Ammann-Kramer-Neri Tiling

An Ammann-Kramer-Neri tiling is a three-dimensional nonperiodic tiling by copies of an acute golden rhombohedron and an obtuse golden rhombohedron. Also called a three-dimensional Penrose tiling, it generalizes the planar Penrose tiles construction (Kramer and Neri 1984, Dietl and Eschenburg 2017). Its edges lie in the six directions joining opposite vertices of an icosahedron.

The construction uses an orthogonal projection from six-dimensional Euclidean space. Let be a three-dimensional affine space in , oriented so that the six coordinate directions project to the six icosahedral directions, and let . The vertices are the projected integer points in the strip ,

where denotes orthogonal projection onto . For in general position, projecting the three-dimensional cubes of the six-dimensional cubic tessellation whose vertices belong to the strip gives the two types of golden rhombohedra and a tiling of all space (Dietl and Eschenburg 2017).

The tiling admits tile deflation with lengths reduced by , where is the golden ratio. This relates it to substitution tilings (Dietl and Eschenburg 2017). Nonperiodicity depends on the construction or suitable matching conditions. Each unmarked rhombohedron also tiles periodically by translations, so the two shapes alone do not force it.