利用正多面体的联系解题

问题 已知一个正四面体和一个正八面体的棱长相等，把它们拼起来，使一个表面重合，所得的多面体有多少个面?     

颗粒增强复合材料的几何模型数值模拟方法研究

为了在细观层次上模拟混凝土和土石混合体等颗粒增强复合材料,假设颗粒为凸多面体.首先研究由随机八面体随机变形得到任意凸多面体及其参数方程的方法,然后研究凸多面体内部与外部的判定条件、点到多面体的距离和两多面体之间距离的计算方法,从而得到了一个生成具有大量多面体随机分布区域的方法.为了提高模拟区域中多面体的含量,还给出了下降算法.实验表明:可以按二级配生成多面体含量达35%（体积比）的模拟区域,为从细观层次研究混凝土、土石混合体等颗粒增强复合材料,提供了创建几何模型的方法.     

R~3中的一个Heilbronn型问题

本文给出Ｈｅｉｌｂｒｏｎｎ型问题的结果．设Ｓ是Ｒ~３中六点组成的集合．直径为Ｄ．若ｄ表示Ｓ中任意两点距离的最小值,则Ｄ≥２ｄ．等号当且仅当Ｓ是由正八面体的六个顶点或多面体面△×△１的六个顶点组成时才成立（△１,△２分别表示一维、二维正则单形,且其棱长相等）．     

R~3中的一个Heilbronn型问题

本文给出Ｈｅｉｌｂｒｏｎｎ型问题的结果．设Ｓ是Ｒ~３中六点组成的集合．直径为Ｄ．若ｄ表示Ｓ中任意两点距离的最小值,则Ｄ≥２ｄ．等号当且仅当Ｓ是由正八面体的六个顶点或多面体面△×△１的六个顶点组成时才成立（△１,△２分别表示一维、二维正则单形,且其棱长相等）．     

棱柱上的小覆盖的等变配边分类

多面体上的小覆盖的等变配边类是由它的切表示集所决定的.本文通过将棱柱上的小覆盖的切表示集约化到一种素形式,来确定其等变配边分类.     

高中数学小单元自测题

高中数学小单元自测题多面体与旋转体（高一）童克西（湖北省荆州中学４３４１００）第一套棱柱、棱锥、棱台１．斜四棱柱的矩形面最多有（）（Ａ）２个．（Ｂ）３个．（Ｃ）４个．（Ｄ）以上都不对．２．侧面都是直角三角形的正三棱锥，底面边长为ａ时，该三棱锥的全面积...     

为什么正多面体只有五种

我们知道,平面上的正多边形,可以有正三角形、正方形、正五边形、正六边形等等.对于任意一个正整数n,都有正n边形存在.平面上的多边形,类比到空间,就是多面体——由若干个平面多边形围成的封闭的空间图形.围成多面体的各个多边形叫多面体的面,两个面的公共边叫多面体的棱,棱和棱的公共点叫多面体的顶点.把多面体的任一面伸展成平面,如果其余的面都位于这个平面的同一侧,这样     

漫谈特殊化方法解题

例1（1999年全国高考）如图1，已知多面体ABCDEF中，面ABCD是边长为3的正方形，EF//AB，EF=3/2，EF与面AC的距离为2，该多面体的体积为（ ）．     

“分割”与“补形”

“分割”与“补形”宋遗先（湖北蒲圻市高中４３７３００）《立体几何》中为了简化多面体中的有些计算或证明，我们常将已知多面体分割成几部分或以已知的多面体为基础补成一些便于计算的多面体．我们不妨分别称为分割法和补形法，现举几例，供参考．图１一、分割法：例１...     

多面体对称性的探索

如果一个多面体关于某个平面对称，我们就称它具有对称性．正棱柱、正棱锥这些基本的几何体都具有对称性．利用对称来研究多面体，是一个容易被大家忽视的重要方法．从近几年的高考来看，立体几何题所给出的多面体很多都具有对称性．利用对称性质解题，所体现的思维过程更加完美．所以，我们要加强对多面体对称性的研究．     

Equifacetted 4-polytopes

Many polyhedra (convex 3-polytopes) are known which occur as facets (3-faces) of convex 4-polytopes. The purpose of this paper is to determine the combinatorial types of infinitely many more polyhedra with this property. This is achieved by determining the facets of polar bicyclic polytopes.     

