Project Origami Hyperbolic Paraboloid (Hypar) Violeta Vasilevska Department of Mathematical Sciences, University of South Dakota, 414 E. Clark Street, Vermillion, SD 57069-2307, USA Phone: (605) 677-5472; Fax: (605) 677-5263 e-mail: [email protected] September 20, 2008 2 Project 1: Constructing a Hypar This project shows how to construct an origami hyperbolic paraboloid - hypar. This shape is also sometimes called a saddle or saddle surface. It’s not clear where this model originates, though it has been claimed that it was discovered by the Bauhaus artists in Germany. This unusual fold has been rediscovered by numerous people over the years. Instructions for this model can be found in Paul Jackson’s book The Complete Origami Course,W.H. Smith. New Yoirk 1989. The instructions for this model used in this project are combined from the ones listed in [1] and [2]. Hypar resembles a 3D surface that you would be studying in Calculus III as a college student. 1. Take a square and crease both diagonals. Then turn over. 2. Fold the top edge to the center point, but creasing only in the middle - between the diagonals. Then unfold. 3. Repeat step (2) on the bottom. 3 4. Bring the top edge to the top crease line, creasing only between the diagonals. Unfold. Then bring the top edge to the bottom crease line. Again, do not crease all the way across. Unfold. 5. Repeat step (4) on the bottom. 6. Repeat steps (4) and (5) on the left and right sides. Turn over. 4 7. Fold the top edge to the first (closest) crease fold, creasing only between the diagonals.Unfold. 8. Repeat step (7) by folding the top edge to the the third, forth, and the bottom crease fold, respectively. 9. Repeat steps (7) and (8) on the bottom. 5 10. Repeat steps (7) and (8) on the left and right sides. 11. Final crease pattern. −−− −·−·− Valley fold Mountain Fold 12. Now make all the creases at once. It may help to fold the creases on the outer ring first and work your way in. 13. Once the creases are folded, the paper will twist into the shape below, and you’re done. 6 14. Folding the crease pattern completely forms an ”X” shape. 15. You can make a larger hypar by folding more divisions in the paper. The key is to have the concentric squares alternate mountain-valley-mountain in the end. The model with 1/16 divisions is shown on the picture below [3]. http://www.math.lsu.edu/ verrill/ Question: Is the hypar a rigid origami model or not? (Could it be made out of rigid sheet metal, with hinges at the creases?) 7 Project 2: Constructing a Hyparhedra Hyparhedra is a closed, curved surface that can be obtained by joining (gluing, taping) hypars together along edges. In this project, we want to construct a hyparhedron that corresponds to the dodecahedron. Recall that the dodecahedron has 12 pentagon faces. This means that to make the corresponding hyparhedron we will need to make 12 corresponding hyparhedron ”faces” as follows: • Start by making 60 hypars. • Glue 5 hypars together get a ”face” of the hyparhedron in such a way that each hypar corresponds to an edge of a face of the dodecahedron. • Repeat the previous step until you form 12 ”faces”. • Glue the ”faces” together to form the hyparhedron. Figure 1 shows a dodecahedron and corresponding hyparhedra and Figure 2 shows the origami hyparhedra [2]. Figure 1 Figure 2 8 References: 1. Eric D. Demaine, Polyhedral Sculptures with Hyperbolic Paraboloids, (University of Waterloo, Canada, 2006) 2. Thomas Hull, Project Origami-Activities for Exploring Mathematics, (A K Peters, Ltd, 2006) 3. Helena A. Verrill, World Wide Web, http://www.math.lsu.edu/ verrill/origami/parabola/

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