arXiv subjects

Thomas C. Hull

Publications and source records attributed to Thomas C. Hull.

At least 19 recordsLinked to original sources

arXiv2026

Tunable Snapping and Rigid Foldability in the Mars Origami Pattern

Origami-inspired metamaterials exploit the interplay between geometry and elasticity to achieve programmable mechanical responses. Yet the origin and tunability of snap-through instabilities in non-rigidly foldable patterns remain poorly understood. Here we show that the Mars tessellation, a degree-4 vertex origami pattern composed of alternating square and rhombic faces, is not rigidly foldable because the folding-speed ratios required for vertex compatibility cannot be propagated consistently across neighboring units. This geometric incompatibility forces the facets to bend during folding, giving rise to a reproducible snap-through discontinuity in the force-displacement curve with a mean force drop of about 92.6 +/- 5.5 %, marking a transition between metastable states. Laser scoring of additional diagonal creases, guided by strain-field simulations, enables continuous tuning of the snap magnitude. These results reveal a general mechanism by which geometric frustration can be harnessed to program multistability in thin-sheet metamaterials.

cond-mat.soft↗

arXiv2025

The Stamp Folding Problem From a Mountain-Valley Perspective

A strip of square stamps can be folded in many ways such that all of the stamps are stacked in a single pile in the folded state. The stamp folding problem asks for the number of such foldings and has previously been studied extensively. We consider this problem with the additional restriction of fixing the mountain-valley assignment of each crease in the stamp pattern. We provide a closed form for counting the number of legal foldings on specific patterns of mountain-valley assignments, including a surprising appearance of the Catalan numbers. We describe results on upper and lower bounds for the number of ways to fold a given mountain-valley assignment on the strip of stamps, provide experimental evidence suggesting more general results, and include conjectures and open problems.

math.CO↗

arXiv2025

The Origami flip graph of the $2\times n$ Miura-ori

Given an origami crease pattern $C=(V,E)$, a straight-line planar graph embedded in a region of $\mathbb{R}^2$, we assign each crease to be either a mountain crease (which bends convexly) or a valley crease (which bends concavely), creating a mountain-valley (MV) assignment $\mu:E\to\{-1,1\}$. An MV assignment $\mu$ is locally valid if the faces around each vertex in $C$ can be folded flat under $\mu$. In this paper, we investigate locally valid MV assignments of the Miura-ori, $M_{m,n}$, an $m\times n$ parallelogram tessellation used in numerous engineering applications. The origami flip graph $OFG(C)$ of $C$ is a graph whose vertices are locally valid MV assignments of $C$, and two vertices are adjacent if they differ by a face flip, an operation that swaps the MV-parity of every crease bordering a given face of $C$. We enumerate the number of vertices and edges in $OFG(M_{2,n})$ and prove several facts about the degrees of vertices in $OFG(M_{2,n})$. By finding recurrence relations, we show that the number of vertices of degree $d$ and $2n-a$ (for $0\leq a$) are both described by polynomials of particular degrees. We then prove that the diameter of $OFG(M_{2,n})$ is $\lceil \frac{n^2}{2}\rceil$ and find lower bounds on the diameter of $OFG(M_{m,n})$ using techniques from $3$-coloring reconfiguration graphs.

math.CO↗

arXiv2025

Quasigeodesics on the Cube

A quasigeodesic is a curve on the surface of a convex polyhedron that has $\le π$ surface to each side at every point. In contrast, a geodesic has exactly $π$ to each side and so can never pass through a vertex, whereas quasigeodesics can. Although it is known that every convex polyhedron has at least three simple closed quasigeodesics, little else is known. Only tetrahedra have been thoroughly studied. In this paper we explore the quasigeodesics on a cube, which have not been previously enumerated. We prove that the cube has exactly $15$ simple closed quasigeodesics (beyond the three known simple closed geodesics). For the lower bound we detail $15$ simple closed quasigeodesics. Our main contribution is establishing a matching upper bound. For general convex polyhedra, there is no known upper bound.

