Paper explores folding patterns of curved creases preserving their geometric properties.
problem Investigating rigid-ruling folding motions of curved crease-rule patterns.
method Deriving conditions for rigid-ruling foldability and analyzing combinations of creases.
result Constant fold-angle creases are only compatible with other constant fold-angle creases.
The paper explores the existence of multiple curved foldings with a common crease pattern.
problem Determining the number of distinct curved foldings with a given crease and crease pattern.
method Analyzing origami maps and their singular sets, focusing on the crease and crease pattern.
result For a non-closed simple arc crease, there are exactly 4 distinct non-congruent curved foldings.
Researchers solved a geometry paradox for creased tubes.
problem Resolving the paradox of Gaussian curvature in creased tubes.
method Calculated Gaussian curvature in terms of rate of change of solid angle, dependent on fold angle and curvature.
result Gaussian curvature is zero overall despite the surface being doubly-curved.
Consider a curve Γ in a domain D in the plane R2. Thinking of D as a piece of paper, one can make a curved folding P in the Euclidean space R3. The singular set C of P as a space curve is called the crease of P and the initially given plane curve Γ is called the crease patt…
Paper classifies pillow box isometric deformations preserving crease patterns.
problem Classifying pillow box isometric deformations preserving crease patterns.
method Continuous isometric deformations from pillow boxes to double rectangles, preserving crease patterns.
result Such deformations necessarily change pillow box topology.
We describe a general family of curved-crease folding tessellations consisting of a repeating "lens" motif formed by two convex curved arcs. The third author invented the first such design in 1992, when he made both a sketch of the crease pattern and a vinyl model (pictured below). Curve fitting suggests that this init…
Classifies and analyzes the stability of black hole event horizon birth points using contact geometry.
problem Classifying and understanding the structural possibilities of black hole crease sets.
method Contact geometry approach, focusing on BigFronts and their Legendrian projections.
result Refined stability discussion of the event horizon birth component and identification of additional components.
The paper proves the existence of a folded annulus with multiple creases.
problem Existence of a folded annulus with multiple creases.
method Analyze developable surfaces, use normal curvature and relative torsion, compute geometric descriptors, prove propagation of folds.
result Proves the existence of a folded annulus with multiple creases.
This paper explores non-periodic folding of Spidron units, revealing nonlinear dynamics.
problem Understanding the kinematics and nonlinear phenomena of Spidron units.
method Analysis of single unit cell kinematics and recursive construction of multiple cells.
result Non-periodic folding restricts isotropic folding as the number of unit cells increases.
A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, t…
Periodic surfaces have a limited number of bending modes, equal to their membrane modes.
problem Understanding the limitations of bending modes in periodic surfaces.
method Analyzing deformation modes of periodic, piecewise smooth, simply connected surfaces.
result Effective membrane modes and bending modes are orthogonal, limiting the total number of modes to 3.
A flat Klein bottle is visualized using origami.
problem Visualizing a Klein bottle's flatness and topology.
method Curved-crease origami with inelastic film.
result The sculpture illustrates both flatness and non-orientability.
Origami can create complex knots, with minimum creases defining a new knot invariant.
problem Creating complex knots using origami folds.
method Developed a new knot invariant called the fold number, defined as the minimum number of creases required to obtain an equivalent knot.
result No proper foldings can produce nontrivial knots, but improper foldings can.
New game defined on origami patterns, linking number introduced.
problem Defining a game on origami patterns.
method Introduced Region Select on origami crease patterns.
result Defined a new unlinking number.
The Positive Mass Theorem for special singular initial data.
problem Proving the positive mass theorem for data with a codimension one singularity.
method Using asymptotically flat spin initial data sets with matching Bartnik data condition involving spacetime rotations.
result Established a spacetime positive mass theorem and rigidity statement.
Classifies embeddings of surfaces in 3D space with product structure.
problem Classifying embeddings of surfaces in Euclidean 3-space with product structure.
method Investigates embeddings of a surface in R2imesR, focusing on critical points and isotopy classes. result Provides necessary and sufficient conditions for realizing certain configurations of curves as crease sets.
The study explores isometric deformations of surfaces of translation.
problem Determine the ways surfaces of translation bend isometrically.
method Analyzes existence conditions and provides closed-form expressions for infinitesimal and finite bendings of surfaces of translation.
result Surfaces of translation admit various infinitesimal and finite bendings, including purely torsional and torsion-free.
New framework reveals limits of flexible, periodic thin surfaces.
problem Understanding the mechanical behavior of thin, periodic surfaces.
method Developed a duality between surface rotations and in-plane stresses.
result Exactly three out of six possible strain states are isometries.
Study finds how periodic surfaces can bend without stretching.
problem Understanding isometric deformations of periodic surfaces.
method Characterization of isometric deformations using a constraint derived from Gauss theorem.
result Relates surface stretching to bending and twisting.
Shells resist three out of six possible loads if simply connected.
problem Understanding the load resistance of shells.
method Formal mathematical analysis of shell strains and deflections.
result The space of strains is three-dimensional for simply-connected shells.
Proves Gromov's conjecture on total mean curvature using surgery and positive mass theorems.
problem Proving Gromov's conjecture on total mean curvature of fill-ins.
method Surgery to reduce to fill-ins of spheres, positive mass theorems, and quantitative surgery process.
result Proves Gromov's conjecture on total mean curvature in various cases.
Clinical decision support systems (CDSS) will play an in-creasing role in improving the quality of medical care for critically ill patients. However, due to limitations in current informatics infrastructure, CDSS do not always have com-plete information on state of supporting physiologic monitor-ing devices, which can …
Unified theory solves strain compatibility and elasticity of origami metamaterials.
problem Understanding and controlling the morphing paths of origami metamaterials.
method Unified theory for a wide array of origami tessellations, solving strain compatibility and elasticity.
result Origami metamaterials exhibit equal but opposite in-plane and out-of-plane Poisson's ratios and bending energy depends on strain gradient.
New solutions found for bending of flat surfaces and origami structures.
problem Understanding the energy-efficient bending modes of origami tessellations and corrugated shells.
method Direct construction of closed-form solutions for surfaces of translation.
result Three inextensional modes identified for surfaces of translation, including stretching, bending, and twisting.
New concept of effective isometries for compliant shells.
problem Inadequate classification of isometric deformations for compliant shells.
method Introduce effective isometric deformations defined by first-order isometry in a small scale separation parameter.
result Effective isometries are solutions to a quasilinear second-order PDE.
An immense class of physical counterexamples to the four dimensional strong cosmic censor conjecture---in its usual broad formulation---is exhibited. More precisely, out of any closed and simply connected 4-manifold an open Ricci-flat Lorentzian 4-manifold is constructed which is not globally hyperbolic and no perturba…