Note the vertex of the G3 at which three pentagons meet. A dodecahedron can be made from four such sections. So four G3s can be joined together to form a dodecahedral shell with the squares and triangles inside. Removing the shell leaves the above polyhedron with a tetrahedral symmetry (left image). The right image shows the shell as a frame.
Image reproduced with permission
The image above is of a physical model
of four G3s created by Ulrich Mikloweit. His
wonderful models are unique in that he incorporates every part of each
polygonal face, including the internal facets. More images of this
polyhedron and of Ulrich's many other models can be found on his website
A 'great' G3 also exists with the pentagons of the G3 replaced by pentagrams. A vertex of the great stellated dodecahedron now appears. Four such polyhedra can be joined to form another polyhedron with a tetrahedral symmetry. This is shown above left, the right image shows the internal great stellated dodecahedron.
Alex Doskey (2003) has expanded the G3 by placing square faces between the pentagons and the triangle-triangle edges and completing the polyhedron with triangular and hexagonal caps, the expanded G3 is shown above left, Alex terms this a H3 (OFF). Four of these can be excavated from a rhombicosidodecahedron to produce the figure on the right (OFF).
I am indebted to Alex Doskey
for bringing the G3
(and the whole search for a '3-4-5') to my attention.
Alex has more on this and related polyhedra at http://www.doskey.com/polyhedra/
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