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Symmetry-breaking constraints for packing identical rectangles within polyhedra

R. Andrade, E. G. Birgin
2011 Optimization Letters  
Two problems related to packing identical rectangles within a polyhedron are tackled in the present work.  ...  We illustrate the impact of the symmetrybreaking constraints in the context of two problems: (i) packing as many identical rectangles within an equilateral triangle; and (ii) finding the smallest equilateral  ...  for the particular case of packing rectangles within rectangles).  ... 
doi:10.1007/s11590-011-0425-9 fatcat:kdcpcby7mrhpnaiujbxevlg4yq

A course in the mathematics of design

Jay Kappraff
1986 Computers and Mathematics with Applications  
space and spacefilling polyhedra; similarity, proportion and the golden mean with application to architectural design; transformations; mirrors and symmetry; and vectors applied to analysis of polyhedra  ...  A project-oriented course on the Mathematics of Design taught for the past six years to freshman architecture students at the New Jersey Institute of Technology is described.  ...  Tiling a rectangle with noncongruent squares. Fig. 11 . 11 The cuboctahedron as a figure of cubic close packing of spheres (courtesy Dover [8] ).  ... 
doi:10.1016/0898-1221(86)90434-7 fatcat:353kkpvcxvhvnoen4mlyeqgr5y

Mini-Workshop: Exploiting Symmetry in Optimization

Volker Kaibel, Leo Liberti, Achill Schürmann, Renata Sotirov
2010 Oberwolfach Reports  
The exchange of state-of-the-art knowledge between these areas lead to identification of important directions for future research activities.  ...  The feasible regions of mathematical optimization models quite often exhibit a high degree of symmetry.  ...  Static Symmetry Breaking Constraints for circle packing in a square problem Consider the following problem. Circle Packing in a Square (CPS).  ... 
doi:10.4171/owr/2010/38 fatcat:n2zcqzooonex7d3kcknkfsxpxa

A new binary decagonal Frank-Kasper quasicrystal phase

Johannes Roth, Christopher L. Henley
1997 Philosophical Magazine A  
Idealized structure models can be described as decorations of triangle-rectangle and rhombus tilings.  ...  Lennard-Jones potentials and simulated by molecular dynamics, was observed to freeze into a decagonal quasicrystal dominated by Frank-Kasper coordination shells and closely related to the Henley-Elser model for  ...  Then Cockayne's quasiperiodic T/R packing (Cockayne, 1994) satisfies the constraint just mentioned; it satisfied a second constraint, in that rectangles don't adjoin along b edges, either.  ... 
doi:10.1080/01418619708207207 fatcat:3pho3uepingazpvgqr6psefp5y

Mutations and short geodesics in hyperbolic 3-manifolds [article]

Christian Millichap
2016 arXiv   pre-print
If anything, this will only further break any possible symmetries.  ...  These identical polyhedra are transformed into ideal polyhedra by collapsing strands of J 2n+1 to ideal vertices.  ... 
arXiv:1406.6033v4 fatcat:msn6edb6zrennjllshv6btax3i

Mutations and short geodesics in hyperbolic 3-manifolds

Christian Millichap
2017 Communications in analysis and geometry  
If anything, this will only further break any possible symmetries.  ...  These identical polyhedra are transformed into ideal polyhedra by collapsing strands of J 2n+1 to ideal vertices.  ... 
doi:10.4310/cag.2017.v25.n3.a5 fatcat:6g3h2tbzlndc3bvvngxjqi6hlq

Influence of shape on ordering of granular systems in two dimensions

I. C. Rankenburg, R. J. Zieve
2001 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
The onset of ordered domains occurs near a packing density of 0.8, a phenomenon observed previously for disks.  ...  Since random packings are unstable for disks and many other shapes in two dimensions, trapezoid packings provide a new approach to studying two-dimensional randomness.  ...  Lawton for help in setting up the apparatus. This work is supported by the National Science Foundation under DMR-9733898.  ... 
doi:10.1103/physreve.63.061303 pmid:11415088 fatcat:4poayopfnnhinjeihfirtiigiq

Phase diagram of a system of hard cubes on the cubic lattice [article]

N. Vigneshwar, Dipanjan Mandal, Kedar Damle, Deepak Dhar, R.Rajesh
2019 arXiv   pre-print
Within each slab, the cubes are disordered; translation symmetry is thus broken along exactly one principal axis.  ...  Within each column, the system is disordered, and thus translational symmetry is broken only along two principal directions.  ...  Within each column, the system is disordered, and the columns break translational symmetry along along two principal directions.  ... 
arXiv:1902.06408v1 fatcat:rmvwtvqbdfaptcrrrcowryjzim

