A method for deriving packings of icosahedral polyhedra from periodic and non-periodic space grids based on icosahedral symmetry is presented. Corresponding to the three axes of symmetry, three classes of packings exist. For each class, an inter-transforming family of 64 (theoretically) possible packings is suggested. Their details remain to be worked out and examples are illustrated with packings of icosahedra.
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References
1.
First presented by the author at the conferenceGeometric Problems in Crystallography, Bielefeld University, August1985.
2.
LalvaniH., Non-periodic Space structures, Space Structures2(2) (1986/87), 93-108.
3.
Compare with the projection methods proposed by De Bruijn (1981), Kramer (1984) and Levine and Steinhardt (1984), cited in Ref. 2.
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HaaseR., KramerL., KramerP. and LalvaniH., Polyhedra of three quasilattices associated with the icosahedral group (submitted for publication).
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During the Spring 1986 term, a model of a space frame based on Figs 3(a)-7(a) using stretched octahedral struts was built by students in the author's studio at Pratt Institute and was presented at the Symmetry Symposium, Darmstadt, June 1986; during that visit, the work of M. Audier and P. Guyot (Phil. Mag. Letters B, 53(1) (1986), L43-51) on the quasi-crystal Al4Mn, suggesting a face-to-face co-ordination of icosahedra, was brought to the author's attention by P. Kramer.
10.
Compare with plane tessellations using convex and non-convex polygons in Ref. 2.
