四面体与立方体的地貌：多面体上最短路径探究

Geodesic trajectories on regular polyhedra

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern--Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.

Width Distributions for Convex Regular Polyhedra

The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the regular tetrahedron and the cube. Higher-order moments of f_tetra and f_cube have not been examined until now. Assume that each polyhedron has edges of unit length. We deduce that the mean square width of the regular tetrahedron is 1/3+(3+sqrt(3))/(3*pi) and the mean square width of the cube is 1+4/pi.

由勒洛多面体得到的一类新常宽体

A new family of constant width bodies obtained from Reuleaux polyhedra

该研究提出一种从单位勒洛多面体构造常宽体的新方法，其保留原多面体对称性，无需选择对偶边，对勒洛四面体可得到特定常宽体。

Shortest Paths and Convex Hulls in 2D Complexes with Non-Positive Curvature

Globally non-positively curved, or CAT(0), polyhedral complexes arise in a number of applications, including evolutionary biology and robotics. These spaces have unique shortest paths and are composed of Euclidean polyhedra, yet many algorithms and properties of shortest paths and convex hulls in Euclidean space fail to transfer over. We give an algorithm, using linear programming, to compute the convex hull of a set of points in a 2-dimensional CAT(0) polyhedral complex with a single vertex. We explore the use of shortest path maps to answer single-source shortest path queries in 2-dimensiona…

Acute triangulations of polyhedra and R^n

We study the problem of acute triangulations of convex polyhedra and the space R^n. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n>=4. Further, we prove that acute triangulations of the space R^n do not exist for n>= 5. In the opposite direction, in R^3, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of R^4 if all dihedral…

可能并非鲁珀特的星状四面体

A stellated tetrahedron that is probably not Rupert

研究提出寻找最简单的非鲁珀特多面体，并提供证据表明特定星状四面体可能非鲁珀特，采用线性规划计算最大可缩放多边形。