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 rhombostatic.
The centre figure is a rhomboflexible
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 rhombostatic, 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 2^{10} 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)*(n1)
 Total Faces = (n)*(n1)*(n2)
 Ratio of surface faces to total faces = 1/(n2)
10 Axes
Axes: ABCDEFGHIJ
Symmetry: icosahedral
Rhombi: (R,r) = (60,30)
Baer: (10,20,30,30,30) 
VRML, OFF


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


8 Axes
Axes: ABCDEFGH
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (38,18)
Compliment: BC
Baer: (5,10,14,13,14) 
VRML, OFF


Axes: ABCDEFHI
Symmetry: 2fold dihedral
Rhombi: (R,r) = (36,20)
Compliment: AB
Baer: (4,8,14,16,14) 
VRML, OFF


7 Axes
Axes: ABCDEFG
Symmetry: 3fold pyramidical
Rhombi: (R,r) = (30,12)
Compliment: CDE
Baer: (4,7,9,6,9) 
VRML, OFF


Axes: ABCDEFH
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (28,14)
Compliment: ADE
Baer: (3,6,9,9,8) 
VRML, OFF


Axes: ABCDEGH
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (26,16)
Compliment: ABC
Baer: (2,4,9,11,9) 
VRML, OFF


Axes: ABCDFGH
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (28,14)
Compliment: ACE
Baer: (3,6,8,9,9) 
VRML, OFF


Axes: ABDEFGH
Symmetry: 3fold 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: 2fold pyramidical
Rhombi: (R,r) = (18,12)
Compliment: ABDE
Baer: (1,2,6,7,4) 
VRML, OFF


Axes: ABCDFH
Symmetry: 2fold dihedral
Rhombi: (R,r) = (20,10)
Compliment: BCEF
Baer: (2,4,4,6,4) 
VRML, OFF


Axes: ABCDGH
Symmetry: 2fold 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)
Rhomboflexible 
VRML, OFF


Axes: ABDEGH
Symmetry: 3fold 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: 5fold dihedral
Rhombi: (R,r) = (10,10)
Compliment: ACDGH
Baer: (0,0,5,5,0) 
VRML, OFF


Axes: ACDGH
Symmetry: 5fold dihedral
Rhombi: (R,r) = (10,10)
Compliment: ACDEF
Baer: (0,0,0,5,5) 
VRML, OFF


Axes: ABEFG
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (16,4)
Compliment: BCDEF
Baer: (1,4,2,1,2) 
VRML, OFF


Axes: BCDEF
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (16,4)
Compliment: ABDEF
Baer: (2,2,3,0,3)
Rhomboflexible 
VRML, OFF


Axes: ABDEF
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (14,6)
Compliment: ABDEF
Baer: (1,2,2,2,3) 
VRML, OFF


Axes: ACDEF
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (14,6)
Compliment: ABEFG
Baer: (1,2,3,2,2) 
VRML, OFF


4 Axes
Axes: ABCD
Symmetry: 3fold dihedral
Rhombi: (R,r) = (6,6)
Compliment: ABDEGH
Baer: (1,0,0,3,0) 
VRML, OFF


Axes: ABCE
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (6,6)
Compliment: ABCDGH
Baer: (0,0,2,2,0) 
VRML, OFF


Axes: ABDE
Symmetry: 2fold 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: 2fold dihedral
Rhombi: (R,r) = (8,4)
Compliment: ABCDFH
Baer: (0,0,2,0,2) 
VRML, OFF



3 Axes
Axes: ABC
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (2,4)
Compliment: ABCDEGH
Baer: 'D'
(0,0,0,1,0)

VRML, OFF


Axes: ACE
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (4,2)
Compliment: ABCDFGH
Baer: 'C'
(0,0,1,0,0) 
VRML, OFF


Axes: ADE
Symmetry: 2fold pyramidical
Rhombi: (R,r) = (4,2)
Compliment: ABCDEFH
Baer: 'E' (0,0,0,0,1) 
VRML, OFF


Axes: BCD
Symmetry: 3fold dihedral
Rhombi: (R,r) = (6,0)
Compliment: ABDEFGH
Baer: 'A'
(1,0,0,0,0) 
VRML, OFF


Axes: CDE
Symmetry: 3fold dihedral
Rhombi: (R,r) = (6,0)
Compliment: ABCDEFG
Baer: 'B'
(0,1,0,0,0) 
VRML, OFF



Rhomboflexible cases.
For cases with more than 4 axes only two cases
are rhomboflexible:
 The 6axis case BCDEFG
(above) is rhomboflexible, 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 5axis case BCDEF
is also rhomboflexible between the same limits with the rhombic
icosahedron as an intermediate form. The animated VRML is linked here.
All 4axis and 3axis cases are rhomboflexible.
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.