3 research outputs found
Nonexpansive directions in the Jeandel-Rao Wang shift
We show that is the set of
slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao
Wang shift, where is the golden mean. This set is a
topological invariant allowing to distinguish the Jeandel-Rao Wang shift from
other subshifts. Moreover, we describe the combinatorial structure of the two
resolutions of the Conway worms along the nonexpansive directions in terms of
irrational rotations of the unit interval. The introduction finishes with
pictures of nonperiodic Wang tilings corresponding to what Conway called the
cartwheel tiling in the context of Penrose tilings. The article concludes with
open questions regarding the description of octopods and essential holes in the
Jeandel-Rao Wang shift.Comment: v1: 24 pages, 19 figures; v2: 30 pages, 23 figures, new section with
open questions on octopods and essential holes; v3: small fixe
Minimal knotting numbers
This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18. © 2009 World Scientific Publishing Company