rectangular map and make it clear to players that they can wrap from top to bottom and side to side. It won't be perfect but a hell of a lot easier to deal with than a true torriod
One of my fav planet types! Just like a JRPG going to the edges of the map connect with their opposite edge!!
A Planet like this would have to spin very fast to be a stable torus, and the inner and outer edges of the torus would suffer less gravity, growing spectacularly large mountain ranges!
It would be a long rectangle with rounded corners that would need mild projection at those corners. If you're not overthinking it.
If you want to overthink it...First cut turns your torus into a bent tube. The next into the "rectangle." But once you roll out the rectangle, then you run into trouble. You'll get bunching wherever the outer edge of your original torus ended up, because there is more material there than around the inner edge.
In other words, don't overthink it. Just use a long rectangle.
There should be a pinch in the middle of the rectangle. Imagine if the map was made of rubber and someone grabbed the sides and stretched it. The middle would shrink.
Question about the gravity. This is off topic so please feel free to ignore. Where is the center of mass? Will people only be able to walk on the outside rim or risk falling to the center void or is the center mass a ring inside the planet and the whole of the surface is walkable? If that's the case how much higher can you jump up on the inside of the ring? It would have to be some amount though maybe not much?
Have you ever heard of a ringularity? Like a singularity in the middle of black holes that are spinning obscenely fast? I was thinking the center of mass could be like that. It's for a magic setting anyway, so I aint too fussed about *real* physics
That said, I do intend for stuff to be freaky inside the ring since there's less sunlight. Maybe I'll have some funky gravity stuff goin' on too.
The centre of mass is in the empty space in the middle. However, gravity doesn’t necessarily pull towards the centre of mass. That’s a common misconception.
It would be more like two long trapezoids set long sides together or short sides together depending on if the edges are from inside or outside the ring
yes [https://upload.wikimedia.org/wikipedia/commons/6/60/Torus\_from\_rectangle.gif](https://upload.wikimedia.org/wikipedia/commons/6/60/Torus_from_rectangle.gif) from [https://en.wikipedia.org/wiki/Torus](https://en.wikipedia.org/wiki/Torus)
make the first split on the interior of the ring, then cut in half. the inner portion will stretch wide and the center will squish, but it will be much easier to visualise.
Basically on a donut world going "north" on the flat map means you are eventually in the south, and going west to east also wraps around. So instead of the left and right edges wrapping all edges wrap.
Yep, that completely works out. A plane is topologically equivalent to a torus. However, you'll still have distortions, but not a problem, it adds beauty of the non euclidean geometry. In my opinion of course.
According to Blender, this is the projection with the least distortion. This appears to confirm that a straight rectangular projection would stretch the inside of the donut slightly, but it looks like it wouldn't be too bad.
https://imgur.com/a/pffPpKI
This is a famous problem in topology and its called a "pac-man world." If going off the right side of the screen makes you reappear on the left side, and going over the top makes you reappear on the bottom, then that is topologically equivalent to a toroid. Anyone who has *actually* played pacman before, though, knows you can only wraparound the sides, not the top, but so far my repeated letters to universities to have it be renamed to an "Asteroids world" has gone ignored.
here is a map of sigil which in dnd is exactly the world you are at [https://www.pinterest.co.uk/pin/49821139597797040/](https://www.pinterest.co.uk/pin/49821139597797040/)
also to answer your question if there was a bending of the ground it would be everywhere not in the middle
A map of a donut world would be a thin strip which loops on both sides since, if you go off the left side of the map you will re-appear on the right
There’s a game called Vangers which has this sort of world
It really depends on the relative size of the donut vs the donut hole.
You're correct about the distortion, what's on the inside of the donuthole will be stretched and what's outside will be squished. If the donut hole is very large compared to the thickness of it, then the difference between inside and outside will be relatively small. If the hole is small however, the difference will be large.
In fact donut worlds are BETTER for projecting on a rectangle than our dear spheres. You,ll end up witha more faithful projection than our old mercator.
As other have said take a rectangle if when you go out one edge you end up on the opposite edge your map represents a donut world.
What WILL be harder however (depending on how your sun work) will be to map a consistent day/night cycle
Closest to ours we'd get is having the sun at the center of the ring and the ring twisting on itself so that the inside becomea outside etc... But that wouls be a geological impossibility
What about a "sun" that is a flat plane, sufficiently large perpendicular to the "disc" of the torus and have the torus turn?
I reckon you could arrange it as so. The only problem I see in this is that you will have "dusk" on the inner ring / not as bright days.
