In the rhombic enneacontahedron, which is shown below, there are thirty narrow rhombi (shown in red) which separate twelve panels of five rhombi each (shown in yellow). This polyhedron is familiar to many people:
As you can see, the rhombic enneacontahedron has three of these yellow panels meeting at some of its vertices, along with three of the red, narrow rhombi.
For this new variant, at the top of this post, the five-rhombi panels are rotated until only two of them (rather than three) meet at certain vertices, and the thirty red, narrow rhombi between the yellow five-rhombi panels are replaced by twenty equilateral (but non-equiangular) hexagons, also shown in red.
Both of these polyhedra are related to the Platonic dodecahedron, which is shown below. In the rhombic enneacontahedron, the red, narrow rhombi correspond in position to the thirty edges of a dodecahedron. In the new variant, the red hexagons correspond to the vertices of a dodecahedron, rather than its edges. In both of these red-and-yellow polyhedra, the yellow, five-rhombi panels correspond to the dodecahedron’s faces. To see this more clearly, just compare the polyhedra above with this dodecahedron:
(All polyhedral images here were created with Stella 4d: Polyhedron Navigator, which you can try and/or buy here.)