I really like the logo at Worldbuilding SE; it features a polyhedral planet and a beautiful city with a flying whale ship. That planet is particularly neat. Light glances off its facets, resulting in what would be called "fire" in a giant gem; scattered light scintillates orange and green and overlays of both. It's beautiful and interesting.
But could it exist?
It appears to be a Icosidodecahedron (https://en.wikipedia.org/wiki/Icosidodecahedron), with one round of triangular infilling applied. If you kept subdividing those triangles into more triangles, you eventually end up with a shape indistinguishable from a sphere.
Now, we all know spherical planets can exist; in fact, posting questions positing worlds that are squares, donuts, or, in the case of one special case of idiot, a banana, receive fairly strident criticism for ignoring the effect of gravity. All the same, Earth has "small" irregular features like mountains that are not eliminated by gravity, not in any directly observable timeframe, anyway.
My question is: Let's assume Logoworld (the Weltlogo?) has earthlike mass and composition but no water. Can the planet exist in its depicted shape - an icosidodechedron with one degree of triangular infilling? Or will gravity eliminate the facets? The edges need to be sharp enough to be visible from the moon under optimal conditions, with the naked eye.
Please ignore: 1) Rotation and/or other secondary effects that distort the planet by a percent or two, 2) Logoworld's ring. 3) Atmosphere, oceans, erosion and other planetary features other than gravitational rounding that urge you to shout FRAME CHALLENGE!, 4) The floating whale ship and its city, 5) the orbiting capital letters. EDIT: No explanation of the origin of the facets is necessary, let's take it that it was made that way deliberately by its creator.
These pages may be helpful: Truncated Icosahedron (https://en.wikipedia.org/wiki/Truncated_icosahedron#Orthogonal_projections)
Geodesic polyhedron (https://en.wikipedia.org/wiki/Geodesic_polyhedron)