Data Structures for Higher-Dimensional Rectilinear Packing

Sam D. Allen, Edmund K. Burke
2012 INFORMS journal on computing  
This thesis presents a number of methods for generating high quality solutions for these cutting and packing challenges. It does so in a number of ways.  ...  Finally some mathematical models which can guarantee the optimality of packings for small datasets are given, and the (in)effectiveness of these techniques discussed.  ...  One can also attempt to break symmetries in the problem. If boxes i and j (where i < j) are identical (i.e.  ... 
doi:10.1287/ijoc.1110.0464 fatcat:v67giu2snrb4fnebt7gjsw3nla

The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space

Michael O'Keeffe, Michael M. J. Treacy
2022 Symmetry  
We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space.  ...  Symmetry 2022, 14, 822  ...  the identity and maximum symmetry of an embedding.  ... 
doi:10.3390/sym14040822 fatcat:mibwown2rbbt7flpycwlzydz2i

Cluster-packing geometry for Al-based F-type icosahedral alloys

Nobuhisa Fujita, Hikari Takano, Akiji Yamamoto, An-Pang Tsai
2013 Acta Crystallographica Section A Foundations of Crystallography  
The structure is well described as a dense packing of clusters of two kinds, which are known in the literature as the pseudo-Mackay type and the Bergman type clusters.  ...  B-cluster, the IADP's are set to be identical for the same site type.  ...  Importantly, the site degeneracies of a single shell in I or V generally break the icosahedral symmetry ( Figures 12, 13 and Table VI) , where the minimal point group is m3 (T h ).  ... 
doi:10.1107/s0108767313005035 fatcat:veafwhrfgbewbnrcctapptzlau

Protein structure: geometry, topology and classification

William R Taylor, Alex C W May, Nigel P Brown, András Aszódi
2001 Reports on progress in physics (Print)  
The review concludes with a consideration of the origin of higher-level symmetries in protein structure.  ...  Computer methods for the automatic comparison and classification of these structures are then reviewed with an analysis of the statistical significance of comparing different shapes.  ...  Acknowledgments David Jones and Jaap Heringa are thanked for help with some of the figures.  ... 
doi:10.1088/0034-4885/64/4/203 fatcat:l6fpa5l4v5cz5mwhehc3m5iqyi

Design of Architectured Materials Based on Topological and Geometrical Interlocking

Yuri Estrin, Vinayak R. Krishnamurthy, Ergun Akleman
2021 Journal of Materials Research and Technology  
Initially, monolayers made from identical blocks were considered, but later this concept was extended to multilayer structures and non-identical blocks [11] .  ...  according to wallpaper symmetries ( Fig. 9 (b, e)).  ... 
doi:10.1016/j.jmrt.2021.08.064 fatcat:ynpcm7p3ejfirebradnmoowqgu

Hierarchical self-assembly of patchy colloidal platelets

Carina Natalia Karner, Christoph Dellago, Emanuela Bianchi
2020 Soft Matter  
Anisotropy at the level of the inter-particle interaction provides the particles with specific instructions for the self-assembly of target structures.  ...  Introduction Nowadays, colloids of different shapes can be accurately synthesized within a vast range of symmetries, from complex convex units -such as spheroidal colloidal molecules, rods or polyhedra  ...  The packing fraction for 5-stars at P = 100 is 0.696 ± 0.0214, while for 6-stars at ∆ = 0.3 it is 0.706 ± 0.018.  ... 
doi:10.1039/d0sm00044b pmid:32104867 fatcat:gbmf4fuhnjekbneh4ia7svch4e

Hierarchical self-assembly of anisotropic colloidal platelets [article]

Carina Karner and Christoph Dellago and Emanuela Bianchi
2020 arXiv   pre-print
Anisotropy at the level of the inter-particle interaction provides the particles with specific instructions for the self-assembly of target structures.  ...  rhombic shape and two attractive sites placed along adjacent edges and we run Monte Carlo simulations in two-dimensions to investigate the two-stage assembly of these units into clusters with well-defined symmetries  ...  INTRODUCTION Nowadays, colloids of different shapes can be accurately synthesized within a vast range of symmetries, from complex convex units -such as spheroidal colloidal molecules, rods or polyhedra  ... 
arXiv:2001.00373v1 fatcat:ktmjvgyzcfbuxgj25l5ux6qgpy
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