K Niizeki 2008 J. Phys. A: Math. Theor. 41 175208 doi:10.1088/1751-8113/41/17/175208
Self-similar quasilattices with windows having fractal boundaries
K Niizeki
Show affiliationsThe point inflation rule is presented as a powerful scheme to produce a self-similar quasilattice with a non-crystallographic point symmetry. Alternatively, the conjugate point inflation rule, which is a set map acting on the internal space, has the unique fixed point, which is a compact set, W, and the quasilattice (QL) is obtained with the projection method by taking W as the window. The fixed point, W, is the attractor of the set map. If W is topologically a disc (a ball for a three-dimensional case), its boundary is usually fractal. We investigate conditions for the boundary of the window to be a von Koch curve. A practical method of constructing windows and the relevant QLs is presented. A lot of windows with fractal boundaries are determined on the basis of this method. We introduce isomerism which has priority of rank to the mutual-local-derivability as a classification scheme for self-similar QLs. It is argued that the PIR and isomerism can be the key points of classification of self-similar QLs.
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Issue 17 (2 May 2008)
Received 4 January 2008, in final form 19 March 2008
Published 15 April 2008
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Self-similar quasilattices with windows having fractal boundaries
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