Random Packing of Spheres and Rods

（1）研究了无限高杆的随机有序堆积问题。我们假设杆的轴的方向平行于x或y或z轴，并且指定杆位置的“坐标”总是整数。然后证明了对于无限大系统，杆的堆积密度为3/4，杆的轴向是统计各向同性的。(2)We研究满足以下假设的棒堆积：（A1）棒的常见形状是以无限高的单位半径圆为底的圆柱体，（A2）棒的轴线方向平行于单位向量n1，n2，n3中的一个，（A3）棒的中心轴方向平行于单位向量n1，n2，n3中的一个，（A4）棒的中心轴方向平行于单位向量n1，n2，n3中的一个。（A3）任何两个相同方向的杆不彼此相邻，其中我们说两个杆彼此相邻当且仅当它们的Voronoi细胞彼此相邻。当三个方向n1，n2，n3相互正交时，棒的构型具有最大的堆积密度。(3)It假设轴的方向数大于或等于t ...更多信息 并且杆的可能位置形成“不规则”构造。然后，我们提出了一个有效的算法，可以模拟包装过程。该算法是计算机几何中三个基本算法的混合：直线排列算法、Voronoi图算法和凸多边形求交算法。（4）研究了四维欧氏空间中无限高矩形杆的正则周期填充问题。假设杆的轴的方向平行于空间的坐标轴之一，并且可以指定杆的位置的“坐标”总是整数。结果表明，棒填料只有两种类型，一种是“细”棒填料，其填料密度为1，另一种是“胖”棒填料，其填料密度为3/4。“细”棒的全堆积密度是一种新的现象，在通常的三维空间中从未发生过，而“胖”棒的堆积密度3/4与三维空间中棒堆积的堆积密度一致。(5)We研究以下问题：（i）掷一个非对称（因此非立方体）骰子。它的每个面落在地板上（或桌子的顶面上）的概率是多少？（ii）我们能不能做一个非对称的骰子，但每个骰子面落在上面的概率相等？我们表明，在投掷方法的一些假设下，所质疑的概率与由骰子（多面体）得到的球形拉盖尔图有关。在此基础上，借助于一种高效的著名算法生成球面拉盖尔，我们可以构造一个近似公平的骰子。(6)We研究一类二维图，如著名的彭罗斯三角形，它们在局部看起来像三维图形的投影像，但在全局上不一致。我们的重点是图表，是从阿基米德瓷砖取代其边缘的矩形杆。特别是它被发现只有一个可能的情况下，'不可能'蜂窝结构减

(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