From: "G. A. Edgar" Subject: Re: Most Dense Sphere Package Date: Fri, 23 Apr 1999 07:55:35 -0700 (PDT) Newsgroups: sci.math,alt.math.moderated Keywords: Apollonian gasket In article <7fpvs5$sqp$1@nnrp1.dejanews.com>, wrote: > I have a simple linear algebra problem which should be already solved. > Can somebody give me an advise where I can look it up ? > > Problem: > We consider first spheres with equal diameter. These spheres can be packed > together to create a most dense unit cell. > Within the voids of this unit cell, you can put a certain number of smaller > sheres, which just fit the space available. Within the remaining voids you can > put spheres with a smaller diameter again, and so on and so on ... > > Whats the ratio between the radii the first, second, third...type of spheres ? > Whats the ratio between the remaining voids and the volume of the unit cell > after filling in first, second, third ... type of spheres into the unit cell. After the third step or so, the speres of any step are not all the same size. This makes the analysis interesting (and difficult). In 2 dimensions (packing circles) this is called (following Mandelbrot) the Appolonian gasket. Some early references... Boyd, Mathematika 20 (1973) 170--174 Boyd, Aeq Math 7 (1971) 182--193 Melzak, Math. Comp. 23 (1969) 169 Malzak, Canad. Math. J. 18 (1966) 838--852 Hirst, J. London Math. Soc. 42 (1960) 457--469 -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax)