Abstract
Finite Packings of circles, spheres or other convex bodies are investigated in various fields. We give a broad survey of planar results based on parametric density, because there is no such survey yet. For —dimensional packings we describe the atomistic approach to Wulff shape for general crystals, i.e. via periodic packings of balls of different size. The corresponding approach for quasicrystals is in the survey by[7].
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baake, M., R.V. Moody (1998): ‘Diffractive Point Sets with Entropy’, J. Phys. A31 pp. 9023–9039
Betke, U., M. Henk, J.M. Wills (1994): ‘Finite and infinite packings’, J. reine angew. Math. 453, pp. 165–191
Betke, U., M. Henk, J.M. Wills (1995): ‘A new approach to covering’, Mathematika 42, pp. 251–263
de Boissieu, M., J.L. Verger-Gaugry, R. Currat (Eds.): Aperiodic’ 97, Proc. Intern. Conf. of Aperiodic Crystals World Scientific, Singapore
Böröczky, K., Jr., U. Schnell (1998): ‘Wulff shape for non-periodic arrangements’, Lett. Math. Phys. 45, pp. 81–94
Böröczky, K., Jr., U. Schnell (1999): ‘Quasicrystals and Wulff-shape’, Disc. Comp. Geom. 21, pp. 421–436
Böröczky, K., Jr., U. Schnell, J.M. Wills (1999): ‘Quasicrystals, Parametric Density and Wulff shape’. In: Directions in mathematical Quasicrystals, ed. by M. Baake, R.V. Moody, to appear
Cockayne, E. (1995): ‘The quasicrystalline sphere packing problem’, Proc. 5th. Int. Conf. Quasicr., ed. by C. Janot, R. Mosseri (World Scient.)
Dinghas, A. (1943): ‘Über einen geometrischen Satz von Wulff über die Gleichgewichtsform von Kristallen’, Z. Kristallogr. 105, pp. 304–314
Duncan, M.A., D.H. Rouvray (1989): ‘Microclusters’, Scient. Amer., Dec. 1989, pp. 60–65
Edelsbrunner, TH., E.P. Mücke (1994): ‘Three-dimensional alpha-shapes’, ACM Transact. Graph 13, pp. 43–72
Graham, R.L., H.S. Witsenhausen, H.J. Zassenhaus (1972): ‘On tightest packings in the Minkowski plane’, Pacific J. Math. 41, pp. 699–715
Gritzmann, P., J.M. Wills (1993): ‘Finite Packing and Covering’. In: Handbook of Convex Geometry, pp. 863–897 (North Holland, Amsterdam)
Gruber, P.M., C.G. Lekkerkerker (1987): Geometry of Numbers, 2nd edn. (North Holland, Amsterdam)
Hoare, M.R., J. McInness (1983): ‘Morphology and statistical statics of simple microclusters’, Adv. Phys. 32, pp. 791–821
Jacobs, K., St. Herminghaus, K.R. Mecke (1998): ‘Thin Liquid Polymer Films Rupture via defects’, Langmuir 14, pp. 965–969
Jacobs, K., St. Herminghaus, K.R. Mecke, J. Bischof, A. Fery, M. Ibn-Elhaj, St. Schlagowski (1998): ‘Spinodal dewetting in Liquid Crystal and Liquid Metal Films’, Science 282, pp. 916–919
Janin, J., F. Rodier (1995): ‘Protein Interaction at Crystal Contacts’, Proteins: Structure, Function and Genetics 23, pp. 580–587
Kerscher, M., J. Schmalzing et al. (1997): ‘Minkowski funcionals of Abell/ACO clusters’, Mon. Not. R. Astron. Soc. 284, pp. 73–84
v. Laue, M. (1943): ‘Der Wulffsche Satz für die Gleichgewichtsform von Kristallen’, Z. Kristallogr. 105, pp. 124–133
Löwen, H. (1990): ‘Equilibrium shapes of crystals near the triple point’, Surface Science 234, pp. 315–323
Löwen, H., M. Schmidt (1997): ‘Freezing in confined supensions’, Progr. Colloid Polym. Sci. 104, pp. 81–89
Mecke, J., R. Schneider, D. Stoyan, W.R.R. Weil (1990): Stochastische Geometrie (Birkhäuser, Basel)
Mosseri, R., J.F. Sadoc (1989): ‘Description of metallic and covalent clusters with icosahedral symmetry: the polytope model’, Z. Phys. D-Atoms, Molecules and Clusters 12, pp. 89–92.
