Exact golden arithmetic of H 2 and H 4 model sets

Source abstract

Abstract The non-crystallographic root systems H 2 and H 4 are both governed by the golden ring Z [ φ ] , but they are usually treated as separate topics: the planar Penrose pattern on the H 2 side, and the icosian 600 -cell on the H 4 side. In this work we develop a single exact-arithmetic bridge that ties the decagonal cut-and-project benchmark to the icosian shell and, finally, to the trace-normalized Z -lattice K 8 of the icosian order, which is isometric to the classical 5 -modular icosian lattice Q 8 ( 1 ) . The discriminant group ( Z / 5 Z ) 4 of this lattice is classical; the point of the present work is its arithmetic origin. We show that the icosian order is unimodular over Z [ φ ] for the reduced trace form, so that the reciprocal lattice of K 8 is exactly the codifferent multiple I / 5 of the order, its discriminant group is the residue algebra of the icosians at the ramified prime 5 of the golden field, and the E 8 lattice is the twist of the same Hermitian module by a totally positive generator of the codifferent. This is the precise sense in which the reciprocal module of icosian model-set labels, which is what indexes diffraction, is not preserved by the unimodular E 8 normalization: the layer erased is the residue algebra of the icosians modulo 5 , and the explicit denominator- 5 reciprocal generators are, up to a unit and the prime 5 , the Z [ φ ] -dual basis of the order.