2022/01/10 by Amanda Burcroff, Burcroff, Amanda
Materials Science · Mathematics · #20F55 #51F15 #51M20 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Metric Geometry (math.MG) #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2201.03437
openalex publication_date 2022/01/10 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
We complete the classification of compact hyperbolic Coxeter d-polytopes with d+4 facets for d=4 and 5. By previous work of Felikson and Tumarkin, the only remaining dimension where new polytopes may arise is d=6. We derive a new method for generating the combinatorial type of these polytopes via the classification of point set order types. In dimensions 4 and 5, there are 348 and 51 polytopes, respectively, yielding many new examples for further study. We furthermore provide new upper bounds on the dimension d of compact hyperbolic Coxeter polytopes with d+k facets for k ≤ 10. It was shown by Vinberg in 1985 that for any k, we have d ≤ 29, and no better bounds have previously been published for k ≥ 5. As a consequence of our bounds, we prove that a compact hyperbolic Coxeter 29-polytope has at least 40 facets.