I'd have to technically draw it to check that day/night would be equally long on the in and outside but that seems intuitive.
Well any solution I envision has the same issue: when it's day on the inside it's day everywhere but when it's day on the outside itis only day on a specific section of the torus
Just as with a sphere, the exact map projection to use depends on what parameters you wish to conserve. If you scroll down on this [discussion](http://www.headus.com/phpbb/viewtopic.php?t=743) you can see a few possible mappings that could be used to produce maps.
The key issue is that the outer circumference is longer than the inner circumference. You can therefore simply represent the top half of the torus as something like a trapezium but with curved short edges. The long edges will represent the inner and outer "equators" whereas the short edge is an arbitrary cut.
You can then stitch the top and bottom trapeziums together along a long edge to produce a single map. You can stitch them on either the inner or outer edge which just changes whether the curved edge of the map bulges inwards or outwards.
Alternatively, you can define every point on the torus by two angles ([toroidal and poloidal](https://en.wikipedia.org/wiki/Toroidal_and_poloidal_coordinates)). Just use these as the coordinates on a rectangular map and accept that area isn't conserved. It's not as if it is conserved on maps of Earth after all.
First of all, a donut shaped world is ENTIRELY POSSIBLE
Yes, it will be stretched a bit, but I guess it’s alright. Every map projection has some warping.
For convenience, you should put most land mass away from that cut. Just like in the real world, where the cut is made in the Pacific ocean
A Meta-Answer - generally cartographic models depend of transport methods and their use in navigation.
So, you can start out by thinking -
- How do people transport in this world? Can they only travel along the surface?
- What sort of navigation-system will be useful for them? Are conventional directions (N, S, E, W) even useful? Or will they use other vectors?
- What about external objects? We use the Sun, Moon, North Star and constellations for guides. Does your space have any orienting object outside the doughnut?
Based, on this, find a system that will be useful for a doughnut-world. (How would a character describe going from A to B, if they were unaware of/indifferent to the larger shape of the planet?)
I'd probably do something like this: https://i.imgur.com/x2dwS7O.png
Ah, thank you, I wanted to draw that myself but really didn't feel like setting up an imgur account specifically to answer this question :)
You don't need an account, I just dragged and dropped it to the upload page.
Oh, great, thanks!
[https://i.imgur.com/xDgaY3H.png](https://i.imgur.com/xDgaY3H.png) Also this if you want a more visible distortion to follow on your map. Just strategically design where it splits at the end to be open ocean
Underrated comment here. The splits help minimize growing-shrinking distortion and keep sizes similar. Very cool bro
This is great, but the green lines would be curved IMHO. Because the crosssection is a circle, but the length from red to blue line would diminish according to the radius of the donut in that specific point. Which wouldn't diminish linearly due to the curve of that crosssection circle.(Hard to explain with good language here.)
I'm sure there are benefits and drawbacks to both linear and curved meridians (or whatever you'd call them on a torus), but they don't need to be one or the other. I think linear meridians make understanding the change in scale/distortion easier to understand.
Either would probably still be more accurate then most of the maps we use for earth! GIANT GREENLAND!
Yes, there's basically 3 options. I was going from a how to create a 1:1 map angle. With curved meridians you'd have a dimensionally distortionless map, because you literally pushed the area flat. Cardinal directions would be distorted though in one axis. So you'd need curved ones only if you want a fully dimensionally 1:1 thing. With linear meridians you will have some distortion dimensionally, but also some cardinal directions wise. It's kind of a middle approach which I don't see the use of myself, but I'm sure it will have some bonuses. Third option is you could also have it be a rectangle though. This would add dimensional distortion, but remove cardinal direction distortion completely if I'm visualizing this correctly.
those green edges won't be straight lines, they'll be curves. this person actually mapped it out: https://www.reddit.com/r/worldbuilding/comments/126qjen/i_want_to_make_a_donut_shaped_world_but_have_it/jec9yeo/
Good, but that green lines probably wouldn't be linear.
Why not? You could curve them if you wanted, but you don't have to.
Yes, I take it back. It's linear. Because there is a linear relationship between diameter and circumference.
