2010/04/05 by Matthew D. Finn, Jean-Luc Thiffeault
Engineering · Materials Science · Mathematics · Physics and Astronomy · #Advanced Materials and Mechanics #Quasicrystal Structures and Properties #Structural Analysis and Optimization #cond-mat.soft #math.DS #math.GT #nlin.CD #physics.flu-dyn
paper · pdf · doi:10.1137/100791828
published as SIAM Review 53, 723-743 (2011) · 22 pages, 53 figures. PDFLaTeX with RevTex4 macros.
arxiv created 2010/04/05 · openalex publication_date 2011/01/01 · arxiv updated 2013/09/24 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly knead a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of material lines is conveniently given by the topological entropy of the rod motion. We discuss the problem of optimizing such rod devices from a topological viewpoint. We express rod motions in terms of generators of the braid group and assign a cost based on the minimum number of generators needed to write the braid. We show that for one cost function—the topological entropy per generator—the optimal growth rate is the logarithm of the golden ratio. For a more realistic cost function, involving the topological entropy per operation where rods are allowed to move together, the optimal growth rate is the logarithm of the silver ratio, 1+√(2). We show how to construct devices that realize this optimal growth, which we call silver mixers.