Convex Rhombic Polyhedra with Icosahedral Symmetry

This page collects together the various polyhedra I have come across which meet the following criteria

• Convex
• Icosahedral Symmetry
• All faces are either regular polygons or rhombi
• Some of the faces must be rhombic.
• Rhombi can be in a maximum of three distinct forms

The last criterion is included purely as a measure to limit the number of qualifying polyhedra.

Some of the polyhedra displayed here are also displayed elsewhere on this site in which case a link is provided.  Others are new, in which case a short explanation is provided.

I do not expect this list to be complete - Roger Kaufman discovered four new examples in 2014.  Contributions are welcome.

One form of rhombus Rhombic Triacontahedron (a=63.435°)  LINK Expanded Rhombic Triacontahedron (a=63.435°)  LINK 'Snub' Rhombic Triacontahedron (a=62.967°) Take an expanded rhombic triacontahedron, divide each square into two triangles and allow to relax.  Note the rhombi are different to those in the original rhombic triacontahedron. OFF Rhombi-propello-icosahedron (a=66.140°)  LINK 'Faceted' Rhombic Enneacontahedron (a=53.130°) Take a rhombic enneacontahedron and connect the obtuse vertices of the 'thin' rhombi.  Then allow to relax.  OFF Rhombified Goldberg Fullerene C80 (a=60.961°).  Take the dual of a geodesic sphere of frequency 2, replace the distorted hexagons with rhombus/triangle complexes and allow to relax. Discovered by Roger Kaufman.  OFF Rhombified Petrie Expanded Truncated Dodecahedron (a=60.961°).  Take the Petrie expanded truncated dodecahedron, replace the distorted hexagons with rhombus/triangle complexes and allow to relax.  This can also be regarded as an expanded version of Roger Kaufman's Rhombified Goldberg Fullerene C80.  OFF Rhombified Face Faceted Truncated Dodecahedron (a=48.471°).  Take a truncated dodecahedron, replace the decagons with complexes of a pentagon surrounded by five triangles and five rhombi.  Relax the model.   OFF

Two forms of rhombi Rhombic Enneacontahedron (a1=70.529°, a2=41.810°) LINK Rhombified Stellated Snub Rhombic Triacontahedron  (a1=67.761°, a2=71.276°):  Take the snub rhombic triacontahedron above and stellate the pentagonal faces.  Relax the model so that the subsequent kite faces become rhombi.  OFF Faceted Tetra-linear rhombic polyhedron  (a1=63.783°, a2=60.390°):  Take the tetra-linear rhombic polyhedron (below) and facet off the pentagonal vertices, leaving pentagons surrounded by triangles.  Relax the model.   OFF Faceted Zonish Snub Icosidodecahedron (Type A)  (a1=62.174°, a2=61.393°):  Discovered by Roger Kaufman in 2014.   It is based on a particular zonish snub icosidodecahedron (OFF) as shown in George Hart's Zonish Polyhedra page (bottom left).  The distorted hexagons can be split into two rhombi and subsequently relaxed.  OFF Faceted Zonish Snub Icosidodecahedron (Type B)  (a1=54.957°, a2=65.645°):  Discovered by Roger Kaufman in 2014.  It is based on the same figure as the Type A version.   The distorted hexagons are in this case split into two triangles and on rhombus, then relaxed.  OFF

Three forms of rhombi Rhombified Face Faceted Truncated Icosidodecahedron (a1=73.912°, a2=60.144°, a3=48.786°).  Take a truncated icosidodecahedron, replace the hexagons with complexes of three rhombi, the squares with rhombi, and the decagons with complexes of a pentagon surrounded by five triangles and five rhombi.  Relax the model.  OFF Rhombified Face Faceted Petrie Expanded Truncated Dodecahedron (a1=73.480°, a2=60.885°, a3=49.166°).  Take a rhombified Petrie expanded truncated dodecahedron, replace the decagons with complexes of a pentagon surrounded by five triangles and five rhombi and the squares with rhombi.  Relax the model.  OFF Rhombified Face Faceted Goldberg Fullerene C140 (a1=68.789°, a2=63.435°, a3=47.870°).  Take a snub dodecahedron, triangulate the pentagonal faces and take the dual.  Replace the hexagons with complexes of three distinct rhombi.   Relax the model.  Note that the pale rhombi in this model are golden rhombi.   OFF Tetra-linear rhombic polyhedron. (a1=73.531°, a2=68.789°, a3=60.533°).  Following the sequence of the rhombi-propello-icosahedron (with rhombi in lines of 2) and the rhombified stellated snub rhombic triacontahedron (with rhombi in lines of 3) comes the above polyhedron with rhombi in lines of 4.  The hexagonal faces formed by the subdivision of the triangular sections are themselves subdivided into complexes of three rhombi.   OFF Stellated Faceted Zonish Snub Icosidodecahedron (Type A)  (a1=61.596°, a2=70.529°, a3=68.119°):  Discovered by Roger Kaufman in 2014.Take the Faceted Zonish Snub Icosidodecahedron (Type A) and stellate the pentagonal faces to form complexes of 5 rhombi.  Relax the model.  Note that the 'brown' rhombi are golden rhombi. OFF Stellated Faceted Zonish Snub Icosidodecahedron (Type B)  (a1=54.424°, a2=75.015°, a3=68.500°):  Discovered by Roger Kaufman in 2014.Take the Faceted Zonish Snub Icosidodecahedron (Type B) and stellate the pentagonal faces to form complexes of 5 rhombi.  Relax the model. OFF

A family of Rhombic Polyhedra

A number of the rhombihedra above can be grouped into a family that I informally call 'path and crown' rhombihedra.  The rhombi form 'paths' of size x by y between the triangular 'crowns' of order x-y.  Those generated to date are as below, recoloured to highlight the paths.  These polyhedra are reminiscent of Paul Gailiunas' 'Twisted Domes'.  Higher order examples have not been explored. 2 by 1 : The Rhombi-Propello Icosahedron 3 by 1: The Rhombified Stellated Snub Rhombic Triacontahedron 3 by 2 : The Stellated Faceted Zonish Snub Icosidodecahedron 4 by 1 : The Tetra-linear rhombic polyhedron Note that the crown of order 3 itself contains rhombi.

There exists a similar family with 12 pentagonal crowns, for the order 1 crowns these are topologically dual to the above family.  Of the examples tried to date only the 3 by 1 is convex.  The 'paths' are also not as distinct. 3 by 1: The Rhombified Face Faceted Truncated Icosidodecahedron

Credits

The original dual figures were produced using Great Stella and relaxed using HEDRON.
Thanks to Roger Kaufman for his contributions.

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