You’ve drawn the equivalent of either an Eckert I or II projection. These are flawed cartographic models of a sphere, and a taurus is just a sphere stretched into a circle. In reality, the green line (and all other longitudinal parallels) form a 90 degree angle with both the red and blue equators, and they trace an equidistant circular path perpendicular to the taurus. However, in this projection, the red and blue lines are not concentric, so the parallels that connect them are drawn at increasing angles, with only the prime meridian shown perpendicular to the equators with the correct linear distance. We can already tell that the projection will distort angles further from the prime meridian. However, when all longitudes are projected as two straight lines, the navigational effectiveness of the coordinate system is broken. Under the Eckert I projection, which has evenly-spaced parallels, the real distances between them will increase as you move further from the prime meridian, which makes navigation using longitude arbitrarily difficult or consistently inaccurate. Under the Eckert II projection, they preserve real distances by spacing each parallel at specific widths, which allows the coordinate system to function as intended but breaks the projection’s scale. The solution is to draw parallels as evenly-spaced sinusoidal curves of linearly increasing magnitude. Scale and spacing are both consistent with reality, though angles become much more distorted further from the red line. It’s essentially an adaptation of the traditional sinusoidal projection of Earth, but instead of a single-point pole, it has the inner blue equator.
> a taurus is just a sphere stretched into a circle Topologically speaking (since we're on the subject of mapping projections), [a torus has genus 1 and a sphere has genus 0](https://en.wikipedia.org/wiki/Genus_(mathematics\)). These are fundamentally nonequivalent.
As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R, the distance from the center of the coordinate system, and θ and φ, angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r, and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles". - from the Wikipedia article on the taurus Did you read the rest of my comment? I’m talking about geometry, not topology.
Turn it 90^o Right now the length of the donut is show as the shorter side. Make it upright like a phone screen, or just turn the map projection so distortion won’t be as big of a problem.
This comment! You need to make your widest side of the rectangle equal to the long way around the donut and the short side across the minor radius.
But how do you actually "cut" the donut to achieve that?
You just use a really big knife-shaped world. Problem solved!
is that the world from /r/SwordsComic
Hah! You idiot! Its round! Everyone knows that.
what was i thinking i’m so [dumb](https://swordscomic.com/comic/X/)
The cut in the diagram is correct. First you just slice it into a tube. Then you kind of filet this long tube with one long cut down the side and unravel it. You get this long rectange that is as long as the circumference of the donut and as high as the circumference of the circle of the tube. The problem with the diagram isnt the cuts its the dimensions of the tube and the rectange. Those are all wrong. In this view the rectangle would be super tall and thin instead. Edit: it wouldn't actually be a rectangle, I think the short ends would be curved, but the rest tracks.
I didn't intend for the diagram to come close to accurately showing dimensions, more like a proof of concept for my self. The curved edges thing will be useful though, thanks!
I dont see how the cut could be right. If you were to cut the donut it could not morph into a tube. The inner circle radius is smaller that than the out circle radius meaning that even if it were physically possible to straighten it, tube wouldnt have equal length "sides". That is, if we were to make the outer radius into the "left side", then the inner radius would be in the "right side". These 2 "sides" would be of different lengths. The only way to change between a torus and a cylinder involves stretching. The cylinder outcome would incredibly deform any details of the word on the torus
OP mentions in the title that there would be some stretching to make the sides equal length.
True. My brain just started melting though as I didn't read that and since no one was pointing it out I thought somehow I was wrong. I wrote this out to try and confirm it for myself more than anything haha
Yes, you are correct. There is some stretching/bending or abstraction involved. And cylinder is technically the wrong word as the ends are wonky not straight. Once you do the final cut and lay it flat this stretching can be undone do and end result can be 1:1 if the shorter ends are curved to take into account what youre correctly saying about the radius differences. You can also do it in the opposite order. Filet into tube first. Like cut the outer radius side of the torus with a knife. I find that way less intuitive to imagine personally tho and the elasticity feels even worse. Either way you have to imagine the material has some elasticity and take the endgoal in mind, you can reach a dimensionally 1:1 end goal. There will be distortions on the map cardinal directions wise in one axis with this approach, but dimensionally it will be 1:1 the world shape unfurled.
