How to describe disordered structures

Disordered structures such as liquids and glasses, grains and foams, galaxies, etc. are often represented as polyhedral tilings. Characterizing the associated polyhedral tiling is a promising strategy to understand the disordered structure. However, since a variety of polyhedra are arranged in complex ways, it is challenging to describe what polyhedra are tiled in what way. Here, to solve this problem, we create the theory of how the polyhedra are tiled. We first formulate an algorithm to convert a polyhedron into a codeword that instructs how to construct the polyhedron from its building-block polygons. By generalizing the method to polyhedral tilings, we describe the arrangements of polyhedra. Our theory allows us to characterize polyhedral tilings, and thereby paves the way to study from short- to long-range order of disordered structures in a systematic way.

[1]  Samir Khuller,et al.  Four colors suffice! , 2005, SIGA.

[2]  F. Yonezawa.  Glass Transition and Relaxation of Disordered Structures , 1991 .

[3]  J. Bai,et al.  Atomic packing and short-to-medium-range order in metallic glasses , 2006, Nature.

[4]  Hisao Nakamura,et al.  Universal medium-range order of amorphous metal oxides. , 2013, Physical review letters.

[5]  W. Boschin,et al.  Finding galaxy clusters using Voronoi tessellations , 2001, astro-ph/0101411.

[6]  Gunnar Brinkmann,et al.  Problems and scope of spiral algorithms and spiral codes for polyhedral cages , 1997 .

[7]  Jl Finney,et al.  RANDOM PACKINGS AND STRUCTURE OF SIMPLE LIQUIDS .2. MOLECULAR GEOMETRY OF SIMPLE LIQUIDS , 1970 .

[8]  Robin J. Wilson.  Four Colors Suffice: How the Map Problem Was Solved , 2002 .

[9]  J. L. Finney,et al.  Random packings and the structure of simple liquids. I. The geometry of random close packing , 1970, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[10]  D. Manolopoulos,et al.  Theoretical studies of the fullerenes: C34 to C70 , 1991 .

[11]  J. Finney,et al.  Random packings and the structure of simple liquids II. The molecular geometry of simple liquids , 1970, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[12]  G. Ziegler.  Lectures on Polytopes , 1994 .

[13]  F. Yonezawa,et al.  Confinement-Induced Stable Amorphous Solid of Lennard–Jones Argon , 2004 .

[14]  D. Manolopoulos,et al.  An Atlas of Fullerenes , 1995 .

[15]  P. Lockhart,et al.  Introduction to Geometry , 1940, The Mathematical Gazette.

[16]  L. Weinberg,et al.  A Simple and Efficient Algorithm for Determining Isomorphism of Planar Triply Connected Graphs , 1966 .

[17]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[18]  D. Reinelt,et al.  Structure of random monodisperse foam. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  David J. Srolovitz,et al.  Complete topology of cells, grains, and bubbles in three-dimensional microstructures , 2012, Physical review letters.

[20]  A Hirata,et al.  Geometric Frustration of Icosahedron in Metallic Glasses , 2013, Science.

[21]  Jacek Klinowski,et al.  Systematic enumeration of crystalline networks , 1999, Nature.

[22]  J. D. Bernal,et al.  The Bakerian Lecture, 1962 The structure of liquids , 1964, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.