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Mathematics and Origami: the Art and Science of Folds

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Mathematics and Origami: the Art and Science of Folds Mathematics and Origami: The Art and Science of Folds Natalija Budinski Abstract There is no consensus where origami originated, but it is assumed that its roots are in China associated with the discovery of paper. However, the craft of origami blossomed in Japan, where it is treated as national art. Origami is usually connected to fun and games, and the most common association with origami is a crane which has a special place in Japanese culture. The popularity of modern origami has grown in many aspects, mathematical, scientific, artistic, or even an enjoyable craft. Keywords Origami · Mathematical · Artistic · Crane · Scientific There is no consensus on where origami originated, but it is assumed that its roots are in China associated with the discovery of paper. However, the craft of origami blossomed in Japan, where it is treated as a national art. Origami is usually connected to fun and games, and the most common association with the origami is a crane which has a special place in Japanese culture. The Crane, as it is shown in Fig. 1, can be simply folded from square piece of paper. The popularity of modern origami has grown in many aspects, mathematical, scientific, artistic, or even as an enjoyable craft. Today it is a complex discipline, with a preference for simplicity, where the less is more. By less, we mean the number of folds. The folding process is equally important as the final result. Complicated and tiresome folding results in stiff, messy, and unappealing origami (Kasahara 1973). The two basic rules that provide simplicity were given by Robert N. Budinski () Rusinska 63, Petro Kuzmjak School, Ruski Krstur, Serbia e-mail: [email protected] © Springer Nature Switzerland AG 2019 1 B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences, https://doi.org/10.1007/978-3-319-70658-0_13-1 2 N. Budinski Fig. 1 Origami crane Harbin (1956). The first one is that models are obtained without scissors and glue, only by folding, and the second is that the shape of models should be recognized without additional colors or markings. Those rules are mostly obeyed by enthusiastic origamists, but variations with magnificent results are also possible. The restriction of uncut and unglued paper induces sparks of creativity. Each step can be reversed and studied, changed, or improved which contributes to the freshness and freedom of the expression. Origami is an art that communicates and shares, based on ordinary paper folding, thus making it appealing to common people. There are many kinds of origami with respect to folding. For example, there are flat origami, modular origami, wet origami, or tessellation origami. Flat origami produces models that can be pressed (Hull 2002; Schneider 2004) without additional creases, such as well-known crane or similar. Modular origami consists of assembled pieces (modules) to a model. It is interesting that modules are received by simple folds, while finished models can be very complex. Different types of polyhedrons are usually made with this technique. Wet origami is a type of origami where models are obtained from a paper that is dampened which allows easy mold and gentle curves. Natural looking models, such as animals and plants, are often made by this origami technique. Origami tessellation is an origami technique where models are obtained by folding in a repetition (Verrill 1998). Also, there is pureland origami, a branch of origami proposed by John Smith. He established minimalistic aesthetic of design and made origami accessible and suitable for beginners in folding, as well as for disabled children and people with hands manipulation difficulties. In Fig. 2, we can see a pureland origami model example called samurai hat. These models exhibit their own elegance and harmony. Principles of pureland origami are few, exact, and simple. One is that only a square-shaped paper can be used. The other is that only mountain and valley folds can be used in folding with a permit of unfold and turn model over. Folds are created or manipulated one at a time. It is interesting to note that this set of strict constrains opens new possibilities for origami but also allows for new artistic approach, as well. Each folding result must provide strong suggestion of a final form where the essence of the structure is appreciated. This kind of simplicity provides a new perspective in art of origami (Smith 1993). Mathematics and Origami: The Art and Science of Folds 3 Fig. 2 Pureland origami model of Samurai hat Fig. 3 Mountain and valley folding When talking about origami, we need to mention the father of modern origami Akahira Yoshizawa (1911–2005) who invented around 50,000 origami models and diagrams and described some of them in 18 books. Yoshizawa’s intention was to make