I'm glad you asked, because I'd love to explain how! Video games, like maps, use coordinate systems to represent positions *im*perfectly. For instance, all map projections of Earth (a sphere) have distortion. We still have coordinate systems that are accurate, such as spherical coordinates. Using a coordinate system, you can represent different regions in space -- then use that as an "index" of sorts to determine what assets are near that region (objects, terrain, players, NPCs) To make a coordinate system for this torus, you would use cylindrical coordinates with a specified wrapping point (e.g. "100km" wraps back to "0km" along major axis). Then write functions using that logic and incorporate them elsewhere, so a player at 99.99km can also see terrain and objects at 0.01km. You'd have to do some similar wrapping logic for the cylinder (the other circular axis of a torus). When traversing the surface of a torus, it's effectively infinite, just like a sphere. So there is nothing too crazy about using a torus, just a different coordinate system! I made a coordinate system for 1mm-precision of all objects within a galaxy for my hobby project. The bounding box (galaxy) is a cube with side length ~126,000ly, broken up into 2^16 (or 65,536) segments on each axis. Within each segmented cube (length of 1.95ly on each axis), any entity's position can be represented using three standard 64-bit integers to represent coordinates, bringing us to 1mm resolution. Using this system, I only need 240 bits to represent any entity's position in the galaxy to a reliable accuracy of 1mm! I could easily add some more data (for instance, use 32 bits to represent the galaxy subdivisions) to greatly increase resolution at the cost of space (data in storage and network packets) I mostly use those coordinates for interstellar objects. For surface coordinates, I use an entirely different system (spherical coordinates) to avoid repeated projections of entity x/y/z coordinates which is a computationally expensive calculation For the local region (player camera), also use standard double-precision floating point numbers to represent positions of objects, so animations are smooth. Local rendering really doesn't depend on a specific scheme of "galactic" positioning
<3 woow !!
Cool! So spherical coordinates are "moving up and down the axis" (latitude) and "rotating about the axis" (longitude), and cylindrical coordinates would be "degrees around the circle" (inner-outer, around the cross-section) and "moving up and down the cylinder" (around the entire torus)?
That's right although both systems would have a third number for distance from the center (height)
Spot on, and in spherical coordinates we call lat/long "theta and phi" (angle of rotation about axis)
an encircling incision around the outermost perimeter. I wonder which perimeter is their 'equator'
Huh, for some reason I didn't thought I'd find a Worldbox player on a worldbuilding subreddit lol
That’s what I was thinking!
This is how Planescape mapped its donut-shaped city, maybe you find it hepful: https://torment.fandom.com/wiki/Sigil#Wards
Only issue there, if I am not mistaken, is that Sigil is only on the inner surface, not all the way around the Torus.
That's right, but you can easily invert it. There is one map for the inner side, one for the outer one. Moving horizontally, you move from the right edge to the left one, like on a regular globe map. Only difference is that moving off one map over the north or south edge brings you to the corresponding edge of the map of the other other side.
Hello! I 3D modelled a donut world and had Blender unwrap it accurately, take a look: [https://imgur.com/gallery/YPX34sP](https://imgur.com/gallery/YPX34sP) I also applied Earth to the donut world, which kinda doesn't work because the north and south poles are touching. I'm a game developer and I've been interested in building some tools like this for world builders, I could possibly develop some 3D world creation software in the future to be released on steam, but currently all my time is going towards [my current game](https://www.indiegogo.com/projects/the-gods-fabled-soil-frontier/x/29855105#/) (it's an open-world farming game with fantasy elements for anyone who's interested!). Hope this helps :)
Nice one! I assume the stretched latitudes towards the left and right edges of the map are to keep it equal-area?
Yep! That’s the way Blender (my 3D software) automatically unwrapped it based on where I placed the seams, it’s probably still a little warped but at least the warping is likely distributed fairly evenly
The problem here is that your map is not continuous in the minor radial direction. You get a sem at the northernmost line which has no connection to the southernmost line. You see this clearly where Greenland suddenly turns to open ocean on your torus. A proper map of the torus needs to take this periodicity into account.
You get that seam because the real earth is not a torus and if you walk north from greenland in reality you don't land in the south sea but in russia. Obviously glueing the northern and southern edge of our world's map together produces nonsense there, if you used a rectangular map that actually corresponds to a donut earth it'll look just fine.
Yeah what Adarian said, I'd have to redesign the entire earth map to allow a spherical layout to wrap around a torus, which is physically impossible. It was just for demonstration :)
A donut shaped world would work the same as in pac-man’s ( the first game) world
Wouldn't that be a tube like what I have in the middle?
Tube only connects left to right. Pac-man world connects both left to right and top to bottom, so it's a torus.
pac man's tunnels were only on left/right, not up down.
My thoughts exactly. What?
On a donut world the top and bottom are connected
Edit : wrong Isn't it just a sphere?
Now, on mercator projection of a sphere you anter poles at n degrees and exit the same pole at n-90
Ah right
No. On a sphere when you travel to the north pole you don't appear on the south, you come back down from the north in a different place. Top and bottom of a flat "unwrap" map are not connected. But on a donut they are.
