A Pentagonal Crystal, the Golden Section, alcove packing and aperiodic tilings

2008/11/03 by Anthony Joseph, Joseph, Anthony
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.0811.0336

57 pages, 12 figures

arxiv created 2008/11/03 · openalex publication_date 2008/11/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A Lie theoretic interpretation is given to a pattern with five-fold symmetry occurring in aperiodic Penrose tiling based on isosceles triangles with length ratios equal to the Golden Section. Specifically a B(∞) crystal based on that of Kashiwara is constructed exhibiting this five-fold symmetry. It is shown that it can be represented as a Kashiwara B(∞) crystal in type A₄. Similar crystals with (2n+1)-fold symmetry are represented as Kashiwara crystals in type A_2n. The weight diagrams of the latter inspire higher aperiodic tiling. In another approach alcove packing is seen to give aperiodic tiling in type A₄. Finally 2m-fold symmetry is related to type B_m.

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