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 á Mathematical á Artistic á Crane á 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]

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 (1911Ð2005) 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). The mathematically formal description provided an explanation of possible origami-geometric constructions. The list of axioms and its graphical representations is provided below in Figs. 7, 8, 9, 10, 11, 12, and 13, respectively. The first axiom: Given two different points A1 and A2, there is a unique passing fold through both (ruler operation). 6 N. Budinski

Fig. 7 Graphical representation of the ﬁrst axiom A1 A2

Fig. 8 Graphical A1 representation of the second axiom

Fig. 9 Graphical representation of the third p1 axiom

Fig. 10 Graphical representation of the fourth axiom

The second axiom: Given two different points A1 and A2, there is a unique superposing fold A1 and A2 (perpendicular bisector). The third axiom: Given different lines p1 and p2, there is a super posing fold of p1 onto p2 (bisector of an angle). The fourth axiom: Given a line p1 and a point A1, there is a unique perpendicular fold to p1 and passing through point A1 (perpendicular footing). The ﬁfth axiom: Given two different points A1 and A2 andalinep1, there is a fold placing A1 to p1 and passing through A2(tangent to parabola from a point). Mathematics and Origami: The Art and Science of Folds 7

Fig. 11 Graphical A1 representation of the ﬁfth axiom

Fig. 12 Graphical A1 representation of the sixth axiom

Fig. 13 Graphical representation of the seventh axiom

The sixth axiom: Given two different points A1 and A2 and two different lines p1 and p2, there is a fold placing A1 onto p1 and A2 to p2. The seventh axiom: For a given point A and two different lines p1 and p2, there is a fold placing A onto p1 and perpendicular to p2. The idea of discovering mathematics behind the paper folds appeared before Huzita Hatori’s axioms in a book written by Sundara Row titled “Geometrical Exercises in Paper Folding” published in 1893 in India that suggested explanations of underlying mathematical concepts in paper folding. In Fig. 14, we can see paper folded squaring binomial explanation from the book. In 1930, an Italian mathematician Margherita Beloch Piazzola proposed that paper folding could be used as a tool for solving geometrical problems. She analyzed algebraic aspects of origami in her work and proposed paper folding in order to solve third degree equations. In honor of her Axiom 6 is also called the Beloch fold (Liu 2017) which corresponds to solving third-degree equations. Today it is well 8 N. Budinski

Fig. 14 The part of “Geometric exercises in paper folding” (Row 1893)

Fig. 15 Peter Messer’s instruction for origami solution of Delian problem (Fenyvesi et al. 2014) known that the mathematical basis of origami enables solutions of equations such as quadratic, cubic, and quartic equations with rational coefficients. Also the doubling the cube problem or trisecting an angle can be solved origami. Constructing cube roots are also possible due to origami axioms. The Doubling the cube problem is also known as a Delian problem and is significant in the history of mathematics (Burton 2006; Zhmud 2006). It is based on calculating the volume of geometric√ solids. A unit cube doubling depends upon constructing a line segment of length 3 2 which is impossible to solve within the constraints of Euclidian geometry. The solution based on origami was found in 1986 by Peter Messer (Messer 1986). The instructions for solution based on folding are shown in Fig. 15. There are five steps. Firstly, a square-shaped paper is needed to be folded in half. Than it must be folded to match the line segments AC and BE. The third step would be folding three equal parts, and fourth is folding angle C to match the line AB and to match the point√ I with line FG. The point C will define the line segment AC and CB in the ratio 1: 3 2. The paper folded solution is shown in Fig. 16. Let BC = 1 and let x = AC and y = BR, where AB = x + 1 and CR = 1 + x−y follow. By applying the Pythagorean theorem CR2 = BR2 + BC2, we obtain + − 2 = 2 + = x2+2x (1 x y) y 1. Simplifying the expression, we obtain y 2+2x (1). If we observe triangles IFC and CBR, we can notice that they are similar. That indicates Mathematics and Origami: The Art and Science of Folds 9