models that would be based on simple folding lines that anyone could follow (Smith 2011). His modifications of traditional design made origami creative art with vast potential and numerous followers. Through the work of Akahira Yoshizawa, a folder can discover recommendations for successful paper folding. According to Yoshizawa, a folder should have a sophisticated respect for paper and firms shape sense. The results that would follow should express inner characteristic of subject and emphasize the suggestion rather than explanation (Konjevod 2008). Artistic moments enable folders to feel the spirit of origami. To Yoshizawa, origami was more than diagrams and geometry, even though diagrams of his models were origami introduction to the world of mathematics. Yoshizawa was very systematic in his work and symbolically represented origami folding, which lead to the development of system of origami folds. For example, folding as it is shown in Fig. 3 (on the left) is called a valley fold or crease and diagrammatically it is represented with ———–, while folding as it is shown in Fig. 3 (on the right) is called a mountain fold or crease. Mountain fold is represented with -•-•-•- kind of lines. Mathematically, it can be said that mountain folds are convex, and valley folds are concave (Hull 2003). Valley and mountain folds interchange the view point of the paper face. The two-fold sets can be considered as dual (Dureisseix 2012). 4 N. Budinski Fig. 4 Crease pattern of crane Origami models can be described with crease patterns to some extent. A crease pattern is a representation of crease types on unfolded paper. In Fig. 4 we can see the crease pattern of crane from Fig. 1. The issue is that crease patterns are lacking information that would describe the folded model, and determination of general crease pattern folding ability is an open question (Maehara 2010). Flat origami is loaded with mathematical problems. Flat paper folding follows the rules that can be described mathematically. The flatness of paper allows us to observe an origami model in two dimensions even though it is three dimensional model without compromising any information about layer overlapping (Schneider 2004). There are established mathematical rules that allow us to produce a crease pattern on flat origami models, for example, if we take a piece of paper and mark a point on it somewhere in the center of the paper. If we make one or more folds that pass through that point and then count the folding that represent mountains and valleys, the difference between the number of mountains and valleys is always two. That claim is known and proven as Maekawa’s theorem (Justin 1986a). Maekawa theorem states:|M-V|=2, where M is number of mountain folds and V is number of valley folds at every vertex. That means that the number of creases is even, and if we imagine an origami figure as a crease-pattern as a graph, it can be two faces colorable. The consequence of Maekawa’s theorem is that for each flat origami figure it always possible to color with two colors in a way that the fields obtained by folding and with the same border are colored with a different color. Also it means that each vertex number of creases is even. Figure 5 shows the crane crease-pattern colored in two colors (Hull 1994). The second very important rule of folding is described by the Kawasaki theorem. The Kawasaki theorem says that an origami model is a flat foldable if and only if the alternating sum of the consecutive angles folds around the vertex is zero. An example of this is shown in Fig. 6. It can be seen that sum of the angles Mathematics and Origami: The Art and Science of Folds 5 Fig. 5 Two colored crane crease pattern Fig. 6 Illustration of Kawasaki’s theorem around the vertex (clockwise from bottom) is equal to zero (90◦−45◦ + 22◦30- 22◦30 + 45◦−90◦ + 22◦30−22◦30) so it can be concluded that this crease pattern is flat-foldable. This criterion cannot be easily extended to the crease patterns with more vertexes (Hull 1994). It is also important to state that the paper sheet can never penetrate the fold. During the folding, the length of curves drawn on the surface is preserved on the paper, despite the transformation, due to the absence of cuts. If we overlook the paper thickness, this geometric transformation can be considered as isometric embeddings (Lebee 2015). Besides theorems there are origami axioms. Basically, origami axioms are operations which distinguish the creation of a crease by aligning one or more points of combination and lines on a paper sheet. They are mostly known as Huzita axioms because they were first proposed formally by Humiaki Huzita (1989, 1992), even though other mathematicians worked on the topic, such as Jacques Justin (Justin 1986b). Koshiro Hatori added another axiom in 2001 (Hatori 2001), which was proven by Robert Lang as a complete system of axioms (Lang 2003; Alperin and Lang 2006).
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