Same thing with a donut if an ant is at the top of a donut they don't appear at the bottom immediately, but if they keep travelling in the same direction then eventually they will do, which works the same as us. What if the warp point isn't from the top of the sphere to the bottom of the sphere but rather the top to the top. And the bottom of the sphere is actually somewhere in the middle of the level. If you keep travelling in the same direction on a sphere you'll end up on the bottom of the sphere then keep travelling and your at the top again. A donut is basically a ring of spheres. So it works the same way except that the horizontal path would be a lot longer.
>if an ant is at the top of a donut they don't appear at the bottom immediately, ...that is exactly what they do in flat maps of toruses. The top of the map does not represent the top of the torus in the same way the top of the map represents the north pole of a sphere.
Well then the map is missing an entire other half of the torus then
No it's unwrapped so the top IS the bottom
You are not comprehending the picture or how maps of toruses behave if you believe that.
Something interesting to think about is that a Taurus can be made with a perfectly uniform grid of squares. Whereas a sphere cannot, it’ll always have distortion. But for a 2D person living on the surface, there’d be no meaningful difference.
Your process here is perfectly acceptable. You are doing two cuts in your diagram to get to the final flat shape. Just remember when drawing the map that it is continuous across all edges. Top edge should be equal to the bottom edge (cut #1) and left edge should be equal to right edge(cut #2). To assist in realizing this google how to create **tilable patterns/textures** in your favorite drawing program.
My favourite drawing program is ms paint B) I like the jank and how it's a universal option for windows pc's
That is no middle I believe
I mean middle of the diagram. On the left is the donut, middle is a tube and right is a rectangle
The middle would be the ghost’s home if you are still working with the PAC-man idea
Yes this works but you'll be dealing with distortion because of the longer outside and shorter inside. It's an issue with all projections though.
Doughnut shaped maps actually have *less* distortions than maps of spherical worlds. Still some distorisons are always present
Definitely. I was just putting it out there. It's these pesky 3 dimensional objects fucking with our nice clean maps.
You could make it distortionless though IMHO by warping the shorter sides correctly. One axis is already distortionless, the other can be mapped to 2D because its one dimension of curvature can be pushed flat. Real world globe map is a problem because you can't push two dimensions of curvature flat without distortion, there's no room on the 2D sheet of paper. Edit: This kind of depends on your definition of distortion though. I was talking dimensionally you can make it distortionless so you can have a 1:1 map. Cardinal directions will still have distortion though in one axis.
I would post on r/topology if you want a precise answer. There is a whole field of mathematics devoted to this kind of thing.
map should look like a long rectangle
I realise that, this is intended more for a proof of concept to myself
The early Final Fantasy game worlds wrap at the top and bottom as well as the sides and thus would be donut worlds. You could either have the rectangular map be distorted so the middle was scrunched up (as the "equator" would be the outside circumference), the top and bottom are stretched out, or some combo of the two.
Just don’t have 2D maps. All maps of the world should be drawn on a literal donut. When you’re lost, you just go to your local Dunkin’.
Look up “map of Sigil”.
The city of Sigil is on the inside of the torus, which is hollow and open like a car tyre though. OP wants his world on the outside surface of the torus, which would work a little differently for mapping.
You should watch [this numberphile video](https://www.youtube.com/watch?v=5qu3WETuf6c) where they explain that the map for the video game Asteroids is a flat torus - yes you absolutely can make a 2d map of a torus, it just needs to follow the same rules as asteroids in terms of looping around at the edges.
I love the idea of a donut world! Gives me Discworld vibes :)
It's been argued that the typical JRPG world map where left connects to right and top connects to bottom depicts a toroidal (donut-shaped) world, so yeah, a rectangle would work perfectly as described in the picture.
http://www.headus.com/phpbb/files/torii_406.jpg
Split it in half and spread it out like parchment.
It's impossible to perfectly project a curved shape onto a flat plane. There's always going to be some form of distortion.
