From Folds to Structures, a Review Arthur Lebée To cite this version: Arthur Lebée. From Folds to Structures, a Review. International Journal of Space Structures, Multi- Science Publishing, 2015, 30 (2), pp.55-74. 10.1260/0266-3511.30.2.55. hal-01266686 HAL Id: hal-01266686 https://hal-enpc.archives-ouvertes.fr/hal-01266686 Submitted on 3 Feb 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. From Folds to Structures, a Review Arthur Lebée* Université Paris-Est, Laboratoire Navier (ENPC, IFSTTAR, CNRS). École des Ponts ParisTech, 6 et 8 avenue Blaise Pascal. 77455 Marne-la-Vallée cedex2 (Submitted on 23/05/2015, Reception of revised paper 28/06/2015, Accepted on 02/07/2015) ABSTRACT: Starting from simple notions of paper folding, a review of current challenges regarding folds and structures is presented. A special focus is dedicated to folded tessellations which are raising interest from the scientific community. Finally, the different mechanical modeling of folded structures are investigated. This reveals efficient applications of folding concepts in the design of structures. Key Words: Origami, Folded Tessellations, Structures, Folds 1. INTRODUCTION drawn on the sheet will keep its length constant during The concept of fold covers many significations. In the transformation. This remarkable property nature, it is mostly related to specific shapes such as combined with the empirical observation that many creases or pleats and may be found at many different different shapes may be obtained from paper folding, scales from the folding of graphen layers [1, 2], to the suggests immediately that there are underlying rules unfolding of tree leaves [3] and the mountains and encourages the mathematical formalization of this building (orogeny). Pleated fabrics found on Egyptian process. The connection between origami and frescoes prove that, long ago, mankind also mathematics is deep and a dedicated new theoretical handcrafted folds. However, the abstract idea of a field of research emerged in the last decades: origami folded surface emerged rather recently. mathematics. For instance, noticing that crease lines The triggering ingredient was the invention of are geometric constructions Jacques Justin defined a paper in Asia. Notwithstanding its crucial role in the set of axioms which defines all geometric diffusion of knowledge, it allowed the achievement of constructions accessible by folding. It turns out that the first folded models. Folding paper had a sacred there are more possibilities than with the classical meaning in ancient Japan and was called origami, Euclid axioms: trisecting an angle is not possible with concatenating “oru” (fold) and “kami” (paper) [4]. rule and compass but it is possible with folding. There Because paper was extremely expensive at that time, are many other questions investigated in origami origami models were first confined to religious or mathematics such as flat foldability and continuous outstanding events such as weddings. It is only during rigid foldability and most active researchers gather the twentieth century that origami spread across the every four years at the international meeting of whole Japanese society and eventually worldwide. Origami Science, Math and Education (OSME [5]). Many flat materials may be folded such as fabrics Anyone that has folded paper noticed the extremely or parchment. However, paper is certainly the first rich motion which may come out from a simple material which was thin enough and yet inextensible organization of folds [6]. It is thus not surprising that a in its plane so that it was suitable for origami. wide variety of technological applications are Considering the geometrical transformation endured currently investigated. These many innovations by the sheet of paper, it appears clearly that any line combining mathematical, kinematic or structural *Corresponding author e-mail: A Lebée [email protected] International Journal of Space Structures Vol. 30 No. 2 2015 55 From Folds to Structures, a Review properties of folds are referred to Origamics [7]. There are many advantages to start from a flat sheet of material. For instance, since there are neither cuts and nor offcuts, this forming process might be rather inexpensive. Similarly, the kinematic properties of folds make them extremely good candidates for designing small robots [8] or MEMS (Micro Electro- Mechanical System). Many of these applications are not published in scientific journals and are often simply scattered on the Internet through blogs or shared videos. Because, origami concepts are extremely visual, this new way of sharing knowledge Figure 2. Josef Albers playing with Miura Ori photographed generated a strong enthusiasm in the last decade. by Henri Cartier Bresson, (Magnum Photos, 1968). Many professions are now investigating the fascinating properties of folds. However, architects transformation inspired many architects, designers and and designers were probably the firsts to identify the engineers such as Jean Prouvé or Pier Luigi Nervi. It key role of folding as a form-finding process. At the is an almost impossible task to give a fair account of end of the 19th century, the emergence of new all the applications of origami in design and construction materials and techniques such as concrete architecture. The kinematic properties of folds make and steel put deeply into question the academic them extremely versatile for designing packages and architectural style prevailing at that time. It became furniture with industrial materials such as cardboard or obvious that building and designing objects should be polypropylene. The moment of inertia given to a also driven by the fabrication process and not only by surface with pleats originally inspired many architects formal questions. Following this movement – from for designing wide roofings and recent developments which originated Modernism – Joseph Albers were reviewed [9, 10, 11]. introduced folds in his preliminary course of the Today, the combination of parametric graphical department of design at the Bauhaus in the 20’s tools, new fabrication process based on numerical (Figure 1 and Figure 2). control and 3D printers makes architects and designer Because folding is certainly the most simple and even closer to actual constructions. First, it is much inexpensive process for transforming matter, it is an faster to test the feasibility of a design thanks to rapid excellent starting point for teaching architecture and prototyping. The design cycle is thus strongly design. This process enables fast and easy accelerated. Second, provided the digital format is achievement of three dimensional shapes which compatible with the full scale fabrication process, clearly show better structural properties than the architects and designer are actually directly editing original sheet and defines directly an envelope, working-drawings from the beginning. This strongly separating the “inside” from the “outside”. Gathering encourages them to have direct interactions with so many interesting properties from such a simple manufacturers. Hence it is worth to consider these evolutions as a new construction paradigm providing a new geometrical freedom which remains yet close to the manufacturing process. In this extremely encouraging context for innovations in construction, it is worth to attempt a review focused on two key aspects: the form-finding process related to folding and the behavior as a structure of a folded shape. Starting from developable surfaces, section 2 provides an overview of interesting shapes which may be derived from folding techniques. A particular focus is provided on folded tessellations. Because a fold is the transformation of a surface, it is actually a mechanism. Hence the Figure 1. Josef Albers and his students at Bauhaus 1928 question of the link between folded surfaces and (photo: Bauhaus Archives). actual structures is not so simple and investigated in 56 International Journal of Space Structures Vol. 30 No. 2 2015 Arthur Lebée section 3. Even if new graphical tools offer much Since origamists start from a flat sheet of paper, the more freedom for finding shapes, it is worth to recall original Gauss curvature is zero. Whereas Theorema that in the end a construction will carry a load. Hence, Egregium sets a purely local constraint on the Gauss finding a relevant mechanical modeling for a folded curvature, in case the latter is zero everywhere on the surface will help the understanding of its structural surface this constraint may be integrated. Finally, the behavior. only C2 surfaces which may be obtained keeping constant lengths on the surface are developable 2. FOLDING AS A FORM-FINDING surfaces. Regular developable surfaces are limited to PROCESS three kinds: cones, cylinders and tangent developable Since no cuts are allowed, folding paper preserves the surfaces (Figure 3). Of course, keeping this high length of any curve drawn on its surface. This is regularity restricts strongly accessible shapes and it is actually a very particular