”The local controlled growth of a perfect Cartwheel type tiling called the quasiperiodic succession”

Uli Gaenshirt (1) , Michael Willsch (2)

1 Sculptur workshop head
2 CT PS 6

The modellation of the growth of decagonal quasi crystals by Penrose structures is limited by the matching rules of the tiles or of an elementary cluster cell. The matching rules act only locally and lead to configurations which cannot be extended further with these rules. How can be shown with the substitution rules of the Penrose tiles, perfect matching rules have to operate in all scales of substitution. The quasi periodic succession which is presented here does exactly this. An elementary aperiodic cluster cell Q contains a mechanism which picks up 5 scale values of one neighboring cluster cell, evaluates it and determines 4, 5 or 6 neighbor cell positions with individual scale values each. This procedure is controlled by 5 constant length lconst which can be understood as a sliding ruler with two marks. The length lconst is derived as the optimized, averaged length of the self similar, one dimensional, aperiodic interval structures (Qa, Qb, Qc, Qd, Qe) from which Q is assembled. The paper presents an arithmetic algorithm on the base of this geometric model which allows to calculate the local structure successive out of each cluster cell without consideration of the global structure. However it could be shown that it corresponds perfectly with the global structure which is generated by substitution. The successive algorithm on the base of a single cluster cell and its 5 scale values makes the existence of natural decagonal quasi crystals and the growth of pure synthetic decagonal quasi crystals more imaginable.

Mots clés: Physical Sciences

Uli Gaenshirt, Michael Willsch. ”The local controlled growth of a perfect Cartwheel type tiling called the quasiperiodic succession”. Philosophical Magazine, 2007, 87 (18-21), pp.3055-3065. ⟨10.1080/14786430701355133⟩. ⟨hal-00513832⟩