Fig. 16 Paper folded solution of Delian problem

Fig. 17 Platonic solids made by origami technique that BR:CR = FC:IC (2). The segment FC is consisted in the segment AB, which = + + + = 1 + + + = means that AB AF FC CB which is 1 x 3 (1 x) FC 1, where FC 2x−1 2x−1 y = 3 3 . If we apply that to (2) we obtain 1+x−y 1+x , which by simplifying is 3 = (2x−1)(x+1) 3 + 2 + = 3 + 2 + − y 3 (3). As the result of that we get x 3x 2x 2x 3x 2x 2 and x3 = 2. Besides flat origami, modular origami attracts scientific attention, due to visually attractive models. Modular origami is attractive in the world of mathematical art and there are many books with instructions for folding shapes like polyhedrons (Kasahara 2003; Mukerji 2007). Unlike flat or traditional origami, models of modular origami are made by assembling units. Usually, all units are the same and easy to fold, while the final models are complex and require certain skill to assemble. Figure 17 represents Platonic solids folded by the principles of modular origami. Platonic solids are regular geometrical solids, named after Plato. There are only five of them: tetrahedron, cube, octahedron, dodecahedron, and icosahedron. This is because there are at least three faces at each vertex. The sum of internal angles at the vertex has to be less than 360◦, otherwise the shape would be flattened. Because of their aesthetic beauty and interesting mathematical properties, they are an inspiration to mathematicians and artists. Modular origami can be used for assembling different types of polyhedrons. For example, Fig. 18 shows a stellated icosahedron with 20 triangular faces raises to triangular pyramids. This polyhedron 10 N. Budinski

Fig. 18 Origami made stellated icosahedron

Fig. 19 Instruction for folding sonobe unit was made from 30 units called sonobe. The instructions for sonobe folding are in Fig. 19.InFig.20 we can see objects called the Epcot balls. Each of them is made from 270 sonobe pieces. Figures 17, 18, and 19 show other origami polyhedrons constructible with paper (origami polyhedrons shown in Figs. 17, 18, and 19 were made by high school students from Petro Kuzmjak School in Serbia in their mathematical lessons dedicated to the geometrical properties of polyhedrons.). It is almost an unbelievable fact that simple transformation of paper can produce so many amazing properties, which inspires many mathematicians, artists designers, scientists, and engineers. It is almost impossible to make a comprehensive list of all applications and results inﬂuenced by origami. Anyone who has tried origami might notice that simple fold organization results in rich motion. Many advantages of paper folding, such as simple transformation of matter, the use of inexpensive material without cuts, easy development of three-dimensional shape, form a basis for the investigation of technological and architectural implementation (Peraza- Hernandez et al. 2014; Sorguç et al. 2009; Dureisseix 2012; Lalloo 2014). The latest challenges are in the ﬁeld of robotics (Felton et al. 2014), technology, and space research. Mathematics and Origami: The Art and Science of Folds 11

Fig. 20 Epcot balls, each made from 270 sonobe units

Fig. 21 Miura-ori folding pattern. (Wikipedia commons, source: www. Commons.wikimedia. org/wiki/File%3AMiura- Ori_CP.svg)

Miura folding (also called Miura-ori) is ubiquitous with connections between origami and science. This form of origami folding proposed by Koryo Miura reveals ﬂatness of folded paper or of some other material. The paper version of Miura-ori folding pattern is represented in Fig. 21. 12 N. Budinski

Fig. 22 The space satellite with foldable solar panels. (Courtesy: National Science Foundation, source: www.nsf.gov)

