Why are spherical worlds so rare?

Because spherical maps, compared to rectangular ones, create a lot of additional complexities regarding the technical implementation and the UI design while usually offering very little gameplay advantage.

First, there is the technical problem. With a rectangular map, you just use a 2-dimensional array to represent map positions. But unfortunately there is no proper way to map a 2-dimensional plane to a 3-dimensional sphere. So if you want your world to be tile-based, then that is usually incompatible with a spherical world (unless you are willing to accept some weird artifacts). And even if your game mechanics are not based on tiles: common tasks which are trivial on a plane, like measuring distances, calculating angles or detecting collisions between geometrical shapes, become a lot more complex when you are doing them in a spherical coordinate system.

The science of doing geometry on curved surfaces is called Non-Euclidean Geometry. If you want to dive deeper into the mathematical details, then this would be the search term to start your research with. But be warned: It drove various mathematicians crazy for over a millennium.

Then there are the UI problems. There is no good way to show the player the whole globe at once. The player can only see half of the game world at a time, and everything near the edges becomes distorted. Also, globes are difficult to navigate. If you let the player rotate the globe around the world axis', you encounter gimbal lock around the poles, which results in very awkward controls. If you let the player rotate the globe relative to their viewport, then the globe will end up in strange orientations and the player gets disoriented. A plane is usually far more intuitive for the player.

This UI problem is so prevalent that even real-world applications where it is important to remember that the Earth is round still often decide to visualize the Earth as a distorted plane instead. Like NASA mission control software, for example.

But dealing with those complications would be worth it if a spherical game world has a notable advantage over a rectangular one. But there are very few game concepts where the shape of the world would make a big difference.

If you wonder if the game you are designing right now should have a truly spherical map or if a planar map would suffice, ask yourself this: Do you have any game-mechanical reason besides "realism" why you want the map to be a sphere? If you can't think of any important game mechanic which would require a sphere and couldn't work on a plane, then do yourself a favor and don't torture yourself with trying to pull this off.

For anyone considering this...

A tile-based spherical world is possible, but you need to carefully consider the topology of your world sphere. And you need to decide what to do about the poles of the topology.

For example, there is a rather neat Catalan solid called the Rhombic Triacontahedron, which would give you a "globe" with each face being a rhombus of the same dimension. You can use this polyhedron's topology, but make it more spherical. The way to achieve this is by subdividing the faces into smaller rhombi, and then spherizing the whole mesh again. This results in a higher count of rhombic faces and thus a higher-resolution spheroid.

^{Above: A rhombic triacontahedron created in Blender (using "Extra Objects": Add → Mesh → Math Function → Regular Solid)}

Then you can consider each subdivided rhombic region to be its own x/y grid.

Put them together, and you've got a continuous world.

Where it gets tricky is that the fundamental rhombic topology has twenty 3-poles and twelve 5-poles (30 faces), which are illustrated by the red seams I've marked in the GIF. So where those meet, your illusion of a rectangular grid falls apart, as it becomes noticeable that the neighboring quad is angled. How to work around this? You can either model the structures on your world a little off-grid in those areas, so they retain the appearance of continuity, or you can do some refraction-like fakery with the render engine to render each rhombic region as if it's actually a square grid - and then hide each pole by covering it with impassable geography (mountains?).

Should you decide you want to make an epic LoD "zoom-out" into space, that will probably require additional planning and testing.

As fun as the rhombic triacontahedron is conceptually with its unique uniformity of faces, the skewed grid shape and the large number of poles in the topology make it somewhat less than ideal. What might be a better option is the "roundcube". A roundcube is simply a spheroid with the topology of a cube. It has only six face regions, and eight 3-poles. Many readers are probably familiar with it, as it is a common shape for 3D modelers to use. The faces at each of those poles have noticeable distortion, where they get pinched in one corner. But that's not a deal-breaking limitation.

^{Above: A Roundcube created in Blender (using Add → Mesh → Round Cube)}

Either way, expect experimentation to be required. And maybe a little clever math, so that the world renders correctly (and physics of projectiles etc behave correctly) at the poles. But it's certainly possible, given that the topologies I mentioned allow for sectioned x/y grids to be mapped onto them. And when dealing with something the scale of a planet, the vast scale may make it easier to conceal any anomalies that might crop up at the poles. After all, those are really tiny points in the global scale of things.

They are rare because the coders are unimaginative, or scared of the geoemtry of it all.

A game that uses spherical worlds is Super Mario Galaxy. So, I’m a Nintendo fanboy LOL

You can use spherical triangles, made up by a convex hull of a set of points in 3 dimensions. If you prefer hex-based maps, you can instead use the dual to the triangular mesh: a Voronoi diagram, which is made up mostly of hexagons, sprinkled with some pentagons and heptagons. The entire C++ code is at: https://github.com/sjhalayka/spherical_triangle

Here we show the Delaunay triangulation of a set of points on a 2-sphere. We used spherical (curved) triangles. The points have gone through many iterations of repulsion, to spread the points out more evenly.

Here we show the dual of the Delaunay triangulation -- the Voronoi diagram. We have drawn the edges of the pentagons-hexagons-heptagons in orange. These n-gons can also be drawn using spherical triangles, if one goes with the hex method.

I used TetGen to do the triangulation and Voronoi diagram generation. It's extremely easy to use this TetGen software library, once you get the hang of it. If the end shape is reminiscent of a soccer ball, it’s because they both follow the same mathematical pattern, in regard to Euler’s formula: vertex count - edge count + face count - 2 = 0.

As for difficulties with game mechanics, the sphere is a very special non-Euclidean shape, and it boils down to paths that lie on great circles. So it's not super difficult. Also, Newtonian gravitation is a little trickier, but not super difficult. I can provide further details if required.

Note that there are no singularities nor slightly overlapping patches, like often done in differential geometry. This is the beauty of discretizing the 2-sphere.

Curved hexagons, etc:

Determining neighbours is relatively easy:

The Americas: