This is a rhombicosidodecahedron, one of the Archimedean solids.
If one pentagonal cupola is removed from this polyhedron, the result is the diminished rhombicosidodecahedron, which is one of the Johnson solids (J76).
The next step is to take another J76, and attach it to the first one, so that their decagonal faces meet.
I’m calling the result the “double rhombicosidodecahedron.”
I did these manipulations of polyhedra and their images with a program called Stella 4d: Polyhedron Navigator. There’s a free trial download available, if you’d like to try the program for yourself, and it’s at this website.
RobertLovesPi.net 
“Gyro” double rhombicosidodecahedron — you would get a different shape if you rotated one of the diminished rhombicosidodecahedra 36 degrees.
Cutting off the cupolas is actually a very interesting thing. If we imagine the rhombicosidodecahedron as a spherical tiling instead of a polyhedron, then this is equivalent to saying that edge length at which the angles of equilateral triangle, square, and pentagon are just so that (3,4,5,4) closes perfectly is the same as the edge length where (4,5,10) closes perfectly.
And this is a general rule. (3,4,n,4) has always the same edge as (4,n,2n). For n = 3 to 5, this is spherical, n=6 is Euclidean, so trivial (all Euclidean configurations can be considered to have edge 0), and n > 6 is hyperbolic.
Here are some selected gyrated/diminished tilings for n = 7:
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Fascinating maneuvers. Thank you for explaining it to us.
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