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A geometric graph G is a graph whose vertex set is a set Pn of n points on the plane in general position, and whose edges are straight line segments (which may cross) joining pairs of vertices of G. We say that G contains a convex r-gon if its vertex and edge sets contain, respectively, the vertices and edges of a convex polygon with r vertices. In this… (More)

We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come… (More)

- Erik D Demaine, Martin L Demaine, Jin-Ichi Itoh, Chie Nara
- 2015

We consider continuously folding polyhedral surfaces down to a multilayered flat folded state. It was asked by E. Demaine, M. Demaine and A. Lebiw, and proposed in [4] if there is a continuous motion for flattening any given polyhedron. For example, it was showed by J.-i. Itoh and C. Nara [5] that a box in Fig 1(a) is continuously flat-folded by pushing… (More)

- Zachary Abel, Erik D Demaine, Martin L Demaine, Jin-Ichi Itoh, Anna Lubiw, Chie Nara +1 other
- 2014

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. ABSTRACT We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening… (More)