Computational origami design typically focuses on achieving a desired shape of folding, treating multiple layers of paper like a single layer. In this paper, we study when we can achieve a desired shape with a desired constant number of layers throughout the shape, or a specified pattern of layer thicknesses. Specifically, we study the case of a rectangular strip of paper, which is the setting of the first universal computational origami design algorithm [SoCG’99]. Depending on the generality of the target surface and on the number of layers modulo 4, we give a variety of universal design algorithms, polynomial-time decision algorithms characterizing what is possible to fold, and NP-hardness results.

We analyze the mathematical existence of one of David Huffman’s most prominent curved-crease designs: the Hexagonal Column with Cusps, featuring circular, parabolic, and straight creases. Observations of the physical folded shape suggest that the concave regions between two parabolas form a cylinder, and the regions between the circle and the nearest intersection of the parabolas form a cone. In our analysis, we deduce the remaining rulings that result in a numerically closed hexagonal shape. Finally, we explore other variations of the shape, including those that incorporate only circular creases.

Planar curved creases represent an important family of creases and offer significant potential for application in interactive design tools. Once a folded state is constructed, the question arises: Does a folding motion exist, maintaining the same ruling layout, that connects the flat state to the folded state? This is referred to as a rigid-ruling folding motion. In this paper, we characterize combinations of planar curves that allow a rigid-ruling folding motion and illustrate our theoretical findings with examples.

In this paper, we investigate rigid-ruling folding motions of crease-rule patterns, that is, conjugacy-preserving isometries of developable semi-discrete conjugate nets. We derive two conditions for the rigid-ruling foldability of pairs of curves and consider two applications. First, we introduce computations that enable the sequential construction of rigid-ruling foldable crease-rule patterns. Second, we examine combinations of planar and constant fold-angle creases. In particular, we show that constant fold-angle creases are only compatible with other constant fold-angle creases, and we provide a characterization of rigid-ruling foldable combinations of planar and constant fold-angle creases.

In this paper, the authors will present a new strategy combining folding and kirigami approaches, entitled slit-folding. Cutting slits and removing (or adding) material along foldlines leads to interesting results and new folding mechanisms. The study of the possible slits’ geometries and their relation to the adjacent foldlines allow for better control of these mechanisms that have a 3D minimal energy equilibrium state after stitching their seamlines. This paper will give an overview of this particular system.

A quasigeodesic is a curve on the surface of a convex polyhedron that has ≤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.

For a given non-degenerate triangle , we define the t-affine family to consist of triangles _t whose vertices are pairwise affine combinations of the vertices of , all parameterized by the same value of t. We demonstrate that the first isogonic centers X_13,t and the first isodynamic centers X_15,t of _t lie on two Maclaurin trisectrices and differ by scaling and a rotation about the barycenter of .

Fabricating complex geometries from flat sheets has many practical advantages such as cost-efficient fabrication and space-efficient transportation. In this paper, we explore a family of shapes that consist of two types of curved-crease molecules, which can be composed as a modular design. Alternatively, we explore how to optimize the target shape towards a global origami development, that is, a development that does not require additional slits or holes.

Plate lattices are high-performance lightweight structures, exhibiting up to twice the yield strength and stiffness compared to truss lattices of similar geometric arrangement and relative density. Although they are of great interest for research and structural engineering applications, their complex manufacturing and assembly processes limit their practical use, with sandwich panels being an exception. This paper presents a novel approach to the design and modular assembly of folded custom 3-dimensional plate lattices as structural corrugations for use in structural engineering and robotics applications. The plate lattice structural corrugation uses a building block strategy and incorporates custom modified unit cells based on the Miura-ori. This transformation involves expanding the top and bottom zig-zag crease lines into facets and orienting them in space. The resulting modified pattern is referred to as the Kirigami Expanded Miura. The unique structure of these lattices not only provides exceptional mechanical performance as static structures, but also allows for the design of anisotropies in their flexural stiffness by alternating between the Maxwell criterion on bending-dominated or stretch-dominated cells. These anisotropies can have value differences of up to 24 with the same geometry, making them ideal for robotic morphing applications. We validate our proposed technology by characterizing the mechanical performance of this new building system and comparing it with state-of-the-art corrugations. We demonstrate the potential of this approach by designing, manufacturing, and modularly assembling multiple structures and robots with single and double curvature.