It can be noticed that the mountain and valley folds form congruent parallel- ograms. After making creases, objects can be simultaneously folded or unfolded (Mahadevan and Rica 2005). It gives the impression that folds can be “remem- bered,” so it is also called a surface memory origami (Nishiyama 2012). Flat surfaces that are Miura folded are collapsible, transportable, and deployable objects. This makes them suitable to design a range of objects, from robots to small surgical devices. The future of Miura folded objects is bright, since it can be applied to a wide range of materials, including graphene, which is practically a one-atom thick material (Turner et al. 2016). The Miura-ori can also be applied to rigid materials, which is called rigid origami, and it has great practical importance and influence on technology. It is used in astro-nautical engineering for space satellites and its solar panels. In 1995 a Japanese research vessel was launched into space with a solar array that was folded using Miura-ori pattern. The space satellite with its foldable solar panel is shown in Fig. 22. Rigid origami, unlike paper origami, deals with flat rigid sheets, which cannot bend during the folding process. Rigid origami models are made from sheet metal or some other material connected with hinges that represents crease lines. Many patterns can be folded in the conventional paper way but cannot be folded rigidly (Abel et al. 2016). Besides space exploration, rigid origami has many applications in kinetic architecture (Tachi 2011) and robotics (Balkcom 2002). Origami and Miura-ori find applications in the construction of metamaterials. Metamaterials are engineered materials with properties that cannot be found in the nature. Their unusual properties arise from the design of their structure arranged in smaller units. Origami metamaterials consist of units that are tessellated folding patterns such as Miura-ori. Metamaterials based on Miura-ori exhibit many Mathematics and Origami: The Art and Science of Folds 13

Fig. 23 A page from the ﬁrst origami book published in Japan in 1797. (Wikipedia commons, source: commons.wikimedia.org/ wiki/ Category:Hiden_Senbazuru_Orikata)

interesting properties with vast possibilities for applications. For example, Miura-ori fold has the feature of coexistence of positive and negative Poisson’s ratio (Poisson’s ration is the ratio of the transverse contraction strain to longitudinal extension strain in a simple tension experiment (Lakes and Witt 2002).) (Lv et al. 2014). These unique features are a valuable starting point for innovative metamaterials design, which in general can give us a new perspective to materials. Beside science, the mathematical concepts are very inspiring and lead to unique artistic expression outside of the tradition. In the case of origami, it so happened that mathematical concepts become crucial in origami design and allowed for better and more diverse forms of art (Lang 2009). It is not an easy job to describe connections between mathematics and origami art. The computer scientist David Huffman said he ﬁnds it natural that elegant mathematical theorems joined with paper surfaces should lead to the certain visual elegance (Wertheim 2004). The artistic expression is noticeable from the very beginnings of origami. Figure 23 shows instructions in the ﬁrst origami book Hiden Senbazuru Orikata, (English: Secret to Folding Thousand Cranes), published in Japan in 1797. The boundaries of origami are pushed not just by science, but also by art. Contemporary artists combine traditional origami aesthetics with new concepts, modern materials, and innovative technology, producing marvelous designs. There are numerous ways of artistic expression through paper folding and geometric structures. 14 N. Budinski

Fig. 24 “Computational origami” by Erik and Martin Demaine (Demaine and Demaine 2009, source: www. erikdemaine.org)

The Demaines, father and son, artists and mathematicians, suggest advantages of combining art and mathematics in their approach. Firstly, those two disciplines complement each other. According to them, doing art allows different insights into mathematics, while mathematics inspires new art. Secondly, this approach enables more ﬂuent work. On the one hand, when mathematics gets problematic, one can switch to visual representation, and on the other hand, art can be analyzed in light of mathematical understanding (Demaine and Demaine 2009). Origami is a great medium for developing structural form due to characteristics of development and folding, which are useful for the design of deployable structures. But it has its restrictions. If folding is combined with bending, the result is called curved folding which can be applied to materials that comes in sheets (Demaine et al. 1999). The art models of curved crease are shown in Fig. 24. It is called “Computational origami” and is the work of Erik and Martin Demaine on display at Museum of Modern Art (MoMA) in New York as a part of permanent collection. The curved-crease sculptures were known even earlier at the beginning of the last century as a result of Joseph Albers’ work at the famous art school Bauhaus in Germany, and later in the Black Mountain College. Both, artists and professors, encouraged experimenting with different materials, including paper and held a preliminary course in “paper folding.” In Fig. 25 we can see the work of students of paper studies in Black Mountain College (Adler 2004). This course had great pedagogical value, since paper folding allowed students to explore constructions through hands-on activities. Materials, such as paper, have certain limitations, but according to Albers, such constrains should awaken students’ creativity. His approach greatly inﬂuenced modern architecture, art, and design. Mathematicians and engineers study his models and examine its geometrical and mechanical features (Magrone 2015). The family of pleated origami models such Mathematics and Origami: The Art and Science of Folds 15

