ﻻ يوجد ملخص باللغة العربية
A convex polyhedron $P$ is $k$-equiprojective if all of its orthogonal projections, i.e., shadows, except those parallel to the faces of $P$ are $k$-gon for some fixed value of $k$. Since 1968, it is an open problem to construct all equiprojective polyhedra. Recently, Hasan and Lubiw [CGTA 40(2):148-155, 2008] have given a characterization of equiprojective polyhedra. Based on their characterization, in this paper we discover some new equiprojective polyhedra by cutting and gluing existing polyhedra.
We introduce polyhedra circuits. Each polyhedra circuit characterizes a geometric region in $mathbb{R}^d$. They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of polyhedra.
Starting with the unsolved Durers problem of edge-unfolding a convex polyhedron to a net, we specialize and generalize (a) the types of cuts permitted, and (b) the polyhedra shapes, to highlight both advances established and which problems remain open.
It is unknown whether every polycube (polyhedron constructed by gluing cubes face-to-face) has an edge unfolding, that is, cuts along edges of the cubes that unfolds the polycube to a single nonoverlapping polygon in the plane. Here we construct poly
We prove that every positively-weighted tree T can be realized as the cut locus C(x) of a point x on a convex polyhedron P, with T weights matching C(x) lengths. If T has n leaves, P has (in general) n+1 vertices. We show there are in fact a continuu
We present new examples of topologically convex edge-ununfoldable polyhedra, i.e., polyhedra that are combinatorially equivalent to convex polyhedra, yet cannot be cut along their edges and unfolded into one planar piece without overlap. One family o