Hexahedral (hex) meshing is a long studied topic in geometry processing with many fascinating and challenging associated problems. Hex meshes vary in complexity from structured to unstructured depending on application or domain of interest. Fully structured meshes require that all interior mesh edges are adjacent to exactly four hexes. Edges not satisfying this criteria are considered singular and indicate an unstructured hex mesh. Singular edges join together into singular curves that either form closed cycles, end on the mesh boundary, or end at a singular node, a complex junction of more than two singular curves. While all hex meshes with singularities are unstructured, those with more complex singular nodes tend to have more distorted elements and smaller scaled Jacobian values. In this work, we study the topology of singular nodes. We show that all eight of the most common singular nodes are decomposable into just singular curves. We further show that all singular nodes, regardless of edge valence, are locally decomposable. Finally we demonstrate these decompositions on hex meshes, thereby decreasing their distortion and converting all singular nodes into singular curves. With this decomposition, the enigmatic complexity of 3D singular nodes becomes effectively 2D.

Conic curved creases with reflected rule lines is a style of curved origami design, first explored by David Huffman, that is attractive in that it gives one-DOF folding motions with rigid rule lines (i.e., the rule lines remain the same throughout the motion). We show how to discretize any such curved crease pattern into a similar straight-line crease pattern that has a one-DOF rigid folding motion. We develop two general methods for such discretization, where each curve is replaced by an inscribing or circumscribing polygonal line, respectively, and show in both cases that the resulting discretized crease patterns are rigidly foldable. In the case of the circumscribed discretization, the crease pattern is also locally flat foldable. On the other hand, only careful sampling in the inscribed method results in locally flat-foldable crease patterns.

We introduce a notion we call quasi-twisting that cuts a convex polyhedron P into two halves and re-glues the halves to form a different convex polyhedron. The cut is along a simple closed quasigeodesic. We initiate the study of the range of polyhedra produced by quasi-twisting P , and in particular, whether P can “quasi-twist flat,” i.e., produce a flat, doubly-covered polygon. We establish a sufficient condition and some necessary conditions, which allow us to show that of the five Platonic solids, the tetrahedron, cube, and octahedron can quasi-twist flat. We conjecture that the dodecahedron and icosahedron cannot quasi-twist flat, and prove that they cannot under certain restrictions. Many open problems remain.

When can a polyomino piece of paper be folded into a unit cube? Prior work studied tree-like polyominoes, but polyominoes with holes remain an intriguing open problem. We present sufficient conditions for a polyomino with one or several holes to fold into a cube, and conditions under which cube folding is impossible. In particular, we show that all but five special “basic” holes guarantee foldability.

We present a tool for the design of curved folds of constant angle along curves composed of circular arcs. This tool allows the user to specify the intended shape, and computes its development. In general, the crease curves fold into helices of constant descent angle. If the crease curves are designed to remain planar (zero descent), our method additionally supports multiple pleated creases.

In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of analogies between the smooth and discrete situations, as well as several equivalent characterizations of the principal property. These include being a conical mesh, and being flat-foldable. This structure-preserving discretization is the basis of computation and design. We propose a new method for designing pleated structures and reconstructing reference shapes as pleated structures: we first gain an overview of possible crease patterns by establishing a connection to pseudogeodesics, and then initialize and optimize a quad mesh so as to become a discrete pleated structure. We conclude by showing applications in design and reconstruction, including cases with combinatorial singularities. Our work is relevant to fabrication in so far as the offset properties of principal pleated structures allow us to construct curved sculptures of finite thickness.

Motivated by the art of curved crease origami we study mathematical models for folding ideal paper along curved creases. We approach the problem of finding mathematical descriptions in terms of developable surfaces of those shapes which can be folded from real paper but where its rigorous description is very often unknown. For that we investigate a particular one-parameter family of surfaces isometric to a given planar surface patch. For each such surface we parametrize the crease curves which fold those surfaces into cylinders and cones. We apply our methods to explore curved crease origami designs, such as tessellations of the plane and cylinders.

We demonstrate that every non-tubular channel linear Weingarten surface in Euclidean space is a surface of revolution, hence parallel to a catenoid or a rotational surface of non-zero constant Gauss curvature. We provide explicit parametrizations and deduce existence of complete hyperbolic linear Weingarten surfaces.