Zome is a ball-and-stick modeling system which can be used to make millions of different polyhedra. If you’d like to get some Zome for yourself, just visit http://www.zometool.com.
The only irregular faces in this polyhedron are the quadrilaterals (kites and rectangles). I made it using Stella 4d, which you can try for yourself — for free — at http://www.software3d.com/Stella.php.
This zonohedron is based on the icosidodecahedron / rhombic triacontahedron compound — more specifically, on its edges. Twelve faces are regular decagons, twenty are regular hexagons, sixty are squares, and the only irregular faces are the thirty equilateral octagons. That’s 122 faces in all.
After I’d posted this, a helpful friend on Facebook told me the official name of the first polyhedron shown here — a pentalofted chamfered dodecahedron.
These images were created using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.
This zonish polyhedron was made by adding zones based on the faces and vertices of an icosahedron. Its faces are 30 decagons, twenty equilateral triangles, and twelve panels of five rhombi each. I used Stella 4d to create this; it’s a program you can try for free at http://www.software3d.com/Stella.php.
Both of these images were created using Stella 4d: Polyhedron Navigator, which you can try for yourself, free, at http://www.software3d.com/Stella.php.