2008/09/25 by Franz Gähler, F. Gähler, John Hunton +3 · 6 citations
Materials Science · Mathematics · #Advanced Combinatorial Mathematics #Codimension #Cohomology #Combinatorics #Dimension (graph theory) #Geometry #Icosahedral symmetry #Integer (computer science) #Liquid Crystal Research Advancements #Mathematics #Penrose tiling #Projection (relational algebra) #Pure mathematics #Quasicrystal #Quasicrystal Structures and Properties #Substitution tiling #Torsion (gastropod) #math.AT #math.DS
paper · pdf · doi:10.1524/zkri.2008.1070
published in Zeitschrift für Kristallographie 223(11-12), 801-804 (De Gruyter)
arxiv created 2008/09/25 · openalex publication_date 2008/12/01 · arxiv updated 2012/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract The integer Cech cohomology of canonical projection tilings of dimension three and codimension three is derived. These formulae are then evaluated for several icosahedral tilings known from the literature. Rather surprisingly, the cohomologies of all these tilings turn out to have torsion. This is the case even for the Danzer tiling, which is, in some sense, the simplest of all icosahedral tilings. This result is in contrast to the case of two-dimensional canonical projection tilings, where many examples without torsion are known.