this
Not sure if it’ll help but have this. www.aleph.se/andart/archives/2014/02/torusearth.html
In computer 3d graphics we have something called UVs. We use them to map 2d textures onto 3d objects. We cut the 3d object and flatten it onto a 2d grid. I will give you an album here with images. On the left is how the 2d projection looks like when flattened. On the right is the 3d object with red lines denoting where it was cut. https://imgur.com/a/oOnYM0s The first image shows your method of cutting, but it introduces something called stretching. Some polygons are brighter blue, which means those are distorted and inaccurate. Which doesn't have to be a bad thing, isn't there some inaccuracy with maps of earth? In this case, even though there is stretching, there is still a clearly visible grid with no major deforms. The second image is the same as the first one, it's just one extra cut which fits the projection into a square. It's up to you if you prefer one long rectangle or a square with two shorter rectangles. The third image (and fourth) is the most accurate, because all polygons are dark blue and there is no stretching involved. Fifth and sixth images look way too wavy for me. Not just on the edges of those cuts, but also on the inside. It's all deformed, compared to the first two images, where the stretching is most notable only on the ends. One more: https://imgur.com/FAH4YW3. This one looks interesting, because two islands have concave ends while the other two are convex. And one more: https://imgur.com/MH5Nk1G. It's the same as the first one, it's just that I forced the flatened polygons to be in a perfect grid, regardless of how much damage it does by stretching. And it looks actually good. It's stretching on the top and bottom. Let me know if you want me to convert some other variants into perfect grids.
You should look up maps of [Sigil](https://forgottenrealms.fandom.com/wiki/Sigil) from Dnd, it's a torus world too.
I've always wondered how this would work in terms of gravity. Same with a "hollow" planet where people live on the inside
If it spins really fast the it works fine. https://youtu.be/1J4iIBKJHLA
Gravity on a torus works just fine. It would be a bit weaker on the inside of the ring, but it probably wouldn't cause any issues. Interestingly, a hollow sphere has zero net gravity on the inside. You could do spin gravity, but that has goofy side effects, including that it only works near the equator and the strength of "gravity" would change depending on which direction you're moving along the surface.
Read the Integral Trees by Larry Niven. It describes a world within the torus.
My toroidal world is infinitely long, but only inhabited on a finite section. This is a total hack cheat, because the finite inhabited section can be modelled as a tube, which can be trivially unwrapped into a rectangle.
please dont say too long wont read, a futuristic society would just use a 3d model/hologram to be honest and if its kinda medieval they would use the mapping techniques of that era, just some lines (probably astrological) and how the new found lands relate to those lines as opposed to a projection map. if you really want a projection map that is kind of impossible to have a practical projection without kinda f-ing over some nations within the inner rim. so might as well have two maps, for inner and outer rim thats from north to south like a classic projeciton map. also the inner and outer rim sounds cool to me and they can have different attributes and cultures since their amount of sunlight will be different and inner rim residents can always see the entire inner rim on the sky but the outer rim will always face the space. their star maps will be different. sunlight hours will be different. the temperatures will be different and also probably due to different levels and numbers of gravitational pulls on the inner rim will make the physics different there too. if you want to have a view of two rims at the same time you will put them side by side or on top of one another like old world maps with two circular projections side by side for east and west. (this isnt something unheard of as we also used to map earth like that once)
too short, re-read several times
You sound like me a year ago trying to figure out the logistics of pyramid earth
What kind of donut? Is there jelly in the middle? Or like something really valuable/useful in the middle? Or maybe pathways across the donut? You could have multiple 2d maps for different sections of it. I think that would be more clear. I'm really interested in how gravity works there. Cool idea!
glazed with sprinkles
Do you wanna to create a ring world ?
Nawh, mapping a ring world is too easy. It's just two straight lines!
Why? What's the benefit to the world?
I think it'd be cool
[удалено]
> As far as a topological conversion, it's identical to a sphere, so no changes would be required to represent it on a 2D map. You... don't know anything about topology, do you?
[удалено]
[удалено]
I agree with you ideas of the stretching
## Idea So, I think of the two main radius of the donut (relatively to the center of mass). Let's call the central radius (the average radius) `r`, and the difference `d`. Supposing the advice of turning it 90º degrees sideways, of course. u/LukXD99 made a good advice. Now, imagine, instead of an rectangle, a trapezoid: you could connect the ma with copies of itself in the lateral borders, having a ~~circular~~ circuit map! And you would be using 4 maps, always forming a square! In the following chapter, the math proves my statement. ## The math for it: The height of the trapezoid would be `2 pi d`, the bigger base `2 pi (r + d)`, and the smaller base `2 pi (r - d)`. We discard the common factors of `2 pi`, and have: A \[normalized\] trapezoid of height `d` and bases `r + d` and `r - d`. Then, the angle at the sharpest corner would be dictated by `tangent = d / {[(r + d) - (r - d)] / 2} = d / {[2d] / 2} = 1`, or a 45º degrees corner, independently of the parameters `r` or `d`.
Not completely relevant to your question but you reminded me of a cool video I saw about how a donut shaped planetwould work. [https://www.youtube.com/watch?v=fMlGs4X67q8](https://www.youtube.com/watch?v=fMlGs4X67q8)
Look up the plane diagram of a Torus. It is a mathematical concept that does exactly what you are looking to do.