On simplicial and cubical complexes with short links

We consider closed simplicial and cubicaln-complexes in terms of the links of their (n−2)-faces. Especially, we consider the case when this link has size 3 or 4, i.e., every (n−2)-face is contained in 3 or 4n-faces. Such simplical complexes withshort (i.e., of length 3 or 4) links are completely classified by theircharacteristic partition. We consider also embedding into (the skeletons of) hypercubes of the skeletons of simplical and cubical complexes. Research of the second author was financed by EC’s IIIRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe,” grant IIPRN-CT-2001-00272. The third author acknowledges financial support of the Russian Foundation of Fundamental Research (grant 02-01-00803) and the Russian Foundation for Scientific Schools (grant NSh. 2185.2003.1), Program OMN (Division of Mathematical Sciences) of the Russian Academy of Sciences.     

The number of faces of the tridiagonal Birkhoff polytope

Counting basic objects as the vertices of polyhedra is a demanding problem in general, even for the most basic structured polytope. In this paper, we determine the number of q-faces for some q ≥ 1 of the polytope of tridiagonal doubly stochastic matrices.     

Regularized extragradient method for finding a saddle point in an optimal control problem

We study n-dimensional cubical pseudomanifolds and their cellular mappings. In particular, we consider a discrete n-cube and all of its (n ? 1)-faces. Then, there exist either one or two or four faces of the cube each of which is mapped onto one face.     

Über die minimalen Seitenzahlen von Polytopen ohne dreieckige 2-Seiten

LetP be ad-polytope without triangular 2-faces,K ad-cube andf _n (P),f _n (K) the respective number ofn-faces. It is shown for simpleP or in dimensiond4 thatf _n (P)f _n (K), and for anyn equality holds if and only ifP andK are combinatorially equivalent.     

Extending shelling orders and a hierarchy of functions of unimodal simple polytopes

Let P be a simplicial d-polytope with n facets ((d − 1)-dimensional faces) in R^d. A shelling of P is an ordering of the facets of P such that the intersection of each facet F with the union of all facets that precede it the ordering is a nonempty union of (d − 2)-faces of F. The following open question was raised by Tverberg and is recorded in [4]. Suppose for some k < n, there is an ordering of k of the facets of P so that the intersection of each of these facets with the union of all of the facets that precede it in the ordering is a nonempty union of (d − 2)-faces. Can this initial “segment” be extended to a shelling of all the facets? This question is open even in the case that P is the dual of the d-dimensional hypercube. The question in this case has resurfaced several times since G. Danaraj and V. Klee (1978) in a variety of forms. It is related to the hierarchies of completely unimodal pseudo-Boolean functions studied in P.L. Hammer et al. (1988), the author (1988) and D. Wiedemann (1986). (A pseudo-Boolean function is a function mapping the vertices of the d-dimensional hypercube into the reals). In this paper, the hierarchies are compared and combined. This hierarchy is then extended to general simple polytopes, and the relationship to the above open question is explained.     

Recurrence Relationships for the Mean Number of Faces and Vertices for Random Convex Hulls

This paper studies the convex hull of n random points in R^d\mathsf{R}^{d} . A recently proved topological identity of the author is used in combination with identities of Efron and Buchta to find the expected number of vertices of the convex hull—yielding a new recurrence formula for all dimensions d. A recurrence for the expected number of facets and (d−2)-faces is also found, this analysis building on a technique of Rényi and Sulanke. Other relationships for the expected count of i-faces (1≤i<d) are found when d≤5, by applying the Dehn–Sommerville identities. A general recurrence identity (see (3) below) for this expected count is conjectured.     

Random projections of regular simplices

Precise asymptotic formulae are obtained for the expected number ofk-faces of the orthogonal projection of a regularn-simplex inn-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimensionn tends to infinity.F. Affentranger was supported by a grant from the Swiss National Foundation.