Enumerating All Convex Polyhedra Glued from Squares in Polynomial Time

TL;DR

This paper introduces a polynomial-time algorithm for enumerating and classifying all convex polyhedra formed by edge-to-edge gluings of unit squares, with potential applications to other regular polygons.

Contribution

It provides the first polynomial-time method to enumerate all such convex polyhedra and extends the technique to other regular polygons.

Findings

Number of gluings is polynomial in the number of squares

Algorithm runs in polynomial time, pseudopolynomial if only n is input

Technique applicable to hexagon and triangle gluings

Abstract

We present an algorithm that enumerates and classifies all edge-to-edge gluings of unit squares that correspond to convex polyhedra. We show that the number of such gluings of $n$ squares is polynomial in $n$, and the algorithm runs in time polynomial in $n$ (pseudopolynomial if $n$ is considered the only input). Our technique can be applied in several similar settings, including gluings of regular hexagons and triangles.