small ditrigonal icosidodecahedron

dodecadodecahedron (2x12 pentagons)

compound of two regular dodecahedra

dodecahedron augmented / excavated

"golden dodecahedron" by Jean Pedersen

This compound of two symmetric tetrahedra is a distorted anticube. The figure may be dynamically modified 

The polyhedra of this sequence have 4n faces which are all half squares. They have been discovered by a visitor, J.P. Lhermet, following a mistake in the assemblage of an IsoAxis.
For n=3 we get... the cube. The net is a "zigzag" of squares extracted from the IsoAxis grid.
net for n=5, that is 20 faces 

This sequence of "plexagons" has been described by Paul Bourke: the nets of these polyhedra are made of two identical hexagons. To build the polyhedron, pleat the hexagons as shown below, turn one updown and rotate it by 60° and assemble the two pieces.

references: 
• Build Your Own Polyhedra by Hilton and J.Pedersen, AddisonWesley Publishing Company, 1988 (pages 107110, 123)
• "Plexagon  Pleated Hexagon" web page by Paul Bourke 
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convex polyhedra  non convex polyhedra  interesting polyhedra  related subjects  January 2001 updated 15/12/2015 