The Rhombic Enneacontahedron and relations

The above 90 sided figure is the rhombic enneacontahedron.  It consists of sixty fat rhombi ('R') (gold coloured) as found in the rhombic dodecahedron and thirty thin rhombi ('r') (bronze coloured) as found in the medial rhombic triacontahedron.  It is a zonohedron, and is also referred to as a zonohedrified dodecahedron.  It can only be formed from these particular rhombi, a quality I refer to as being rhombo-static.  

The right hand figure is a rhombo-flexible triacontahedron.

Partial zonohedrifications of the dodecahedron

As a zonohedron bands of parallel edges can be removed from the polyhedron leaving a remnant polyhedron which is also normally rhombo-static, although there are some exceptions.

To define partial zonohedrifications of the dodecahedron it is useful to label the 10 axes of the dodecahedron as A through J.  As an axis of the dodecahedron is equivalent to two opposite faces of an icosahedron, these axes are best shown on an icosahedron as follows:

.

In total there are 2¹⁰ possible combinations of axes.  Any selection of just one axis leads to a line segment.  There are two distinct ways to select two axes, those relating to edge connected triangles (eg AB) and those related to vertex connected triangles (eg BC).  These selections lead to a thin ('r') rhombus and a fat ('R') rhombus respectively.   For three or more axes the figures are given below.  All possible selections of axes are equivalent to one of the selections below as is shown in tabular form here.

The selected axes are shown on the icosahedron next to each figure, note that in most cases this icosahedron has been rotated to show the symmetry of the polyhedron.  Note also that all cases including those where the symmetry is given as 'none' are symmetric through central inversion.

10 Axes

Axes: ABCDEFGHIJ Symmetry: icosahedral Rhombi: (R,r) = (60,30) VRML,  OFF

9 Axes

Axes: ABCDEFGHI Symmetry: 3-fold dihedral  Rhombi: (R,r) = (48,24) Compliment: A VRML,  OFF

8 Axes

Axes: ABCDEFGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (38,18) Compliment: BC VRML,  OFF

Axes: ABCDEFHI Symmetry: 2-fold dihedral  Rhombi: (R,r) = (36,20) Compliment: AB VRML,  OFF

7 Axes

Axes: ABCDEFG Symmetry: 3-fold pyramidical  Rhombi: (R,r) = (30,12) Compliment: CDE VRML,  OFF	Axes: ABCDEFH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (28,14) Compliment: ADE VRML,  OFF
Axes: ABCDEGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (28,14) Compliment: ABC VRML,  OFF	Axes: ABCDFGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (28,14) Compliment: ACE VRML,  OFF
Axes: ABDEFGH Symmetry: 3-fold dihedral  Rhombi: (R,r) = (30,12) Compliment: BCD VRML,  OFF

6 Axes

Axes: ABCDEF Symmetry: None*  Rhombi: (R,r) = (20,10) Compliment: ACDE VRML,  OFF	Axes: ABCDEH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (18,12) Compliment: ABDE VRML,  OFF
Axes: ABCDFH Symmetry: 2-fold dihedral  Rhombi: (R,r) = (20,10) Compliment: BCEF VRML,  OFF	Axes: ABCDGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (18,12) Compliment: ABCE VRML,  OFF
Axes: ABCEFG Symmetry: None* Rhombi: (R,r) = (22,8) Compliment: BCDE VRML,  OFF	Axes: BCDEFG Symmetry: Tetrahedral  Rhombi: (R,r) = (24,6) Compliment: AEFG Rhombo-flexible VRML,  OFF
Axes: ABDEGH Symmetry: 3-fold dihedral  Rhombi: (R,r) = (18,12) Compliment: ABCD VRML,  OFF

5 Axes

Axes: ABCDE Symmetry: None*  Rhombi: (R,r) = (12,8) Compliment: ABCEF VRML,  OFF	Axes: ABCEF Symmetry: None*  Rhombi: (R,r) = (12,8) Compliment: ABCDE VRML,  OFF
Axes: ABCEH Symmetry: 5-fold dihedral  Rhombi: (R,r) = (10,10) Compliment: ACDGH VRML,  OFF	Axes: ACDGH Symmetry: 5-fold dihedral  Rhombi: (R,r) = (10,10) Compliment: ACDEF VRML,  OFF
Axes: ABEFG Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (16,4) Compliment: BCDEF VRML,  OFF	Axes: BCDEF Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (16,4) Compliment: ABDEF Rhombo-flexible VRML,  OFF
Axes: ABDEF Symmetry: 2-fold pyramidical Rhombi: (R,r) = (14,6) Compliment: ABDEF VRML,  OFF	Axes: ACDEF Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (14,6) Compliment: ABEFG VRML,  OFF

4 Axes

Axes: ABCD Symmetry: 3-fold dihedral  Rhombi: (R,r) = (6,6) Compliment: ABDEGH VRML,  OFF	Axes: ABCE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (6,6) Compliment: ABCDGH VRML,  OFF
Axes: ABDE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (6,6) Compliment: ABCDEH VRML,  OFF	Axes: ACDE Symmetry: None*  Rhombi: (R,r) = (8,4) Compliment: ABCDEF VRML,  OFF
Axes: AEFG Symmetry: Octahedral  Rhombi: (R,r) = (12,0) Compliment: BCDEFG VRML,  OFF	Axes: BCDE Symmetry: None*  Rhombi: (R,r) = (10,2) Compliment: ABCEFG VRML,  OFF
Axes: BCEF Symmetry: 2-fold dihedral  Rhombi: (R,r) = (8,4) Compliment: ABCDFH VRML,  OFF

3 Axes

Axes: ABC Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (2,4) Compliment: ABCDEGH VRML,  OFF	Axes: ACE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (4,2) Compliment: ABCDFGH VRML,  OFF
Axes: ADE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (4,2) Compliment: ABCDEFH VRML,  OFF	Axes: BCD Symmetry: 3-fold dihedral  Rhombi: (R,r) = (6,0) Compliment: ABDEFGH VRML,  OFF
Axes: CDE Symmetry: 3-fold dihedral  Rhombi: (R,r) = (6,0) Compliment: ABCDEFG VRML,  OFF

Rhombo-flexible cases.

For cases with more than 4 axes only two cases are rhombo-flexible:

• The 6-axis case BCDEFG (above) is rhombo-flexible, and is convex between limiting cases of R and r having acute angles of 90º and 0º (a frequency two cube) and 60º and 90º  (a truncated octahedron)  with the rhombic triacontahedron as an intermediate form.
  
• The 5-axis case BCDEF is also rhombo-flexible between the same limits with the rhombic icosahedron as an intermediate form.  The animated VRML is linked here.

All 4-axis and 3-axis cases are rhombo-flexible.

Related page: http://www.georgehart.com/virtual-polyhedra/dissection-re.html