Fig. 25 Picture of students work in Black Mountains paper studies course in thesis “A New Unity, the Art and Pedagogy of Joseph Albers” by Esther Dora Adler (2004)

Fig. 26 Paper folded hyperbolic paraboloid

as pleated hyperbolic paraboloids are simple repeated patterns of mountain and valley folds, forming concentric shapes. It is fascinating that paper somehow finds a configuration of equilibrium, where flat parts remain flats, while creased parts remain curved (Demaine et al. 2011). A simple model of paper folded hyperbolic paraboloid is shown in Fig. 26. Paul Jackson highlights paper folds as an excellent point to start teaching design since that is a simple and nonexpensive way of matter transformation (Jackson 2011). He is considered a pioneer of the one crease style. His results come from the exploration of possibilities coming from minimal or even single one-fold of piece of paper. Jackson can be considered as an artistic minimalist. He is considered as the creator of recognizable and appealing models obtained from very few folds (Smith 2011), where intertwined light and shades on a sheet are essential (Konjevod 2008). Figures 27, 28, and 29 show the work of Paul Jackson. Even though there is no solid definition of minimalism, beside using a minimal number of possible folds, Jackson’s challenge of minimalism was accepted by other origamists. Paola Versnic (2004) Santa Claus is an origami minimalism masterpiece. The pose of the model is remarkable, sensitive, and impressive and achieved in 16 N. Budinski

Fig. 27 Minimalism of Paul Jackson. (Used with author’s permission)

Fig. 28 Organic Abstract by Paul Jackson. (Used with author’s permission)

a simple way. Even though it is extremely plain, it has immediate impact and recognition. It attracts instant interest and attention. It is shown in Fig. 30. Origami might seem very simple to be important, but Robert Lang, one of the leading origamists in the world whose passion for paper folding inﬂuenced science and art, made bridges among mathematics, science, and nature. Lang holds a doctorate in physics, but he dedicated his work to origami. With his work he proved that origami is not just a playful activity but a potential problem solver in the ﬁeld of design, fashion, electronics, robotics, or space research (Orlean 2017). His book Origami Design Secrets become an origami bible. Origami- related terminology suggested by Lang, such as circle packing or uniaxial base, has become very well accepted. Besides creating numerous origami diagrams, Lang produced two computer programs for implementing origami designs. The Mathematics and Origami: The Art and Science of Folds 17

Fig. 29 Paper bags by Paul Jackson. (Used with author’s permission)

Fig. 30 Folded Paola Versnic’s Santa Claus origami model. (Source: www. orihouse.com)

ﬁrst is called Tree maker. Tree maker is a computer program that designs origami bases. It produces crease patterns for “tree” resembling forms such as people or bugs. The second computer program is called Reference ﬁnder, also very useful since it gives instructions for the patterns. Robert Lang changed the very meaning of origami and increased its importance dramatically. Origami becomes more complex and practical. According to Lang, origami is developed in three segments, 18 N. Budinski

