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

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 $B(\infty)$ crystal based on that of Kashiwara is constructed exhibiting this five-fold symmetry. It is shown that it can be represented as a Kashiwara $B(\infty)$ crystal in type $A_4$. 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 $A_4$. Finally $2m$-fold symmetry is related to type $B_m$.