Random Packing of Spheres and Rods

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(1)Random sequential packing of rods with infinite height is studied. We assume that directions of axes of rods are parallel to the x or y or z axes, and their 'coordinates' that specify positions of rods are always integers. Then we prove that, for infinitely large system, packing density of rods is 3/4 and directions of axes of rods are statistically isotropic.(2)We study rod packing satisfying the following assumptions : (A1) the common shape of rods is a cylinder whose base is a circle of unit radius and infinite height ; (A2) axes of rods have directions parallel to one of the unit vectors n1, n2, n3 ; (A3) any two rods of the same direction do not neighbour each other, where we say that two rods neighbour each other if and only if their Voronoi cells neighbour each other. Then we show that the configuration of rods has the maximal packing density when three directions n1, n2, n3 are mutually orthogonal.(3)It is assumed that the number of directions of axes is more than or equal t … More o 3, and possible positions of rods form a 'irregular' configuration. Then we propose an efficient algorithm that can simulate the packing process. The algorithm is an amalgam of three fundamental algorithms in computer geometry : that of arrangement of straight lines, that of Voronoi diagram, and that of intersection of two convex polygons.(4)Regular periodic packing of rectangular rods with infinite height is studied in four-dimensional Euclidean spaces. Assume that directions of axes of rods are parallel to one of the coordinates axes of the space, and 'coordinates' that can specify positions of rods are always integers. Then we find there are only two types of rod packing ; one is packing by 'slim' rods, whose packing density is unity; another is that by 'fat' rods, whose packing density s 3/4. The full packing density for 'slim' rods is a new phenomenon that never occur in the usual three-dimensional space, while the packing density 3/4 for 'fat' rods coincides with that for rod packing in the three-dimensional space.(5)We study the following questions : (i)Toss a non-symmetric (thus non-cubic) dice. What is the probability that its each face lands on floor (or on top face of a table)? ; (ii)Can we make a non-symmetric dice but with equal probabilities that its each face lands on? We show that under some assumptions on tossing method the questioned probabilities are related to the spherical Laguerre diagram obtained by the dice (polyhedron). Based on the result, with aid of an efficient well-known algorithm generating the spherical Laguerre, we can construct a dice that is approximately fair.(6)We study a class of two-dimensional diagrams like the famous Penrose triangle that look like projective images of three-dimensional figures in local, but are inconsistent in global. Our focus is on diagrams that are made from the Archimedian tilings by replacing their edges by rectangular rods. In particular it is found there is only one possible case for 'impossible' honeycomb structures Less