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 centre figure is a rhombo-flexible triacontahedron.

There is a isomorph to the rhombic enneacontahedron: the great rhombic enneacontahedron, shown above right.

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.

In 1970 Steve Baer described a dissection of the rhombic enneacontahedron into 120 parallelepiped blocks [1] .  The blocks are in five forms, denoted A B C D and E (my italics).  These are equivalent to the 3 axes cases below.  The reference to Baer gives the number of Baer Cells required to construct the polyhedron in the form (A,B,C,D,E). The data was generated by David Koski and is included here with his permission.  George Hart has an interesting page on dissections of rhombohedra which includes discussion of the rhombic enneacontahedron.

David Koski has also discovered the interesting facts that hold true for all the polyhedra below:

If axes = n then:

• Surface Faces =  (n)*(n-1)
• Total Faces = (n)*(n-1)*(n-2)
• Ratio of surface faces to total faces = 1/(n-2)

10 Axes

 Axes: ABCDEFGHIJ Symmetry: icosahedral Rhombi: (R,r) = (60,30) Baer: (10,20,30,30,30) VRML,  OFF

9 Axes

 Axes: ABCDEFGHI Symmetry: 3-fold dihedral  Rhombi: (R,r) = (48,24) Compliment: A Baer: (7,14,21,21,21) VRML,  OFF

8 Axes

 Axes: ABCDEFGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (38,18) Compliment: BC Baer: (5,10,14,13,14) VRML,  OFF
 Axes: ABCDEFHI Symmetry: 2-fold dihedral  Rhombi: (R,r) = (36,20) Compliment: AB Baer: (4,8,14,16,14) VRML,  OFF

7 Axes

 Axes: ABCDEFG Symmetry: 3-fold pyramidical  Rhombi: (R,r) = (30,12) Compliment: CDE Baer: (4,7,9,6,9) VRML,  OFF
 Axes: ABCDEFH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (28,14) Compliment: ADE Baer: (3,6,9,9,8) VRML,  OFF
 Axes: ABCDEGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (26,16) Compliment: ABC Baer: (2,4,9,11,9) VRML,  OFF
 Axes: ABCDFGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (28,14) Compliment: ACE Baer: (3,6,8,9,9) VRML,  OFF
 Axes: ABDEFGH Symmetry: 3-fold dihedral  Rhombi: (R,r) = (30,12) Compliment: BCD Baer: (3,8,9,6,9) VRML,  OFF

6 Axes

 Axes: ABCDEF Symmetry: None*  Rhombi: (R,r) = (20,10) Compliment: ACDE Baer: (2,3,5,5,5) VRML,  OFF
 Axes: ABCDEH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (18,12) Compliment: ABDE Baer: (1,2,6,7,4) VRML,  OFF
 Axes: ABCDFH Symmetry: 2-fold dihedral  Rhombi: (R,r) = (20,10) Compliment: BCEF Baer: (2,4,4,6,4) VRML,  OFF
 Axes: ABCDGH Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (18,12) Compliment: ABCE Baer: (1,2,4,7,6) VRML,  OFF
 Axes: ABCEFG Symmetry: None* Rhombi: (R,r) = (22,8) Compliment: BCDE Baer: (2,5,5,3,5) VRML,  OFF
 Axes: BCDEFG Symmetry: Tetrahedral  Rhombi: (R,r) = (24,6) Compliment: AEFG Baer: (4,4,6,0,6) Rhombo-flexible VRML,  OFF
 Axes: ABDEGH Symmetry: 3-fold dihedral  Rhombi: (R,r) = (18,12) Compliment: ABCD Baer: (0,2,6,6,6) VRML,  OFF

5 Axes

 Axes: ABCDE Symmetry: None*  Rhombi: (R,r) = (12,8) Compliment: ABCEF Baer: (1,1,2,4,2) VRML,  OFF
 Axes: ABCEF Symmetry: None*  Rhombi: (R,r) = (12,8) Compliment: ABCDE Baer: (0,1,3,3,3) VRML,  OFF
 Axes: ABCEH Symmetry: 5-fold dihedral  Rhombi: (R,r) = (10,10) Compliment: ACDGH Baer: (0,0,5,5,0) VRML,  OFF
 Axes: ACDGH Symmetry: 5-fold dihedral  Rhombi: (R,r) = (10,10) Compliment: ACDEF Baer: (0,0,0,5,5) VRML,  OFF
 Axes: ABEFG Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (16,4) Compliment: BCDEF Baer: (1,4,2,1,2) VRML,  OFF
 Axes: BCDEF Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (16,4) Compliment: ABDEF Baer: (2,2,3,0,3) Rhombo-flexible VRML,  OFF
 Axes: ABDEF Symmetry: 2-fold pyramidical Rhombi: (R,r) = (14,6) Compliment: ABDEF Baer: (1,2,2,2,3) VRML,  OFF
 Axes: ACDEF Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (14,6) Compliment: ABEFG Baer: (1,2,3,2,2) VRML,  OFF

4 Axes

 Axes: ABCD Symmetry: 3-fold dihedral  Rhombi: (R,r) = (6,6) Compliment: ABDEGH Baer: (1,0,0,3,0) VRML,  OFF
 Axes: ABCE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (6,6) Compliment: ABCDGH Baer: (0,0,2,2,0) VRML,  OFF
 Axes: ABDE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (6,6) Compliment: ABCDEH Baer: (0,0,0,2,2) VRML,  OFF
 Axes: ACDE Symmetry: None*  Rhombi: (R,r) = (8,4) Compliment: ABCDEF Baer: (0,1,1,1,1) VRML,  OFF
 Axes: AEFG Symmetry: Octahedral  Rhombi: (R,r) = (12,0) Compliment: BCDEFG Baer: (0,4,0,0,0) VRML,  OFF
 Axes: BCDE Symmetry: None*  Rhombi: (R,r) = (10,2) Compliment: ABCEFG Baer: (1,1,1,0,1) VRML,  OFF
 Axes: BCEF Symmetry: 2-fold dihedral  Rhombi: (R,r) = (8,4) Compliment: ABCDFH Baer: (0,0,2,0,2) VRML,  OFF

3 Axes

 Axes: ABC Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (2,4) Compliment: ABCDEGH Baer: 'D' (0,0,0,1,0) VRML,  OFF
 Axes: ACE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (4,2) Compliment: ABCDFGH Baer: 'C' (0,0,1,0,0) VRML,  OFF
 Axes: ADE Symmetry: 2-fold pyramidical  Rhombi: (R,r) = (4,2) Compliment: ABCDEFH Baer: 'E' (0,0,0,0,1) VRML,  OFF
 Axes: BCD Symmetry: 3-fold dihedral  Rhombi: (R,r) = (6,0) Compliment: ABDEFGH Baer: 'A' (1,0,0,0,0) VRML,  OFF
 Axes: CDE Symmetry: 3-fold dihedral  Rhombi: (R,r) = (6,0) Compliment: ABCDEFG Baer: 'B' (0,1,0,0,0) 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.

Credits

My thanks to David Koski for providing information relating to Baer Cell construction and for his permission to republish his data.

References

[1] Baer, Steve, Zome Primer, Zomeworks, 1970.