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Info Board

Usage of the database

The use of the database starts in most cases from the search form. There you can enter special criteria for the packings you are searching. Suppose you are interested in packings where the number of spheres lies between 50 and 60 and the parameter of your interest is about 1. So you enter into the search form:
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

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:
saus the packing is a so called sausage packing
fcc-gIt is an fcc-packing whose facets are contained in a hexagonal lattice
fcc-g2fcc packing with hexagonal and quadratic facets
bipyrpacking derived from a bipyramid
other non-lattice packing
The last two columns describe the rho-Intervall in which the packing is denser than all the other considered packings.

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
linked to other viewpoints. In case of a tetrahedral packing, this means a packing which can be obtained by cutting off the vertices and edges of a fcc packing, where the convex hull of the midpoints of the spheres is a tetrahedron, you can choose between 14 viewpoints: You can look on the 4 vertices of the former tetrahedron, the 4 facets or the 6 edges. To make it easier to distinguish between the viewpoints the vertices of the tetrahedron are described by different colors (magenta,red,green,yellow). A non-lattice packing can be viewed from 90 directios, which are described by sphere-coordinates in theta and phi.

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
This describes a tetrahedron with edgelengt k+1 spheres or 2k, if the radius of a sphere is 1. And at the vertices tetrahedra are removed with an edgelength of k1 to k4 spheres, or 2k1. The first tetrahedron is removed at the magenta vertce (Nr. 1), then red, green and yellow. The Values t1-t6 describe cuts taken at the edges asociated to the edges of the tetrahedron.
value edge
t1edge from vertice 1 to 2 (magenta to red)
t2from vertice 1 to 3 (magenta to green)
t3from vertice 1 to 4 (magenta to yellow)
t4from vertice 2 to 3 (red to green)
t5from vertice 2 to 4 (red to yellow)
t6from vertice 3 to 4 (green to yellow)
You will soon see, if you simply try. One restriction is done to this. The Value for k should not exceed 39.

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History and development

The first packings I have investigated were the truncations of tetrahedra, whose vertices are latticepoints in the fcc-lattice. At the beginning only truncations at the vertices have been developed, where the slice plane is parallel to the opposing facet of the tetrahedron. We call these polytopes Groemer-Polytopes because every facet of such a polytop is contained in a densest two-dimensional sublattice of the fcc-lattice. In the sequel I added truncations at the edges of Groemer-Polytopes, where the slice planes lie in the second densest sublattice of the fcc-lattice (the Z2-lattice). This is only possible at maximal 6 edges, which lie parallel to one of the edges of the tetrahedron, out of which the Groemer-Polytope was obtained. With the results of these operations I began the database in January 2001.

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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[1] U.Betke, M.Henk and J.M.Wills, Finite and infinite packings, J. reine angew. Math. 453 (1994) 165-191

[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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