The Stewart G3 and its relations

A trawl through the uniform polyhedra and the Johnson Solids reveals that none of them contain a vertex at which a single pentagon, square and triangle meet (a 5-4-3 'acrohedron').  So the question has to be asked; does one exist?  That answer is that one does, more than one in fact.  One of the simplest (imaged above) was originally described by Professor Bonnie Stewart in his book "Adventures Among the Toroids" (more on this on my Toroids pages).  Stewart names this polyhedron a 'G3', it appears in his sectioning of a rhombicosidodecahedron.  More on the search for a '5-4-3' acrohedron and other acrohedra can be found on my Acrohedra page.

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 at

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


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