### Abstract

Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behavior at larger ones. In many systems there is no single, optimal packing that dominates, but rather one must understand the entire set of possible packings. As a step in this direction we enumerate rigid clusters of identical hard spheres for n ≤ 14 and clusters with the maximum number of contacts for n ≤ 19. A rigid cluster is one that cannot be continuously deformed while maintaining all contacts. This is a nonlinear notion that arises naturally because such clusters are the metastable states when the spheres interact with a short-range potential, as is the case in many nano- or microscale systems. We believe that our lists are nearly complete, except for a small number of highly singular clusters (linearly floppy but nonlinearly rigid). The data contains some major geometrical surprises, such as the prevalence of hypostatic clusters: those with less than the 3n - 6 contacts generically necessary for rigidity. We discuss these and several other unusual clusters whose geometries may give insight into physical mechanisms, pose mathematical and computational problems, or bring inspiration for designing new materials.

Original language | English (US) |
---|---|

Pages (from-to) | 229-244 |

Number of pages | 16 |

Journal | SIAM Review |

Volume | 58 |

Issue number | 2 |

DOIs | |

State | Published - 2016 |

### Fingerprint

### Keywords

- Clusters
- Rigidity
- Sphere packing
- Statistical mechanics

### ASJC Scopus subject areas

- Applied Mathematics
- Computational Mathematics
- Theoretical Computer Science

### Cite this

*SIAM Review*,

*58*(2), 229-244. https://doi.org/10.1137/140982337

**Enumerating rigid sphere packings.** / Holmes-Cerfon, Miranda.

Research output: Contribution to journal › Review article

*SIAM Review*, vol. 58, no. 2, pp. 229-244. https://doi.org/10.1137/140982337

}

TY - JOUR

T1 - Enumerating rigid sphere packings

AU - Holmes-Cerfon, Miranda

PY - 2016

Y1 - 2016

N2 - Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behavior at larger ones. In many systems there is no single, optimal packing that dominates, but rather one must understand the entire set of possible packings. As a step in this direction we enumerate rigid clusters of identical hard spheres for n ≤ 14 and clusters with the maximum number of contacts for n ≤ 19. A rigid cluster is one that cannot be continuously deformed while maintaining all contacts. This is a nonlinear notion that arises naturally because such clusters are the metastable states when the spheres interact with a short-range potential, as is the case in many nano- or microscale systems. We believe that our lists are nearly complete, except for a small number of highly singular clusters (linearly floppy but nonlinearly rigid). The data contains some major geometrical surprises, such as the prevalence of hypostatic clusters: those with less than the 3n - 6 contacts generically necessary for rigidity. We discuss these and several other unusual clusters whose geometries may give insight into physical mechanisms, pose mathematical and computational problems, or bring inspiration for designing new materials.

AB - Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behavior at larger ones. In many systems there is no single, optimal packing that dominates, but rather one must understand the entire set of possible packings. As a step in this direction we enumerate rigid clusters of identical hard spheres for n ≤ 14 and clusters with the maximum number of contacts for n ≤ 19. A rigid cluster is one that cannot be continuously deformed while maintaining all contacts. This is a nonlinear notion that arises naturally because such clusters are the metastable states when the spheres interact with a short-range potential, as is the case in many nano- or microscale systems. We believe that our lists are nearly complete, except for a small number of highly singular clusters (linearly floppy but nonlinearly rigid). The data contains some major geometrical surprises, such as the prevalence of hypostatic clusters: those with less than the 3n - 6 contacts generically necessary for rigidity. We discuss these and several other unusual clusters whose geometries may give insight into physical mechanisms, pose mathematical and computational problems, or bring inspiration for designing new materials.

KW - Clusters

KW - Rigidity

KW - Sphere packing

KW - Statistical mechanics

UR - http://www.scopus.com/inward/record.url?scp=84973468684&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84973468684&partnerID=8YFLogxK

U2 - 10.1137/140982337

DO - 10.1137/140982337

M3 - Review article

VL - 58

SP - 229

EP - 244

JO - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 2

ER -