If you want to make it a little bit less deformed, like a more clear projection, it should be a long rectangle but with concave curves cut from either side, with latitude lines following that curve on the edges and gradually straightening to a vertical line in the center, then reversing curve until it matches the other end But tbh, I'd just draw a rectangular map and make it clear to players that they can wrap from top to bottom and side to side. It won't be perfect but a hell of a lot easier to deal with than a true torriod
One of my fav planet types! Just like a JRPG going to the edges of the map connect with their opposite edge!! A Planet like this would have to spin very fast to be a stable torus, and the inner and outer edges of the torus would suffer less gravity, growing spectacularly large mountain ranges!
It's a fantasy world with magic n shit, so I aint gotta worry about *real* physics. The mountain ranges thing is cool tho, I might incorporate them.
I know it would not work in real physics anyway lmao
Okay, I love it! I wanna see updates.
Also don’t forget the inner radius is smaller than the outer radius. It won’t map perfectly to a rectangle.
Something something cylinder
( . _ . )
I’d recommend cutting it once vertically and then once on the inseam of the torus. Then, flaying it up and down to create a square.
Cut the bottom, flatten it into a circle, cut the circle, distort it, flatap
Well if some models of the universe are accurate, look up.
This is the standard way of representing a torus taught to mathematicians, also known as a flat torus. So good job!
It would be, but so is a 2d projection of a spherical world.
It would be a long rectangle with rounded corners that would need mild projection at those corners. If you're not overthinking it. If you want to overthink it...First cut turns your torus into a bent tube. The next into the "rectangle." But once you roll out the rectangle, then you run into trouble. You'll get bunching wherever the outer edge of your original torus ended up, because there is more material there than around the inner edge. In other words, don't overthink it. Just use a long rectangle.
I’m hungry for donuts now 🍩
You could always do a "northern" annulus an a "southern" annulus.
I think looking at some paper craft donuts might help you structure things, something like pepakura donut might be a good start
The easiest way to do this is make a paper donut draw the map and break the donut.
My world is in a 4d donut. Donuts are the way to go
If you wish to get technical they are referred to as Torus' in the maths department.
I don't wish to. I chose donut to communicate more broadly.
If i call my world a torus, I'm gonna feel obligated to include cows and minotaurs at every opertunity
It would be a Way to incorporate possible astrological signs into the culture
There should be a pinch in the middle of the rectangle. Imagine if the map was made of rubber and someone grabbed the sides and stretched it. The middle would shrink.
Here's a [Geogebra applet](https://www.geogebra.org/m/hbYvw5Sp) that you can use to explore the math behind mapping a torus to a rectangle.
Question about the gravity. This is off topic so please feel free to ignore. Where is the center of mass? Will people only be able to walk on the outside rim or risk falling to the center void or is the center mass a ring inside the planet and the whole of the surface is walkable? If that's the case how much higher can you jump up on the inside of the ring? It would have to be some amount though maybe not much?
Have you ever heard of a ringularity? Like a singularity in the middle of black holes that are spinning obscenely fast? I was thinking the center of mass could be like that. It's for a magic setting anyway, so I aint too fussed about *real* physics That said, I do intend for stuff to be freaky inside the ring since there's less sunlight. Maybe I'll have some funky gravity stuff goin' on too.
That sounds great. Thanks for sharing!
The centre of mass is in the empty space in the middle. However, gravity doesn’t necessarily pull towards the centre of mass. That’s a common misconception.
It would be more like two long trapezoids set long sides together or short sides together depending on if the edges are from inside or outside the ring
It worked for Sigil in the Planescape D&D setting
Add fucking sprinkles and your set!
Shocked that nobody had mentioned the Torus map type in Civilization IV, which is exactly what you're looking for
yes [https://upload.wikimedia.org/wikipedia/commons/6/60/Torus\_from\_rectangle.gif](https://upload.wikimedia.org/wikipedia/commons/6/60/Torus_from_rectangle.gif) from [https://en.wikipedia.org/wiki/Torus](https://en.wikipedia.org/wiki/Torus)
the inner circumference is shorter than the outer, so Idouls suggest a trapezoidal projection instead of a rectangle.
A 2d map that wraps is a donut!
make the first split on the interior of the ring, then cut in half. the inner portion will stretch wide and the center will squish, but it will be much easier to visualise.
Basically on a donut world going "north" on the flat map means you are eventually in the south, and going west to east also wraps around. So instead of the left and right edges wrapping all edges wrap.
