The Combinatorics of Flat Folds: a Survey

TL;DR

This survey reviews the combinatorial aspects of flat origami foldability, including conditions for flat folding, crease pattern analysis, and enumeration of fold configurations, highlighting recent theoretical advances and generalizations.

Contribution

It provides a comprehensive overview of flat foldability criteria, generalizes key theorems, and introduces recursive formulas for counting fold configurations.

Findings

Generalizations of Maekawa's and Kawasaki's Theorems

Necessary and sufficient conditions for single vertex foldability

Recursive formulas for counting fold configurations

Abstract

We survey results on the foldability of flat origami models. The main topics are the question of when a given crease pattern can fold flat, the combinatorics of mountain and valley creases, and counting how many ways a given crease pattern can be folded. In particular, we explore generalizations of Maekawa's and Kawasaki's Theorems, develop a necessary and sufficient condition for a given assignment of mountains and valleys to fold up in a special case of single vertex folds, and describe recursive formulas to enumerate the number of ways that single vertex in a crease pattern can be folded.