FULL EQUIVALENCE BETWEEN SOCOLAR TILINGS AND THE (A, B, C, K)-TILINGS LEADING TO A RATHER NATURAL DECORATION

In 1986 Socolar and Steinhardt introduced a family of quasiperiodic tilings of the euclidean 3-space E3 by four rhombic zonohedra, which admits a local matching rule. In 1989 the first of the present authors introduced another family with similar properties and four skew tetrahedra A, B, C, K as prototiles. Now we show, that either family can be obtained from the other by a unique construction; hence they are mutually derivable in the sense of Baake et al. Some additional observations are induced by this equivalence.