Each of the smaller pictures below may be enlarged by clicking on them.

All of these images were produced using Stella 4d, which you may try or buy at http://www.software3d.com/Stella.php.

This one is a variant of the icosidodecahedron.

This one is based on the rhombcuboctahedron.

This one is made of squares, rhombi, and irregular pentagons.

This one is composed entirely of pentagons and hexagons, none of which are regular.

This one has faces which include squares, rhombi, and isosceles triangles.

In this one, the hexagons and squares are regular. Only the isosceles triangles are irregular.

This is the dual of the last one shown here. Its faces are all either kites or rhombi.

I hoped to make this one into a near miss to the Johnson solids, but the octagons of both types, especially, are too far from regularity to get that to work. The only faces which are regular are the green triangles.

This one is a variant of the icosahedron.

I found this one interesting.

And this one is its dual:

Finally, here’s one made of kites and regular hexagons.