Additions to

Zome Geometry

George W. Hart   and  Henri Picciotto

Polyhedra with Concave Pentagons

Recall that twelve concave pentagons can be assembled to make a concave dodecahedron.  (See Zome Geometry, p. 84).  Paul Gailiunas has proposed an interesting general question:
What polyhedra can be made using only concave pentagons and regular polygons?

Above is one example that I really like.  It looks like two regular dodecahedra which crashed together.  Its faces are 18 regular pentagons and 6 concave pentagons.

Here is a second nice example.  It is like a prism with two regular 10-gon bases, but around the sides are 10 concave pentagons and 10 equilateral triangles.  You can extend this, if you like, to make a larger polyhedron which looks like two icosidodecahedra which crashed.

For other interesting examples and a nice analysis, you can look up his paper:

Paul Gailiunas, "A Polyhedral Byway," Proceedings of Bridges 2001: Mathematical Connections in Art, Music, and Science, pp. 115-122.
But work on it on your own first.