Are All Convex Polyhedra Castable?

Title: Are All Convex Polyhedra Castable?

Abstract: This research investigates the castability of convex polyhedra, a concept from the field of manufacturing. Castability refers to the process of creating a three-dimensional object by pouring a molten substance into a mold and allowing it to solidify. The study focuses on a specific type of mold called a sand casting mold, which is divided into two parts. The main question is whether all convex polyhedra can be manufactured using this method.

The research finds that not all convex polyhedra are castable. It presents a general class of convex polyhedra that are not 2-castable, meaning they cannot be manufactured using two cast parts. This class includes polyhedra that are close approximations of a sphere and have a certain number of vertices and facets.

The study also provides useful lemmas and bounds for castable polyhedra, such as the length of their edges, the area of their faces, and the volume of the entire polyhedron. These findings contribute to a better understanding of the castability of convex polyhedra and may have implications for the manufacturing industry.

Implications: The results of this research have implications for the manufacturing industry, particularly in the field of casting. Understanding which polyhedra are castable and which are not can help improve the design and manufacturing process. Additionally, the bounds provided for castable polyhedra can be used to estimate the size and volume of the objects that can be produced using this method.

Methodology: The research uses mathematical proofs and geometric concepts to analyze the castability of convex polyhedra. It starts by defining what it means for a polyhedron to be castable and then presents a general class of polyhedra that are not 2-castable. The study also provides lemmas and bounds to further understand the properties of castable polyhedra.

Results: The main result of this research is the identification of a general class of convex polyhedra that are not 2-castable. The study also provides useful bounds for the length of edges, the area of faces, and the volume of castable polyhedra.

Keywords: Convex polyhedra, castability, sand casting, manufacturing, geometry, bounds

Link to Article: https://arxiv.org/abs/0203025v1 Authors: arXiv ID: 0203025v1