Nemeth, Z.T., H. Löwen (1998): ‘Freezing in Finite: Systems: Hard discs in circular cavities’, J. Phys. Condens. Matter 10, pp. 6189–6204
Northby, J.A. (1987): ‘Structure and binding of Lennard-Jones clusters’, J. Chem. Phys. 87, pp. 6166–6177
Olami, Z., S. Alexander (1988): ‘Quasiperiodic packing densities’, Phys. Rev. 37, pp. 3973–3978
Pach, J. (Ed.) (1991): New Trends in Discr. and Compr. Geom. (Springer, Berlin 1991)
Pach, J., P.K. Agarwal (1995): Combinatorial Geometry (John Wiley, New York)
McPherson, A. (1989): ‘Macromolecular Crystals’, Scient. American, March 1989, pp. 42–49
Rivier, Ch., J.F. Sadoc (1988): ‘Polymorphism and disorder in a close-packed-structure’, Europhys. Lett. 7, pp. 523–528
Sangwine-Yager, J.R. (1993): ‘Mixed volumes’. Ch. 1.2 in: Handbook of Convex Geometry, ed. by P.M. Gruber, J.M. Wills (North Holland, Amsterdam)
Schmalzing, J., K.M. Gorski (1998): ‘Minkowski functionals used in the morphological analysis of cosmic microwave background anisotropy maps’, Mon. Not. R. Astron. Soc. 297, pp. 355–365
Schmidt, M., H. Löwen (1997): ‘Phase Diagram of hard spheres confined between two parallel plates’, Phys. Review E 55, pp. 7228–7241
Schnell, U. (1999): ‘Periodic sphere packings and the Wulff-shape’, Beitr. Allg. Geom. 40, pp. 125–140
Schnell, U. (2000): ‘FCC versus HCP via Parametric Density’, to appear in Discrete Math
Smith, A.P. (1993): ‘The sphere packing problem in quasicrystals’, J. of Non-Cryst. Solids 153/154, pp. 258–263
Verger-Gaugry, J.L. (1988): ‘Approximate icosahedral periodic tilings with pseudo-icosahedral symmetry in reciprocal space’, J. Phys. (France) 49, pp. 1867–1874
Wills, J.M. (1990): ‘A quasicrystalline sphere-packing with unexpected high density’, J. Phys. (France) 51, pp. 1061–1064
Wills, J.M. (1993): ‘Finite sphere packings and sphere coverings’, Rend. Semin. Mat., Messina, Ser. II 2, pp. 91–97
Wills, J.M. (1996): ‘Lattice packings of spheres and Wulff-shape’, Mathematika 86, pp. 229–236
Wills, J.M. (1997a): ‘On large lattice packings of spheres’, Geom. Dedicata 65, pp. 117–126
Wills, J.M. (1997b): ‘Parametric density, online packings and crystal growth’, Rendic. di Palermo 50, pp. 413–424
Wills, J.M. (1998): ‘Crystals and Quasicrystals, Sphere Packings andWulff-Shape’, Proc. Aperiodic’ 97, ed. by J.L. Verger-Gaugry World Scientific, Singapore
Wills, J.M. (1999): ‘The Wulff-shape of large periodic packings’, Proc. DIMACS workshop Discrete Math. Chem., to appear
Woodcock, L.V. (1997): ‘Entropy difference between the face centered cubic and the hexagonal close-packed crystal structures’, Nature 385, pp. 141–142
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Wills, J.M. (2000). Finite Packings and Parametric Density. In: Mecke, K.R., Stoyan, D. (eds) Statistical Physics and Spatial Statistics. Lecture Notes in Physics, vol 554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45043-2_12
Download citation
DOI: https://doi.org/10.1007/3-540-45043-2_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67750-5
Online ISBN: 978-3-540-45043-6
eBook Packages: Springer Book Archive