math.MG↗

arXiv2025

Flat origami is Turing Complete

"Flat origami" refers to the folding of flat, zero-curvature paper such that the finished object lies in a plane. Mathematically, flat origami consists of a continuous, piecewise isometric map $f:P\subseteq\mathbb{R}^2\to\mathbb{R}^2$ along with a layer ordering $λ_f:P\times P\to \{-1,1\}$ that tracks which points of $P$ are above/below others when folded. The set of crease lines that a flat origami makes (i.e., the set on which the mapping $f$ is non-differentiable) is called its "crease pattern." Flat origami mappings and their layer orderings can possess surprisingly intricate structure. For instance, determining whether or not a given straight-line planar graph drawn on $P$ is the crease pattern for some flat origami has been shown to be an NP-complete problem, and this result from 1996 led to numerous explorations in computational aspects of flat origami. In this paper we prove that flat origami, when viewed as a computational device, is Turing complete, or more specifically P-complete. We do this by showing that flat origami crease patterns with "optional creases" (creases that might be folded or remain unfolded depending on constraints imposed by other creases or inputs) can be constructed to simulate Rule 110, a one-dimensional cellular automaton that was proven to be Turing complete by Matthew Cook in 2004.

math.CO↗

arXiv2025

On random locally flat-foldable origami

We develop a theory of random flat-foldable origami. Given a crease pattern, we consider a uniformly random assignment of mountain and valley creases, conditioned on the assignment being flat-foldable at each vertex. A natural method to approximately sample from this distribution is via the face-flip Markov chain where one selects a face of the crease pattern uniformly at random and, if possible, flips all edges of that face from mountain to valley and vice-versa. We prove that this chain mixes rapidly for several natural families of origami tessellations -- the square twist, the square grid, and the Miura-ori -- as well as for the single-vertex crease pattern. We also compare local to global flat-foldability and show that on the square grid, a random locally flat-foldable configuration is exponentially unlikely to be globally flat-foldable.

math.PR↗

arXiv2025

A rigid origami elliptic-hyperbolic vertex duality

The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the interior of the folded material and that are adjacent to four crease lines, i.e. degree-4 vertices, have a single degree of freedom and can be chained together to make flexible polyhedral surfaces. Degree-4 vertices that can fold to a completely flat state have folding kinematics that are very well-understood, and thus they have been used in many engineering and physics applications. However, degree-4 vertices that are not flat-foldable or not folded from flat paper so that the vertex forms either an elliptic or hyperbolic cone, have folding angles at the creases that follow more complicated kinematic equations. In this work we present a new duality between general degree-4 rigid origami vertices, where dual vertices come in elliptic-hyperbolic pairs that have essentially equivalent kinematics. This reveals a mathematical structure in the space of degree-4 rigid origami vertices that can be leveraged in applications, for example in the construction of flexible 3D structures that possess metamaterial properties.

math.MG↗

arXiv2024

Nonlinear Kinematics of Recursive Origami Inspired by the Spidron

Non-periodic folding of periodic crease patterns paves the way to novel nonlinear phenomena that cannot be feasible through periodic folding. This paper focuses on the non-periodic folding of recursive crease patterns generalized from Spidron. Although it is known that the Spidron has a $1$-DOF isotropic rigid folding motion, its general kinematics and dependence on the crease pattern remain unclear. Using the kinematics of a single unit cell of the Spidron and the recursive construction of the folded state of multiple unit cells, we consider the folding of the Spidron that is not necessarily isotropic. We found that as the number of unit cells increases, the non-periodic folding is restricted and the isotropic folding becomes dominant. Then, we analyze the three kinds of isotropic folding modes by constructing $1$-dimensional dynamical systems governing each of them. We show that these systems can possess different recursive natures depending on folding modes even in an identical crease pattern. Furthermore, we show their novel nonlinear nature, including the period-doubling cascade leading to the emergence of chaos.

cond-mat.soft↗

arXiv2022

Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean

We derive new algebraic equations for the folding angle relationships in completely general degree-four rigid-foldable origami vertices, including both Euclidean (developable) and non-Euclidean cases. These equations in turn lead to novel, elegant equations for the general developable degree-four case. We compare our equations to previous results in the literature and provide two examples of how the equations can be used: In analyzing a family of square twist pouches with discrete configuration spaces, and for proving that a new folding table design made with hyperbolic vertices has a single folding mode.

