I find it a funny geometric coincidence since 12 pentagons makes a dodecahedron which is a 'sphere' (okay, spherical polyhedron, sue me.) So you need a sphere to finish the hexagon sphere!
Also fun fact: you can make a toroid out of regular hexagons:
Although displaying this in a 3d fashion in game would start to get trippy.
i know like 10 people have already replied, but certain shapes can never make closed objects in 3+ dimensions. For triangles, they always can form some kind of shape. Imaging trying you have 3 (equilateral) triangles, and you want to put them together so they all meet at a single point, and touch to form a sort of cone. Well that's basically the top of a pyramid. If you use 4 triangles you will also get the top of a pyramid, but now its a wider cone. If you use 5 triangles, you get the top of an icosahedron (a 20 sided die) and it's even shallower. To fit 6 triangles around a point, you end up making a hexagon, which is perfectly flat! In the same way, this is why you can almost make a sphere of hexagons but you can never make a perfectly closed one. For those who are nerds, look up Euler's Polyhedral Formula for a real explanation.
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Anyways...
If they brought the map itself to be able to be translated between flat and spherical - like civ4 had- then you could get a lot of what people want (zoom out and see a globe) without making a mess of the internals.
The bigger problem is so much of the game is built on using a grid of (X,Y) which perfectly maps a cylinder, and a la civ4 you can stretch it a little to cover the areas between the poles of a sphere.
But in a true spherical map, the grid breaks down because different "latitudes" have different numbers of tiles in each "row". (they get thinner/fatter depending on how north/south you are.)
Okay, so, we need to introduce a new table to tell the game what tiles touch what tiles. With that go-between, there's no reason you couldn't have pentagons on the map; of course, that would add some issues with graphics- you would need hexagon and pentagon supported designs for districts, for example- but I don't think I can emphasize enough how much extra work that "translation" step to connect tiles would make for virtually every subsystem. You can't really just say "range of 2 tiles," you need to search for every tile within two tiles, for example. It loses the intuition aspect.
This might work better in a beyond earth style game than a historical civ game, though, where you have flying ships and satellites and such. And in a space setting, a 3d map space would even allow you to add in other geometries. For example, a moon or second celestial body that players could get to and play on. A binary system of planets would make for quite the "terra" or continents map... A ringworld map. A ringworld encircling the planet! Etc.
I would totally be a fan of a improved civ4 style "zoom out and you see a sphere" thing though. Imagine zooming and seeing hurricanes from space...