![]() ![]() Now we are ready to understand an even more complicated unfolding. one without self-intersection) with the least number of vertices - view the animated version The Császár torus is an embedded polyhedral torus (i.e. Lutz produced the model of the unfolding below. A torus with the least number of vertices, 7, was found by Császár. The Clifford torus shown above uses about 200 vertices, which is surely more than necessary. It is a challenging task in combinatorial geometry to find polyhedral models of a given topological shape which use the minimal number of faces or vertices. The torus unfolds nicely despite the negative curvature of the inner region - view the animated version The Clifford torus is generated by a family of circles. (For experts, this polyhedral torus is the Clifford torus obtained from a non-standard parameterization using Hopf fibers in the 3-sphere.) For example, the torus shown below has a planar unfolding to a single component such that no two faces overlap. Unfoldings exist for many more polyhedral surfaces. ĭodecahedron with a different unfolding - view the animated version Perhaps the largest collection of mathematical paper models was produced by Father Magnus Wenninger, a mathematician and priest at St. There are many folding kits which allow you to cut and fold paper models of polyhedra. Unfoldings of Platonic and Archimedean solids are very wellknown. Regions you can't see - makes it much easier to understand a complicated geometric structure. Being able to touch a model - in particular, being able to touch the For example, the Boy surface mentioned in the introduction can be understood much more easily when you have a paper model in your hands that you can turn and view from different directions. Since Dürer's time, mathematicians have made intensive use of paper models to study geometric surfaces, in both education and research. ![]() The art of paper folding: Origami - view the animated version There are many references on origami, for example, the website, but in this article we focus on the folding, or more precisely, on the unfolding, of polyhedral surfaces. Origami, the Japanese art of paper folding, is the best-known application of unfoldings - the word literally means "to fold a paper". Unfolding of a cube - view the animated version Process, but the final flattened surface must be free of overlaps. Self-intersections are allowed during the unfolding Edge unfolding, which is what we shall consider in this article, only allows cuts along edges, and not through the interiors of faces. Unfolding is the process of cutting a polyhedral surface along certain curves and then flattening the surface onto the plane, without overlaps and without distorting the individual faces. But even with modern software tools, a number of unsolved geometric problems remain. Nowadays software has advanced and allows us to produce "cut drawings" by automatically computing the unfolding of geometric shapes: to the right you can see the unfolding of the Boy model, or view an animation. ![]()
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