I made this using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

To make this, I augmented the pentagonal faces of an icosidodecahedron with pyramids, while the triangular faces of this icosidodecahedron had pyramids excavated from them. The result, as you can see, looks like an icosahedron with tetrahedral indentations is the middle of each face. I made it using Stella 4d, which you can try for free at http://www.robertlovespi.net/Stella.php.

Here’s a second view, in “rainbow color mode.”

I made these using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.

This isn’t quite a near-miss to the Johnson solids, so I’m calling it a “near near-miss.” It could also be classified as a symmetrohedron. The pentagons and triangles are regular, but the red triangles have edges which are about 12% longer than those of the yellow triangles. The blue faces are isoceles trapezoids.

I found it while playing around with the great rhombicosidedcahedron using Stella 4d, which you can try for free right here.