Sufficiently Fat Polyhedra are not 2-castable

TL;DR

This paper investigates the limitations of manufacturing convex polyhedra through a simplified casting process, demonstrating that certain fat polyhedra cannot be produced with only two cast parts due to geometric constraints.

Contribution

It introduces a generalized casting model and proves that convex polyhedra with a specific fatness ratio are not 2-castable, highlighting fundamental geometric limitations.

Findings

Convex polyhedra with a fatness ratio less than 1.07 cannot be 2-castable.

The paper establishes a geometric constraint for the casting process.

It extends understanding of manufacturability of convex shapes in casting.

Abstract

In this note we consider the problem of manufacturing a convex polyhedral object via casting. We consider a generalization of the sand casting process where the object is manufactured by gluing together two identical faces of parts cast with a single piece mold. In this model we show that the class of convex polyhedra which can be enclosed between two concentric spheres of the ratio of their radii less than 1.07 cannot be manufactured using only two cast parts.