Projection of a Pentagonally-tiled Sphere

Question

I know that a regular pentagonal tiling does not work in Euclidian space, but does work on a sphere. But this got me wondering something that I hope people can help with here, because I can't find any way to do it myself.

All projection software I can find requires you to give the projection and it reverses it to apply it to the sphere, instead of vice versa. And looking it up I can't find any papers on the projection of a sphere tessellated with regular polygons that cannot tesselate Euclidian space.

So what would projections of a Sphere tiled with 12 regular pentagons look like? And around how much of the pentagonal shape can you keep by projecting. It would be helpful if you can find a way to show images, if that is not too big of a bother.

Edit: For those that might misread this, I am perfectly aware that pentagons can tile the sphere, as I have already stated. My question is about projections of said spherical tiling onto the plane, where I know regular pentagonal tiling doesn't work. And how much is preserved upon such projections. I hope this clarification helps.

Edit: I particularly want to see a graph that displays the distortions in size, angle, and distance between points from this. To see how much is actually preserved by the projection.

Answer 1

Perhaps the reason that there are no papers on this topic is because they are very classical topics that are explained in (sometimes very old) mathematical books or taught in mathematical courses.

For example, Euclid's Elements contains the theorem which, in modern language, produces all possible regular tilings of a sphere, including the regular dodecahedron tiling which can be represented as a spherical tiling.

Here is another wikipedia link depicting lots of spherical tilings. A key sentence you can read at that link:

All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings...