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:

10 Axes


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


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9 Axes


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


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8 Axes


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


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


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7 Axes


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


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


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


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


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Axes: ABDEFGH
Symmetry: 3-fold dihedral 
Rhombi: (R,r) = (30,12)
Compliment: BCD
Baer: (3,8,9,6,9)


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6 Axes


Axes: ABCDEF
Symmetry: None* 
Rhombi: (R,r) = (20,10)
Compliment: ACDE
Baer: (2,3,5,5,5)


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Axes: ABCDEH
Symmetry: 2-fold pyramidical 
Rhombi: (R,r) = (18,12)
Compliment: ABDE
Baer: (1,2,6,7,4)


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


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Axes: ABCDGH
Symmetry: 2-fold pyramidical 
Rhombi: (R,r) = (18,12)
Compliment: ABCE
Baer: (1,2,4,7,6)


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Axes: ABCEFG
Symmetry: None*
Rhombi: (R,r) = (22,8)
Compliment: BCDE
Baer: (2,5,5,3,5)


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Axes: BCDEFG
Symmetry: Tetrahedral 
Rhombi: (R,r) = (24,6)
Compliment: AEFG
Baer: (4,4,6,0,6)
Rhombo-flexible


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Axes: ABDEGH
Symmetry: 3-fold dihedral 
Rhombi: (R,r) = (18,12)
Compliment: ABCD
Baer: (0,2,6,6,6)


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5 Axes


Axes: ABCDE
Symmetry: None* 
Rhombi: (R,r) = (12,8)
Compliment: ABCEF
Baer: (1,1,2,4,2)


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Axes: ABCEF
Symmetry: None* 
Rhombi: (R,r) = (12,8)
Compliment: ABCDE
Baer: (0,1,3,3,3)


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Axes: ABCEH
Symmetry: 5-fold dihedral 
Rhombi: (R,r) = (10,10)
Compliment: ACDGH
Baer: (0,0,5,5,0)


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


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


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Axes: BCDEF
Symmetry: 2-fold pyramidical 
Rhombi: (R,r) = (16,4)
Compliment: ABDEF
Baer: (2,2,3,0,3)
Rhombo-flexible


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


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


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4 Axes


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


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


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Axes: ABDE
Symmetry: 2-fold pyramidical 
Rhombi: (R,r) = (6,6)
Compliment: ABCDEH
Baer: (0,0,0,2,2)


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Axes: ACDE
Symmetry: None* 
Rhombi: (R,r) = (8,4)
Compliment: ABCDEF
Baer: (0,1,1,1,1)


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Axes: AEFG
Symmetry: Octahedral 
Rhombi: (R,r) = (12,0)
Compliment: BCDEFG
Baer: (0,4,0,0,0)


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Axes: BCDE
Symmetry: None* 
Rhombi: (R,r) = (10,2)
Compliment: ABCEFG
Baer: (1,1,1,0,1)


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Axes: BCEF
Symmetry: 2-fold dihedral 
Rhombi: (R,r) = (8,4)
Compliment: ABCDFH
Baer: (0,0,2,0,2)


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3 Axes


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

 


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


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


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


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


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Rhombo-flexible cases.

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

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.