ABSTRACT
Building rules are examined for an icosahedral quasiperiodic packing of fibres with axes parallel to the ten threefold axes, first employing an experimental construction and afterwards a mathematical demonstration using the cut-and-project method applied in hyperspace. As a result of this latter approach, very simple two-dimensional (2D) building rules are proposed. Similar simple 2D rules have also been proposed for the case of an icosahedral quasiperiodic packing with fibre axes parallel to the six fivefold axes [Duneau & Audier (1999). Acta Cryst. A55, 746-754]. Finally, the construction of another icosahedral quasiperiodic packing resulting from a combination of two groups of fibres respectively parallel to six fivefold and ten threefold axes is reported. A brief discussion is given on different particular mechanical behaviours which might a priori be expected from the various enantiomorphic properties of these packings.
ABSTRACT
Different icosahedral packings of fibres have been experimentally realized. A packing construction with straight fibres of the same circular cross section, only parallel to fivefold icosahedral axes and respecting the closest packing condition, is reported. Its characteristics of point-group symmetry and related two-dimensional tilings are analysed. But for determining unambiguously all the fibre positions it appears that a mathematical construction has to be made from the cut and projection of a five-dimensional space. Through such a method, the volume fraction of fibrous reinforcement in a composite material can be calculated. The related two-dimensional tiling can be proved to be different from a Penrose tiling. Finally, the characteristics of other icosahedral packings where fibres are parallel to threefold axes or to both threefold and fivefold axes are briefly discussed and a few further experiments on their elasticity properties and photonic band-gap structure are suggested.