cond-mat.soft↗

arXiv2022

Maximal origami flip graphs of flat-foldable vertices: properties and algorithms

Flat origami studies straight line, planar graphs $C=(V,E)$ drawn on a region $R\subset\mathbb{R}^2$ that can act as crease patterns to map, or fold, $R$ into $\mathbb{R}^2$ in a way that is continuous and a piecewise isometry exactly on the faces of $C$. Associated with such crease pattern graphs are valid mountain-valley (MV) assignments $μ:E\to\{-1,1\}$, indicating which creases can be mountains (convex) or valleys (concave) to allow $R$ to physically fold flat without self-intersecting. In this paper, we initiate the first study of how valid MV assignments of single-vertex crease patterns are related to one another via face-flips, a concept that emerged from applications of origami in engineering and physics, where flipping a face $F$ means switching the MV parity of all creases of $C$ that border $F$. Specifically, we study the origami flip graph ${\rm{OFG}}(C)$, whose vertices are all valid MV assignments of $C$ and edges connect assignments that differ by only one face flip. We prove that, for the single-vertex crease pattern $A_{2n}$ whose $2n$ sector angles around the vertex are all equal, ${\rm{OFG}}(A_{2n})$ contains as subgraphs all other origami flip graphs of degree-$2n$ flat origami vertex crease patterns. We also prove that ${\rm{OFG}}(A_{2n})$ is connected and has diameter $n$ by providing two $O(n^2)$ algorithms to traverse between vertices in the graph, and we enumerate the vertices, edges, and degree sequence of ${\rm{OFG}}(A_{2n})$. We conclude with open questions on the surprising complexity found in origami flip graphs of this type.

math.CO↗

arXiv2022

Rigid folding equations of degree-6 origami vertices

Rigid origami, with applications ranging from nano-robots to unfolding solar sails in space, describes when a material is folded along straight crease line segments while keeping the regions between the creases planar. Prior work has found explicit equations for the folding angles of a flat-foldable degree-4 origami vertex and some cases of degree-6 vertices. We extend this work to generalized symmetries of the degree-6 vertex where all sector angles equal $60^\circ$. We enumerate the different viable rigid folding modes of these degree-6 crease patterns and then use $2^{nd}$-order Taylor expansions and prior rigid folding techniques to find algebraic folding angle relationships between the creases. This allows us to explicitly compute the configuration space of these degree-6 vertices, and in the process we uncover new explanations for the effectiveness of Weierstrass substitutions in modeling rigid origami. These results expand the toolbox of rigid origami mechanisms that engineers and materials scientists may use in origami-inspired designs.

math.MG↗

arXiv2019

Face flips in origami tessellations

Given a flat-foldable origami crease pattern $G=(V,E)$ (a straight-line drawing of a planar graph on a region of the plane) with a mountain-valley (MV) assignment $μ:E\to\{-1,1\}$ indicating which creases in $E$ bend convexly (mountain) or concavely (valley), we may \emph{flip} a face $F$ of $G$ to create a new MV assignment $μ_F$ which equals $μ$ except for all creases $e$ bordering $F$, where we have $μ_F(e)=-μ(e)$. In this paper we explore the configuration space of face flips for a variety of crease patterns $G$ that are tilings of the plane, proving examples where $μ_F$ results in a MV assignment that is either never, sometimes, or always flat-foldable for various choices of $F$. We also consider the problem of finding, given two foldable MV assignments $μ_1$ and $μ_2$ of a given crease pattern $G$, a minimal sequence of face flips to turn $μ_1$ into $μ_2$. We find polynomial-time algorithms for this in the cases where $G$ is either a square grid or the Miura-ori, and show that this problem is NP-hard in the case where $G$ is the triangle lattice.

math.CO↗

arXiv2019

Counting locally flat-foldable origami configurations via 3-coloring graphs

Origami, where two-dimensional sheets are folded into complex structures, is proving to be rich with combinatorial and geometric structure, most of which remains to be fully understood. In this paper we consider \emph{flat origami}, where the sheet of material is folded into a two-dimensional object, and consider the mountain (convex) and valley (concave) creases made by such foldings, called a \emph{MV assignment} of the crease pattern. We establish a method to, given a flat-foldable crease pattern $C$ under certain conditions, create a planar graph $C^*$ whose 3-colorings are in one-to-one correspondence with the locally-valid MV assignments of $C$. This reduces the general, unsolved problem of enumerating locally-valid MV assignments to the enumeration of 3-colorings of graphs.

