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Curved Crease Folds of Spherical Polyhedra with Regular Faces
| Content Provider | Semantic Scholar |
|---|---|
| Author | Mundilova, Klara |
| Copyright Year | 2019 |
| Abstract | Based on the art of curved crease origami, we give an example of the construction of a family of shapes from mathematically ideal paper that consists of cylinders and cones and reassembles the structure of a spherical polyhedron with regular faces. We offer explicit formulas to parametrize the crease curves. Moreover, we illustrate this method on the five Platonic solids, which can be folded from one single sheet of paper. Introduction This paper is a continuation of our work on mathematical curved crease paper folding [6, 7]. Motivated by the properties of real paper, we model an ideal paper as an infinitesimally thin shape that can be obtained from a planar patch without stretching or tearing, i.e., as a composition of developable surfaces. The fundamentals for mathematical paper folding are given by Huffman [5] and Fuchs and Tabachnikov [4]. Further properties were investigated by Demaine et al. [2, 3]. However, given a folded shape and its development, it is most of the times unknown whether the given shape would exist in the mathematical world as real paper seems to allow little imperfections. For example, Demaine et al. [1] show that the so-called pleated hyperbolic paraboloid does not exist without additional creases. A positive result on the other hand is the folded Vesica Piscis, cf. [6]. In this paper we give a construction method for a family of shapes that consist of planar, cylindrical and conical patches and are based on spherical polyhedra with regular faces, such as the Platonic andArchimedean solids. Experimental studies of folded Platonic solids were pursued by Schling and Otterson [8]. However, their approach is just an approximation of a sphere. Our method on the other hand starts with right circular cylinders whose profile curves are the resp. great circles on the sphere, i.e., the geodesics connecting two points of a spherical polyhedron. Those cylinders are then folded into cones with appropriately chosen apices. Moreover, we then can add additional planar creases that reflect the interior cones, see Figure 1. Figure 1: A folded Icosidodecahedron and a part of its development (darker gray depicts the mountain, lighter gray the valley folds). Bridges 2019 Conference Proceedings |
| Starting Page | 423 |
| Ending Page | 426 |
| Page Count | 4 |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://archive.bridgesmathart.org/2019/bridges2019-423.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
