Papers:

*Counting locally flat-foldable origami configurations via 3-coloring graphs*by Alvin Chiu, William Hoganson, Thomas C. Hull, and Sylvia Wu; preprint.*Connectivity in origami flip graphs for flat-foldable vertices*by Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya Ter-Saakov; in preparation.*Maximal-size origami flip graphs of flat-foldable vertices: properties and algorithms*by Thomas C. Hull, Manuel Morales, Sarah Nash, and Natalya Ter-Saakov; in preparation.

**2019 Research Description:**
In 2019 the research topic of MathILy-EST will be **the combinatorics of flat origami**, an area that combines discrete mathematics with geometry, under the direction of Dr. Thomas Hull.
Origami, otherwise known as paper folding, is modeled by a *crease pattern* \((P,C)\) where \(P\) is a compact region of \(\mathbb{R}^2\) (the paper) and \(C\) is a straight-line planar graph drawn on \(P\). A *flat origami* with crease pattern \((P,C)\) is a function \(f:P\to\mathbb{R}^2\) that is an isometry on every face of \(C\), continuous, and non-differentiable on the edges and vertices of \(C\).

Intuitively, flat origamis are folded paper models that can be pressed in a book without crumpling or adding new creases; see Figure 1 for some examples. We model how the faces of \(C\) get folded above or below neighboring faces via a *mountain-valley (MV) assignment* on the creases. Formally, a MV assignment is a function \(\mu:E(C)\to\{-1,1\}\), where \(E(C)\) is the edge set of \(C\) and \(-1\) denotes mountain (convex) crease while \(1\) denotes a valley (concave) crease.

Figure 1: (a) Mountain and valley creases, with the classic crane crease pattern. (b) The Miura-ori crease pattern, with MV assignment, and the folded result.

A MV assignment is called *valid* if it can be realized in a flat origami \(f\) without causing any self-intersections of the paper. (For example, a crease pattern where all the creases are mountains would not be possible to fold flat, and thus is invalid.)

The major open problem in this area, which has applications in materials science and engineering, is the following: *Given a flat-foldable crease pattern \(C\), how many different valid MV assignments exist on \(C\)?* A recent advance has been the discovery that for some flat-foldable crease patterns this question can be translated into a graph-coloring problem. Developing a general theory of the link between valid MV-assignments and graph colorings (or other combinatorial structures) is the main thrust of this REU. Many smaller open problems exist, and will be explored, as we work towards this goal.

**2019 Research Mentor:** Dr. Thomas Hull is an Associate Professor of Mathematics at Western New England University in Springfield, MA. He has been researching the mathematics of paper folding since 1990 and is considered one of the world experts on the topic. His book *Project Origami* explores the use of origami in math education at the college level, and he is currently finishing *Origametry: Mathematical methods in paper folding*, a research-level monograph on origami mathematics forthcoming from Cambridge University Press.

*MathILy, MathILy-Er, and MathILy-EST are projects of the nonprofit organization Mathematical Staircase, Inc..*