Contents
Stellation diagrams
Homohedral stellations
Compounds of a left and right chiral stellation
Table
Stellation diagrams
The most simple stellation diagram is provided by stellating a dual uniform polyhedron, because you get only 'one' diagram.
I will limit myself to this kind of stellation diagrams.
There are two kinds of stellation diagrams: symmetrical and asymmetrical ones.
I selected six duals of which I give the stellation diagram and stellations:
1 Asymmetrical Diagrams
s34d
Pentagonal Icositetrahedron
s34d-ext
Pentagonal Icositetrahedron-extended
s35d
Pentagonal Hexecontahedron
W113d
Great Pentagonal Hexecontahedron
2 Symmetrical Diagrams
trc
Rhombic Triacontahedron
r34d
Deltoidal Icositetrahedron
Most of my attention will go to the stellation diagram of s34d and its extended stellation diagram s34d-ext. A scholar rarely pays attention to an extended stellation diagram and the stellations involved.
Abbreviations
c
chiral
co
compound
d
dual in s34d
diagram in d_CBF.gif
le
left
m
monohedral (better: homohedral)
mr
monohedral & reflexible
ri
right
W
Wenninger : Polyhedron Models & Dual Models
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Homohedral stellations
All the faces of a homohedral stellation are congruent.
An asymmetrical stellation diagram is produced by a chiral polyhedron as its core. This stellation has one facet in each prolongated plane of the core and this facet has in the stellation always the same chiral orientation. Thereafter you can deduce only one kind of homohedral stellations: chiral ones.
I try to give an exhaustive enumeration of homohedral stellations of two asymmetrical stellation diagrams. I found 23 chiral homohedral stellations in s35d and 24 chiral homohedral stellations in W113d. The reader is invited to verify if those enumerations are complete.
There are five homohedral stellations of the Pentagonal Icositetrahedron (dual to the Snub Cube) in my article of which three are in the Gallery of Robert Webb. He found a sixth one (4 chiral homohedral stellations).
A symmetrical stellation diagram has three kinds of homohedral stellations:
- The facet of a chiral homohedral stellation is to find on the left or on the right side of the central symmetry axis in the stellation diagram.
- One facet in the stellation diagram is situated on the central symmetry axis of the stellation diagram. We call this a reflexible stellation.
- Two facets in the stellation diagram are symmetrically situated regarding to the central symmetry axis of the stellation diagram. We call this a reflexible stellation.
Note 1 : the stellation diagram of the Rhombic Triacontahedron has two axes of symmetry. Some monehedral stellations correspond to four facets in the stellation diagram.
I guess that there are no chiral monohedral stellations.
Note 2 : Most of the facets correspond to the 'primary regions' which Peter Messer recognizes in a symmetrical stellation diagram.
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Compounds of a left and right chiral stellation
From every chiral polyhedron a left and a right model can be constructed; both are enantiomorphous.
A pair of enantiomorphous polyhedra is generated by mirroring both in the planes of symmetry. Thereafter the lines of intersection of both constituents of the compound lie in the planes of symmetry and they are on the lines of symmetry in the stellation diagram.
If the stellation diagram has an axis of symmetry, then the symmetry lines and the compounds of an enantiomorphous pair are situated in that stellation diagram.
If the stellation diagram containing the chiral stellation is asymmetrical, then the intersection of both, the left and right chiral stellation, forms a new asymmetrical core of which a prolongated face contains the "extended" stellation diagram. And this asymmetrical stellation diagram contains the symmetry lines and the facets of the compound.
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Table
From my article, 1989, page 177, I quote:
"The levels of the tetragonal and trigonal pyramids and digonal points may, in the case of s (3/4)*, be combined in 3*4*6 = 72 possibilities. If our exploration is exhaustive then only 49 stellated forms may be built. Corresponding to each combination there is a set of stellated forms. Each set can be denoted by three capital letters. On the axis of four-fold rotation, there are three levels, to which correspond the capital letters A, B and C, A being the lowest level. On the axis of three-fold rotation, there are four levels with the letters A-D and on the axis of two-fold rotation, there are six levels with the letters A-F.
In the symbol, belonging to a set of stellated forms, the first letter refers to the level on the axis of four-fold rotation, the second letter refers to the level on the axis of three-fold rotation and the third letter refers to the level on the axis of two-fold rotation. If different stellated forms of the same set, denoted therefore by the same symbol, have to be distinguished, a number may be added to the reference symbol."
By these symbols the models may be represented in a table comparable to a matrix. The table in this chapter is a bit more extensive than in my article of 1989. Three new models are added at the right of the table and form a series: CAC, CBC and CCC; CAC is the sixth homohedral stellation published by Robert Webb.
Gratefully acknowledgements. All images of the Table are generated by the program STELLA of Robert WEBB and by the POVRAY-program.
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