On shelling \(E_ 8\) quasicrystals (Q1328376)

The icosian ring, a subring of the ring \(\mathbb{H} (\mathbb{R})\) of quaternions, can be seen as a faithful projection of the root lattice \(E_ 8\) to a 4D subspace that is invariant under the Coxeter group H4. As such, it is a 4D module of rank 8 (over \(\mathbb{Z}\)) that underlies the 4D quasicrystal of \textit{V. Elser} and \textit{N. J. A. Sloane} [J. Phys. A 20, 6161-6168 (1987; Zbl 0637.20024)]: it is generated by linear combinations of the vertex points of this quasicrystal. The authors replace the usual acceptance domain (a 4D, H4-invariant, regular polytope) by a sphere and investigate various statistical properties, in particular the ``shelling'' structure of the vertex points. Here they correct and prove a conjecture of \textit{J. F. Sadoc} and \textit{R. Mosseri} [J. Non-Cryst. Solids 153-154, 247-252 (1993)].