math.CO↗

arXiv2018

Rigid Foldability is NP-Hard

In this paper, we show that deciding rigid foldability of a given crease pattern using all creases is weakly NP-hard by a reduction from Partition, and that deciding rigid foldability with optional creases is strongly NP-hard by a reduction from 1-in-3 SAT. Unlike flat foldability of origami or flexibility of other kinematic linkages, whose complexity originates in the complexity of the layer ordering and possible self-intersection of the material, rigid foldability from a planar state is hard even though there is no potential self-intersection. In fact, the complexity comes from the combinatorial behavior of the different possible rigid folding configurations at each vertex. The results underpin the fact that it is harder to fold from an unfolded sheet of paper than to unfold a folded state back to a plane, frequently encountered problem when realizing folding-based systems such as self-folding matter and reconfigurable robots.

cs.CG↗

arXiv2018

Rigid foldability of the augmented square twist

Define the augmented square twist origami crease pattern to be the classic square twist crease pattern with one crease added along a diagonal of the twisted square. In this paper we fully describe the rigid foldability of this new crease pattern. Specifically, the extra crease allows the square twist to rigidly fold in ways the original cannot. We prove that there are exactly four non-degenerate rigid foldings of this crease pattern from the unfolded state.

math.MG↗

arXiv2018

Self-foldability of monohedral quadrilateral origami tessellations

Using a mathematical model for self-foldability of rigid origami, we determine which monohedral quadrilateral tilings of the plane are uniquely self-foldable. In particular, the Miura-ori and Chicken Wire patterns are not self-foldable under our definition, but such tilings that are rotationally-symmetric about the midpoints of the tile are uniquely self-foldable.

math.MG↗

arXiv2017

Double-line rigid origami

In this paper, we will show methods to interpret some rigid origami with higher degree vertices as the limit case of structures with degree-4 supplementary angle vertices. The interpretation is based on separating each crease into two parallel creases, or \emph{double lines}, connected by additional structures at the vertex. We show that double-lined versions of degree-4 flat-foldable vertices possess a rigid folding motion, as do symmetric degree-$2n$ vertices. The latter gives us a symbolic analysis of the original vertex, showing that the tangent of the quarter fold angles are proportional to each other. The double line method is also a potentially useful in giving thickness to rigid origami mechanisms. By making single crease into two creases, the fold angles can be distributed to avoid $180^\circ$ folds, when panels can easily collide with each other. This can be understood as an extension of the crease offset method of thick rigid origami with an additional guarantee of rigid-foldability.

math.MG↗

arXiv2017

Sculpting the Vertex: Manipulating the Configuration Space Topography and Topology of Origami Vertices to Design Mechanical Robustness

The geometric, aesthetic, and mathematical elegance of origami is being recognized as a powerful pathway to self-assembly of micro and nano-scale machines with programmable mechanical properties. The typical approach to designing the mechanical response of an ideal origami machine is to include mechanisms where mechanical constraints transform applied forces into a desired motion along a narrow set of degrees of freedom. In fact, to date, most design approaches focus on building up complex mechanisms from simple ones in ways that preserve each individual mechanism's degree of freedom (DOF), with examples ranging from simple robotic arms to homogenous arrays of identical vertices, such as the well-known Miura-ori. However, such approaches typically require tight fabrication tolerances, and often suffer from parasitic compliance. In this work, we demonstrate a technique in which high-degree-of-freedom mechanisms associated with single vertices are heterogeneously combined so that the coupled phase spaces of neighboring vertices are pared down to a controlled range of motions. This approach has the advantage that it produces mechanisms that retain the DOF at each vertex, are robust against fabrication tolerances and parasitic compliance, but nevertheless effectively constrain the range of motion of the entire machine. We demonstrate the utility of this approach by mapping out the configuration space for the modified Miura-ori vertex of degree 6, and show that when strung together, their combined configuration spaces create mechanisms that isolate deformations, constrain the configuration topology of neighboring vertices, or lead to sequential bistable folding throughout the entire origami sheet.

cond-mat.soft↗