- Usage of the database
- parametric density
- Packings derived from Tetrahedra
- Packings derived from bipyramids
- deltahedra
- non-lattice packings
- history and development
- technical details
- References
- Links

number min = 50

number max = 60

rho min = 1

rho max = 1

and press the submit butten. Then you get a table of packings, like this:

Nr.: |
Anzahl |
M |
O |
V |
Typ |
rho_u |
rho_o |
---|---|---|---|---|---|---|---|

1156 | 50 | 307.876080 | 0.000000 | 0.000000 | saus | 0.000000 | 1.000636 |

3945 | 51 | 314.159265 | 0.000000 | 0.000000 | saus | 0.000000 | 1.002966 |

2171 | 52 | 320.442451 | 0.000000 | 0.000000 | saus | 0.000000 | 1.001990 |

1408 | 53 | 326.725636 | 0.000000 | 0.000000 | saus | 0.000000 | 1.001048 |

429 | 54 | 333.008821 | 0.000000 | 0.000000 | saus | 0.000000 | 1.001432 |

4661 | 55 | 339.292007 | 0.000000 | 0.000000 | saus | 0.000000 | 1.000428 |

1647 | 56 | 48.392582 | 159.348674 | 137.650120 | fcc-g | 0.999577 | 1.059269 |

4630 | 57 | 351.858377 | 0.000000 | 0.000000 | saus | 0.000000 | 1.000005 |

2476 | 58 | 48.392582 | 160.420471 | 148.963829 | fcc-g2 | 0.999205 | 1.597342 |

1311 | 59 | 49.495153 | 163.348674 | 150.849447 | fcc-g2 | 0.998430 | 1.394002 |

2129 | 60 | 50.597724 | 166.276878 | 152.735065 | fcc-g | 0.997679 | 1.358493 |

The table contains 8 columns. The first gives the ID of the packing, which is sometimes a hyperlink. The second column gives the number of spheres and the third to the fifth contain the values for mean width, surface and volume. The next column supplies a label, which can be one of the following:

label | meaning |

saus | the packing is a so called sausage packing |

fcc-g | It is an fcc-packing whose facets are contained in a hexagonal lattice |

fcc-g2 | fcc packing with hexagonal and quadratic facets |

bipyr | packing derived from a bipyramid |

other | non-lattice packing |

If you follow the hyperlink given by the ID of the packing you will be lead to a page which provides you with further Information about that packing and a picture. The picture is divided into six areas

top left | top middle | top right |

bottom left | bottom middle | bottom right |

I hope from the pictures one is able to imagine the packing.

Another feature of the database is to generate pictures for tetrahedral packings which you provide. If this is in your mind you should vist the tetrahedra page. For example you are interested in a packing cutted out of a tetrahedron with an edge length of 8 spheres. From one vertice you want to cut off a tetrahedron with edgelength of 2 spheres and also from the other three vertices. So you enter:

k |
k1 |
k2 |
k3 |
k4 |
t1 |
t2 |
t3 |
t4 |
t5 |
t6 |
view |
---|---|---|---|---|---|---|---|---|---|---|---|

7 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

value | edge |

t1 | edge from vertice 1 to 2 (magenta to red) |

t2 | from vertice 1 to 3 (magenta to green) |

t3 | from vertice 1 to 4 (magenta to yellow) |

t4 | from vertice 2 to 3 (red to green) |

t5 | from vertice 2 to 4 (red to yellow) |

t6 | from vertice 3 to 4 (green to yellow) |

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The main interest of my research project is the possible connection of densest packings to microclusters. So I took good clusters found by physicians and calculated their quermassintegrals to compare them to the packings in my database. These were the next packings added to the database (Gold Clusters and Lennard Jones Clusters).

Another Important structure in Physics are lamminated hexagonal packings e.g. the hexagonal closest packing (hcp). If such a packing has only three layers it can be treated as a piece from a tetragonal bipyramid. So I added my results for these packings to the database. A special candidate of these is the packing of 61 spheres, because it is the only packing of those investigated which is for Parameter 1 denser than the linear sausage-arrangement.

Following an observation of Prof. Wills, that some of the densest packings are deltahedra (Polytopes, where all the faces are regular triangles), we tested all these packings (February 2001) and observed that they are very dense. Unfortunately there are only deltahedra for 4-12 spheres with exception of 11. But when there exists a deltahedra it is specially dense.

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[2] K.Böröczky jr. U.Schnell and J.M.Wills, Quasicrystals, Parametric Density and Wulff-Shape, in: Directions in Math. Quasicrystals, edit. M.Baake, R.V. Moody, AMS, CRM-Series, Montreal 2000

[3] J.H.Convay and N.J.A.Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer, New York 1999

[4] B. Grünbaum, Convex Polytopes, Wiley & Sons, London 1967

[5] R.L.Graham and N.J.A.Sloane, Penny-Packing and two-simensional codes, Discrete Comp. Geom. 5 (1990) 1-11

[6] P.Gritzmann and J.M.Wills, Finite packing and covering, in: Handbook of Convex Geometry, eds. P.M.Gruber and J.M.Wills, North-Holland, Amsterdam 1993

[7] N.W.Johnson, Convex polyhedra with regular faces, Canad. J. Math. 18 (1966) 169-200

[8] J.R.Sangwine-Yager, Mixed volumes, in: Handbook of Convex Geometry, eds. P.M.Gruber and J.M.Wills, North-Holland, Amsterdam 1993

[9] U.Schnell, Periodic Sphere packings and the Wulff-Shape, Beitr. Alg. Geom. 40 (1999) 125-140

[10] U.Schnell, FCC versus HCP via parametric density, Discrete Math. 211 (2000) 269-274

[11] U.Schnell, J.M.Wills, Densest packings of more than three d-spheres are nonplanar, Discrete Comp. Geom. 24 (2000) 539-549

[12] P. Scholl, Microcluster im fcc-Gitter, Diploma-Thesis [pdf, ps]

[13] N.J.A.Sloane, R.H.Hardin, T.D.S.Duff and J.H.Convay, Minimal-Energy Clusters of Hard Spheres, Discrete Comp. Geom. 14 (1995) 237-259

[14] J.M.Wills, Finite sphere Packings and Sphere Coverings, rendic.Sem.Matem.Messina, Ser. II 2 (1993) 91-97

[15] N.T.Wilson and R.L.Johnston, Modelling gold clusters with an empirical many-body potential, Europ. Phys. J. D 12 (2000) 161-169

[16] V.A.Zalgaller, Convex polyhedra with regular faces, Sem. in Math., Steklov MI; Amer. Transl. New York 1969

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