# Rules for Folding Polyminoes from One Level to Two Levels

## TL;DR

This paper explores folding rules for polyominoes to transform a single-layer shape into a two-layer shape, providing algorithms for some shapes and proving impossibility results for others.

## Contribution

It introduces specific folding algorithms and impossibility proofs for polyominoes under constrained folding moves.

## Key findings

- Certain polyominoes can be folded into two layers using the proposed rules.
- Some polyominoes cannot be folded into two layers with the allowed moves.

## Abstract

Polyominoes have been the focus of many recreational and research investigations. In this article, the authors investigate whether a paper cutout of a polyomino can be folded to produce a second polyomino in the same shape as the original, but now with two layers of paper. For the folding, only "corner folds" and "half edge cuts" are allowed, unless the polyomino forms a closed loop, in which case one is allowed to completely cut two squares in the polyomino apart. With this set of allowable moves, the authors present algorithms for folding different types of polyominoes and prove that certain polyominoes can successfully be folded to two layers. The authors also establish that other polyominoes cannot be folded to two layers if only these moves are allowed.