For the purpose of game development, I'm trying to figure out what uniform shapes can be used to create a approximated spherical object.
To give an example of a shape that does not fit my criteria, consider the shapes that compose a soccer ball:
Because the soccer ball has 12 regular pentagonal faces (5 sides) and 20 regular hexagonal faces (6 sides), the shapes are not uniform (they have a different number of sides and a different geometric area).
After a few hours of research online (both here on math.stackoverflow.com and on Google / Wikipedia) the only shape I was able to come across that does match my criteria is the "regular dodecahedron" because it is a sphere-like object composed of only pentagonal faces (5 sides) each with an equal geometric area.
How can I find more geometric shapes that fit my criteria, or what terms could I use to research more in this area?
You may want to look into Catalan solids. These are polyhedra in which every face is identical, and the larger ones make pretty good approximations to spheres. The faces are not regular, however.
Note however that there are only 13 Catalan solids. If you decide that none of them are a good enough approximation to a sphere for your purposes, you will likely have to relax your uniformity requirement.