Edge-Unfolding Prismatoids: Tall or Rectangular Base

TL;DR

This paper introduces new methods for edge-unfolding certain convex prismatoids without overlap, expanding previous results and highlighting challenges with more complex bases.

Contribution

It presents novel edge-unfolding techniques for specific classes of prismatoids, including those with tall or rectangular bases, extending prior work and identifying limitations.

Findings

Edge-unfolding of prismatoids with tall or rectangular bases achieved without overlap.

Extension of petal unfolding technique to new classes of convex polyhedra.

Counterexample demonstrating limitations for general quadrilateral bases.

Abstract

We show how to edge-unfold a new class of convex polyhedra, specifically a new class of prismatoids (the convex hull of two parallel convex polygons, called the top and base), by constructing a nonoverlapping "petal unfolding" in two new cases: (1) when the top and base are sufficiently far from each other; and (2) when the base is a rectangle and all other faces are nonobtuse triangles. The latter result extends a previous result by O'Rourke that the petal unfolding of a prismatoid avoids overlap when the base is a triangle (possibly obtuse) and all other faces are nonobtuse triangles. We also illustrate the difficulty of extending this result to a general quadrilateral base by giving a counterexample to our technique.