Acute angle 90°
Acute angle 70.529°
Acute angle 63.435°
Isohedral Rhombohedra
This page is the result of a collaboration between Jim McNeill and Mason Green
In this page we explore the
possibilities for isohedral
rhombihedra, i.e.
polyhedra which are face transitive and consist of rhombic faces.
In total
we present 13 examples.
This category of polyhedron has also
been
examined by Professor
Branko
Grünbaum, to whom I am indebted for providing me with a copy
of his draft (and unpublished) manuscript of a systematic approach to
this
topic. Whilst a number of the above polyhedra were independently
re-discovered by ourselves, cross references to this manuscript are
provided in the form 60°n.
Professor Grünbaum's paper also covers examples with doubled faces and
coincident
vertices
(which explains the gaps in the 60°n notation on this page).
The linked 3D models to the figures
below all have one face highlighted for clarity.
Duals to Uniform Polyhedra.
Acute angle 90°
|
Acute angle 70.529°
|
Acute angle 63.435°
|
Three examples are well known: the cube (and
it's range of rhombic isomorphs, examples 1
and 2), the
rhombic dodecahedron (dual to
the cuboctahedron) and the rhombic triacontahedron (dual to the
icosidodecahedron). Note that
the
acute angle of the rhombic dodecahedron is 2arctan(1/√2) and
is referred to as θ below. That of the
rhombic triacontahedron (63.435°) is arctan(2) and
is referred to as φ below
 Acute angle 63.435° (=φ)
|
Acute angle 41.810°
|
Two further examples occur amongst the duals to the uniform polyhedra, the great rhombic triacontahedron (dual to the great icosidodecahedron) and the medial rhombic triacontahedron (dual to the dodecadodecahedron). The acute angle of the medial rhombic triacontahedron is arccos(√5 /3).
The above five examples (not the rhombic
isomorphs to the cube) are all also
edge transitive.
Rhombified Duals to Uniform Polyhedra
A number of other uniform polyhedra which
have tetragonal vertices (four faces
meeting at each vertex) have duals containing irregular quadrilateral
faces. In a number of cases, these
faces can be deformed to rhombi whilst retaining the same polyhedral
net. The face transitivity of the faces is preserved, but edge
transitivity is lost.
Those uniform polyhedra meeting the
above criteria and the rhombified
isomorphs of their duals are given in the table below.
| 1 | Icosified Dodecadodecahedron |  Rhombified Medial Icosacronic Hexecontahedron 60°13 Acute angle 60° |
| 2 | Rhombicosidodecahedron |  Rhombified Hexecontahedron a.k.a "Unkelbach polyhedron" 60°1 Acute angle 63.435° (=φ)
|
| 3 | Great Icosified Icosidodecahedron |  Rhombified Great Icosacronic Hexecontahedron 60°8 Acute angle 63.435° (=φ)
|
| 4 | Small Icosicosidodecahedron |  Rhombified Small Icosacronic Hexecontahedron 60°2 Acute angle 63.435° (=φ)
|
| 5 | Rhombidodecadodecahedron |  Rhombified Medial Trapezoidal Hexecontahedron 60°6 Acute angle 70.529° (=θ)
|
| 6 | Great Dodekicosidodecahedron |  Rhombified Great Dodecacronic Hexecontahedron 60°19 Acute angle 70.529° (=θ)
|
| 7 | Small Dodekicosidodecahedron |  Rhombified Great Ditrigonary Dodecacronic Hexecontahedron 60°7 Acute angle 70.529° (=θ)
|
One further example cannot be generated from the dual of a uniform polyhedron. It is shown below, and is included in Grünbaum as 60°12.
Acute angle 63.435° (=φ)
Notes:
(1) above, has the rhombi coplanar in sets of three and is a stellation of the icosahedron that superficially resembles a great icosahedron, it is not degenerate.
(2) to (4) and 60°12 are all stellations of the rhombic triacontahedron, (2) and (3) are superficially similar but differ in their internal structure, as can be seen from exploring the 'translucent' and 'frame' modes of the 3D images.
 |
 |
(5) to (7) are all stellations of a strombic figure that is dual to a rhombicosidodecahedron (both shown above) where the ratio of the lengths of the triangular and pentagonal edges is phi2 to 1.
Credits
The original dual figures were produced
using Great
Stella and relaxed using HEDRON.
My thanks to Professor
Branko
Grünbaum to whom I am indebted for providing me with a copy
of his draft (and unpublished) manuscript of a systematic approach to
this
topic.
I am grateful to Roger Kaufman for bringing 60°12 to my attention.