The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability, and the use of paper folds to solve up-to cubic mathematical equations.
In geometry, a net of a polyhedron is an arrangement of non-overlapping edge-joined polygons in the plane which can be folded to become the faces of the polyhedron. Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for physical models of polyhedra to be constructed from material such as thin cardboard.
Kawasaki's theorem or Kawasaki–Justin theorem is a theorem in the mathematics of paper folding that describes the crease patterns with a single vertex that may be folded to form a flat figure. It states that the pattern is flat-foldable if and only if alternatingly adding and subtracting the angles of consecutive folds around the vertex gives an alternating sum of zero. Crease patterns with more than one vertex do not obey such a simple criterion, and are NP-hard to fold.
Joseph O'Rourke is the Spencer T. and Ann W. Olin Professor of Computer Science at Smith College and the founding chair of the Smith computer science department. His main research interest is computational geometry.
Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike in traditional origami, the panels of the paper cannot be bent during the folding process; they must remain flat at all times, and the paper only folded along its hinges. A rigid origami model would still be foldable if it was made from glass sheets with hinges in place of its crease lines.
Martin L. (Marty) Demaine is an artist and mathematician, the Angelika and Barton Weller artist in residence at the Massachusetts Institute of Technology (MIT).
In the mathematics of paper folding, map folding and stamp folding are two problems of counting the number of ways that a piece of paper can be folded. In the stamp folding problem, the paper is a strip of stamps with creases between them, and the folds must lie on the creases. In the map folding problem, the paper is a map, divided by creases into rectangles, and the folds must again lie only along these creases.
The fold-and-cut theorem states that any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include polygons, which may be concave, shapes with holes, and collections of such shapes.
Bidimensionality theory characterizes a broad range of graph problems (bidimensional) that admit efficient approximate, fixed-parameter or kernel solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, bounded-genus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the graph minor theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. The theory was introduced in the work of Demaine, Fomin, Hajiaghayi, and Thilikos, for which the authors received the Nerode Prize in 2015.
Stefan Langerman false Swarzberg is a Belgian computer scientist and mathematician whose research topics include computational geometry, data structures, and recreational mathematics. He is professor and co-head of the algorithms research group at the Université libre de Bruxelles (ULB) with Jean Cardinal. He is a director of research for the Belgian Fonds de la Recherche Scientifique (FRS–FNRS).
Mihai Pătrașcu was a Romanian-American computer scientist at AT&T Labs in Florham Park, New Jersey, USA.
The EATCS–IPEC Nerode Prize is a theoretical computer science prize awarded for outstanding research in the area of multivariate algorithmics. It is awarded by the European Association for Theoretical Computer Science and the International Symposium on Parameterized and Exact Computation. The prize was offered for the first time in 2013.
Anna Lubiw is a computer scientist known for her work in computational geometry and graph theory. She is currently a professor at the University of Waterloo.
Mohammad Taghi Hajiaghayi is a computer scientist known for his work in algorithms, game theory, social networks, network design, graph theory, and big data. He has over 200 publications with over 185 collaborators and 10 issued patents.
In geometry, Steffen's polyhedron is a flexible polyhedron discovered by and named after Klaus Steffen. It is based on the Bricard octahedron, but unlike the Bricard octahedron its surface does not cross itself. With nine vertices, 21 edges, and 14 triangular faces, it is the simplest possible non-crossing flexible polyhedron. Its faces can be decomposed into three subsets: two six-triangle-patches from a Bricard octahedron, and two more triangles that link these patches together.
Geometric Folding Algorithms: Linkages, Origami, Polyhedra is a monograph on the mathematics and computational geometry of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by Cambridge University Press (ISBN 978-0-521-85757-4). A Japanese-language translation by Ryuhei Uehara was published in 2009 by the Modern Science Company (ISBN 978-4-7649-0377-7).
In the mathematics of paper folding, the big-little-big lemma is a necessary condition for a crease pattern with specified mountain folds and valley folds to be able to be folded flat. It differs from Kawasaki's theorem, which characterizes the flat-foldable crease patterns in which a mountain-valley assignment has not yet been made. Together with Maekawa's theorem on the total number of folds of each type, the big-little-big lemma is one of the two main conditions used to characterize the flat-foldability of mountain-valley assignments for crease patterns that meet the conditions of Kawasaki's theorem. Mathematical origami expert Tom Hull calls the big-little-big lemma "one of the most basic rules" for flat foldability of crease patterns.
Tomohiro Tachi is a Japanese academic who studies origami from an interdisciplinary perspective, combining approaches from the mathematics of paper folding, structural rigidity, computational geometry, architecture, and materials science. His work was profiled in "The Origami Revolution" (2017), part of the Nova series of US science documentaries. He is a professor at the University of Tokyo.
In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement of the net in a plane. As in rigid origami, the polygons of the net must remain individually flat throughout the motion, and are not allowed to intersect or cross through each other. A blooming, reversed to go from the flat net to a polyhedron, can be thought of intuitively as a way to fold the polyhedron from a paper net without bending the paper except at its designated creases.
In computational geometry, the source unfolding of a convex polyhedron is a net obtained by cutting the polyhedron along the cut locus of a point on the surface of the polyhedron. The cut locus of a point consists of all points on the surface that have two or more shortest geodesics to . For every convex polyhedron, and every choice of the point on its surface, cutting the polyhedron on the cut locus will produce a result that can be unfolded into a flat plane, producing the source unfolding. The resulting net may, however, cut across some of the faces of the polyhedron rather than only cutting along its edges.
