Generalized D-Forms Have No Spurious Creases

TL;DR

This paper proves that generalized D-forms and similar seam forms have highly restricted crease patterns, with no spurious creases in flat regions, resolving open problems in geometric origami theory.

Contribution

It establishes that convex seam forms have no creases in flat parts except at vertices or along seams, solving longstanding open problems.

Findings

Flat components of D-forms have no creases.

Flat component of pita-forms has at most one crease.

Creases only occur at vertices or tangent to seams.

Abstract

A convex surface that is flat everywhere but on finitely many smooth curves (or "seams") and points is a seam form. We show that the only creases through the flat components of a seam form are either between vertices or tangent to the seams. As corollaries we resolve open problems about certain special seam forms: the flat components of a D-form have no creases at all, and the flat component of a pita-form has at most one crease, between the seam's endpoints.