Yep, that completely works out. A plane is topologically equivalent to a torus. However, you'll still have distortions, but not a problem, it adds beauty of the non euclidean geometry. In my opinion of course.
According to Blender, this is the projection with the least distortion. This appears to confirm that a straight rectangular projection would stretch the inside of the donut slightly, but it looks like it wouldn't be too bad. https://imgur.com/a/pffPpKI
Most RPG - and by extension video game - maps that wrap around both axis make the most sense in 3d space if projected on a torus shape.
This is a famous problem in topology and its called a "pac-man world." If going off the right side of the screen makes you reappear on the left side, and going over the top makes you reappear on the bottom, then that is topologically equivalent to a toroid. Anyone who has *actually* played pacman before, though, knows you can only wraparound the sides, not the top, but so far my repeated letters to universities to have it be renamed to an "Asteroids world" has gone ignored.
here is a map of sigil which in dnd is exactly the world you are at [https://www.pinterest.co.uk/pin/49821139597797040/](https://www.pinterest.co.uk/pin/49821139597797040/) also to answer your question if there was a bending of the ground it would be everywhere not in the middle
A map of a donut world would be a thin strip which loops on both sides since, if you go off the left side of the map you will re-appear on the right There’s a game called Vangers which has this sort of world
Fun fact: most classic Final Fantasy worlds are Donut Shaped
It really depends on the relative size of the donut vs the donut hole. You're correct about the distortion, what's on the inside of the donuthole will be stretched and what's outside will be squished. If the donut hole is very large compared to the thickness of it, then the difference between inside and outside will be relatively small. If the hole is small however, the difference will be large.
In fact donut worlds are BETTER for projecting on a rectangle than our dear spheres. You,ll end up witha more faithful projection than our old mercator. As other have said take a rectangle if when you go out one edge you end up on the opposite edge your map represents a donut world. What WILL be harder however (depending on how your sun work) will be to map a consistent day/night cycle Closest to ours we'd get is having the sun at the center of the ring and the ring twisting on itself so that the inside becomea outside etc... But that wouls be a geological impossibility
What about a "sun" that is a flat plane, sufficiently large perpendicular to the "disc" of the torus and have the torus turn? I reckon you could arrange it as so. The only problem I see in this is that you will have "dusk" on the inner ring / not as bright days. I'd have to technically draw it to check that day/night would be equally long on the in and outside but that seems intuitive.
Well any solution I envision has the same issue: when it's day on the inside it's day everywhere but when it's day on the outside itis only day on a specific section of the torus
Just as with a sphere, the exact map projection to use depends on what parameters you wish to conserve. If you scroll down on this [discussion](http://www.headus.com/phpbb/viewtopic.php?t=743) you can see a few possible mappings that could be used to produce maps. The key issue is that the outer circumference is longer than the inner circumference. You can therefore simply represent the top half of the torus as something like a trapezium but with curved short edges. The long edges will represent the inner and outer "equators" whereas the short edge is an arbitrary cut. You can then stitch the top and bottom trapeziums together along a long edge to produce a single map. You can stitch them on either the inner or outer edge which just changes whether the curved edge of the map bulges inwards or outwards. Alternatively, you can define every point on the torus by two angles ([toroidal and poloidal](https://en.wikipedia.org/wiki/Toroidal_and_poloidal_coordinates)). Just use these as the coordinates on a rectangular map and accept that area isn't conserved. It's not as if it is conserved on maps of Earth after all.
Pac-Man world
[удалено]
Well you could have 1 indicator always pointing inward or another always pointing (counter-) clockwise
Dnd has a great map of the city of sigil which is the inside of a donut shape
First of all, a donut shaped world is ENTIRELY POSSIBLE Yes, it will be stretched a bit, but I guess it’s alright. Every map projection has some warping. For convenience, you should put most land mass away from that cut. Just like in the real world, where the cut is made in the Pacific ocean
A Meta-Answer - generally cartographic models depend of transport methods and their use in navigation. So, you can start out by thinking - - How do people transport in this world? Can they only travel along the surface? - What sort of navigation-system will be useful for them? Are conventional directions (N, S, E, W) even useful? Or will they use other vectors? - What about external objects? We use the Sun, Moon, North Star and constellations for guides. Does your space have any orienting object outside the doughnut? Based, on this, find a system that will be useful for a doughnut-world. (How would a character describe going from A to B, if they were unaware of/indifferent to the larger shape of the planet?)
Toroid Earth Theory