Variations on a Rhombic Theme

In addition to the Convex Polyhedra containing Rhombi as found in the Rhombic Dodecahedron and Convex Polyhedra containing Golden Rhombi described previously, there are some interesting non-convex variations that can be formed.

1. 'Imploded' polyhedra

Expanded Rhombic Dodecahedron (VRML, OFF)

Imploded Rhombic Dodecahedron (VRML, OFF)

Expanded Rhombic Triacontahedron (VRML, OFF)

Imploded Rhombic Triacontahedron (VRML, OFF)

Tetrahedrally Expanded Rhombic Dodecahedron (VRML, OFF)

Tetrahedrally Imploded Rhombic Dodecahedron (VRML, OFF)

As well as the 'Expanded' forms of the rhombic dodecahedron and rhombic triacontahedron, the bands can be 'inverted'.  Each face of the original polyhedron is moved inwards towards (and if necessary past) the centre of the polyhedron until there is a distance of one unit between each originally adjacent face.  These gaps are filled by the insertion of square faces.  The remaining holes, corresponding to the original vertices of the polyhedron can be filled by triangles and either squares or pentagons as appropriate. 

The rhombic dodecahedron can also be tetrahedrally imploded.

2. The Medial and Great Rhombic Triacontahedra

Expanded Medial Rhombic Triacontahedron (VRML, OFF)

Imploded Medial Rhombic Triacontahedron (VRML, OFF)

Expanded Great Rhombic Triacontahedron (VRML, OFF)

Imploded Great Rhombic Triacontahedron (VRML, OFF)

The same treatment can also be applied to the medial rhombic triacontahedron (dual to the dodecadodecahedron) and the great rhombic triacontahedron (dual to the great icosidodecahedron) generating both expanded and imploded forms. 

Related Pages 

George Hart: 'Zonish Polyhedra'
Convex Polyhedra containing Rhombi as found in the Rhombic Dodecahedron 
Alex Doskey's: Diamond Regular Polyhedra and Steve Waterman's: Convex Hulls having Regular Diamonds
Roger Kaufman's
: 108 Degree Rhombohedra

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