Symmetry Structure of the Elser-sloane Quasicrystal

T . This can now be used to find all patterns in LI(P) with special symmetries (relative to the origin) which is a standard procedure in crystallography. For repetitive, but non-crystallographic P , however, the LI-class has a much richer structure and contains uncountably many (20) translation classes. It is thus much more difficult to classify the complete symmetry structure of such classes, and no general answer is known. If the pattern happens to be quasi-crystallographic (in the sense that it stems from a standard projection scheme), the key to parametrizing its LI-class is to use the fundamental domain of the embedding lattice. This is the so-called torus parametrization that has been introduced recently 1 and then applied to some of the most frequently used quasicrystallographic tilings 1,2 in two and three dimensions. It is the aim of this contribution to extend this set of examples to the ElserSloane quasicrystal in four dimensions. 3 It is constructed by the projection method from the root lattice E8 and has the Coxeter group H4 of order 14400 as its point symmetry group, together with an inflation/deflation symmetry with scaling factor τ = (1 + √ 5)/2. It is of interest due to its role in the hierarchy of quasicrystals with τ inflation.