Origami Hyperbolic Paraboloid Violeta Vasilevska USD Math Days for Women September 18, 2008 Origami math web sides and books • Tom Hull - Merrimack College g in North Andover,, MA. http://mars.wnec.edu/~th297133/ • Erik D. Demaine http://erikdemaine.org/ • Project Origami – activities for exploring mathematics by Thomas Hull Origami and math • Origami is a Japanese word which means paperfolding: g • Ori (oru) - meaning "folded," and • Kami - meaning "paper." • The art of paper folding (origami), has received a considerable amount of mathematical study in the recent years. Project 1: Constructing Hyper This hi project j shows h h how to construct an origami hyperbolic paraboloid-hyper. Hyperbolic Paraboloid • A hyperbolic paraboloid is an infinite surface discovered in the 17th century. Dr. Jose Flores and the Hyperbolic Paraboloid Maplet http://usd http://usdmaplenet.usd.edu/maplenet/FloresJSP/HyperPar.jsp Properties of Hyperbolic Paraboloid • Mathematically, y, a hyperbolic yp pparaboloid is defined byy the equation z/c=x2/a2 - y2/b2 • The name "hyperbolic paraboloid" comes from the property that the • the xy cross-sections are hyperbolas, and • the yz cross-sections are translated copies of a common parabola P. • Note also that the zx cross-sections are translated upside down copies of the same parabola P. Hypar • The term hypar is used to mean a hyparbolic paraboloid shape, or more formally a partial hyparbolic paraboloid, cut from the full infinite surface. • The term hypar was introduced by the architect Engel (Structure Systems, Frederick A. Praeger Publishers, New York, 1967). Hypars in Architecture • Hypars and joining hypars have been used extensively in architecture since the 1950's. A few examples can be found in Curt Siegel's 1962 book Structure and Form in Modern Architecture : • Philips ps ppavilion v o at thee 1958 9 Brussels usse s eexhibition; b o ; • The roof of the Girls Girls'ss Grammar School in London. London Project 2: Constructing a Hyparhedra In this project you will be joining 5 hypars together to get a “face” of a hyparhedra, and then use 12 “faces” to construct a hyparhedra yp that corresponds p to the dodecahedron. Hyparhedra Demaine at el. developed algorithms for building hypar "sculptures" based on polyhedra. They called these structures hyparhedra. Hyparhedra that corresponds to Dodecahedron Project 3: The Miura Map Fold In this project you will be constructing a rigid origami model. model Rigid Origami "Which origami g models can be opened p (to ( a completely flat, unfolded state) and closed (to the completely folded state) in a rigid manner?" By rigid origami we mean origami that can be folded while keeping all regions of the paper flat and all crease lines straight (i.e. the regions of paper between crease lines do not bend or twist in the folding process). Why care about rigid origami? • Those seeking to use origami in industrial designs often want to know that their chosen fold is a rigid fold before devoting the resources for manufacturing, say, stiff cardboard boxes that you hope will fold up properly. • The problem of rigid origami, treating the folds as hinges joining two flat, rigid surfaces, such as sheet metal, has great practical importance. The Miura Map Fold • The Miura mapp fold is a famous rigid g fold that has applications to space science – it has been used to deploy large solar panel arrays for space satellites. • Koryo Miura invented this fold while searching for a way to collapse a large solar panel into a package that could be attached to a space satellite and fit inside a rocket capsule. The Miura Map Fold • Also Also, Miura discovered that since this model opens and closes so easily, it makes an ideal map fold. In fact fact, one can buy Tokyo subway maps that are folded in this way. Video presentation of the fold http://www.math.lsu.edu/~verrill/ Reference: • E. D. Demaine, M. L. Demaine, and A. Lubiw, “Polyhedral Sculptures with Hyperbolic Paraboloids”, in Proceedings of the 2nd Annual Conference of BRIDGES: Mathematical Connections in Art, Music, and Science (BRIDGE’99) Winfield, (BRIDGE’99), Wi fi ld Kansas, K July J l 30–August 30 A t 1, 1999, pages 91–100. • E. D. Demaine, World Wide Web. http://erikdemaine org/hypar/ http://erikdemaine.org/hypar/ Reference: • T. Hull, Project Origami Origami-Activities Activities for exploring mathematics, A. K. Peters, Ltd., 2006. • T. Hull, World Wide Web. http://kahuna.merrimack.edu/~thull/rigid/rigid.html p g g • H.. Ve Verrill,, Algebraic geb c geometry geo e y origami, o g , World Wo d Wide Web. http://www.math.lsu.edu/~verrill/