Fig. 31 Dogwood blossom, Opus 688 by Robert Lang. (Used with author’s permission, source: www. langorigami.com)

even though not strictly divided: mathematical, computational, and engineering. Besides developing the mathematical, computational, and engineering foundations of origami, Lang left his mark in the art of origami. According to him, origami can be compared with music. Lang compares folding instructions to a performance guide which still allows the performer to express oneself. Figures 31, 32, and 33 show the art work of Robert Lang. For each displayed artwork, Lang uses mediums such as Korean hanji, Japanese paper, or even an American foil, respectively. Regardless the medium, each piece is breathtaking. Robert Lang is recognized, among his other famous paper ﬁgures, for origami insects which are very complex, colorful and realistic. Before Lang, there were very few origami models of insects. He raised insects folding to an artistic level by ﬁnding inspiration in spiders, scorpions, and other artphomodes, which can be both fascinating and disturbing (Lang 1995). Lang’s models are beautiful and elegant and somehow absorb negative emotions. Some of his extraordinary passion for art involving paper folded insects is shown in Figs. 34, 35, and 36. Wet origami is a widely accepted technique established by the Yoshizawa. Wet folding gives origami models a more realistic appearance. In Figs. 37 and 38, we can see the work of origami artist David Chain made by this technique. Animals made by this origamist take on a very life like expression. His artistic expression is detached from any references, he creates his design by himself and his models look like they are clay models rather than paper folded. Paper puts many limitations to those who fold, but origami artists such as Daniel Chang overcome these limitations with their lavish imagination which result in astonishing artwork. Chang is distinguished Mathematics and Origami: The Art and Science of Folds 19

Fig. 32 White Rhinoceros, Opus 714 by Robert Lang. (Used with author’s permission, source: www. langorigami.com)

Fig. 33 Elevated Icosahedron (gold) by Robert Lang. (Used with author’s permission, source: www. langorigami.com)

for paper folded face sculptures. Catching a facial expression in paper is quite challenging, but his results bring inner emotions to the surface accurately. In Fig. 39, we can see his art work called “Female Hair Style.” Origami tessellations are an aspect of ﬂat origami that represents paper folding into tessellated patterns. Tessellations are known as patterns consisting of shapes arranged side by side without gaps. Patterns can repeat itself as long as they continue with the folding. Each origami tessellation consists of the following patterns: the crease, front, back, and light pattern. Light pattern can be seen when origami is put up to the light (Verrill 1998). The most common types of origami tessellations are 20 N. Budinski

Fig. 34 Allomyrina dichotoma, Opus 655 by Robert Lang. (Used with author’s permission, source: www.langorigami.com)

Fig. 35 Yellow Jacket, Opus 624 by Robert Lang. (Used with author’s permission, source: www.langorigami. com)

classic and corrugation. Classic tessellations are based on square or hexagonal grids which are folded into an odd number of layers forming patterns. The corrugation tessellations are based on a layer where tessellations are made into waves and wrinkles in the paper. The ﬂat origami tessellations discovery is attributed to Shuzo Fujimoto (1976; Lister 1997). The world of origami tessellation is enriched by the Joel Cooper’s striking three-dimensional sculptures of origami tessellations with exciting elements of tessellated nets. His artistic style is distinguished by folded masks inspired by bronze sculptures. Face shapes are made from a single paper sheet folded in tessellations. In Figs. 40 and 41, we can see Cooper’s female and male mask made with origami tessellation technique. Mathematics and Origami: The Art and Science of Folds 21

Fig. 36 Stag Beetle, Opus 220 by Robert Lang. (Used with author’s permission, source www.langorigami. com)

Fig. 37 Protector by David Chang. (Used with author’s permission source: www. ﬂickr.com/photos/mitanei)

The work of David Huffman has been recently revealed to the public and among many things he did, he is recognized for exceptional tessellations which are very different, not only from the most modern paper folded tessellations, but also from Fujimoto’s which are considered historical. Those tessellations are three dimensional and unlocked, which means that they can be rigidly folded by bending materials as crease lines into ﬁnal form (Davis et al. 2013). Another important reference in origami tessellation folding is Chris Palmer, whose designs of Queen Box Decoration and Five Points Flower Tower folded are shown in Figs. 42 and 43, respectively. When analyzing the connections between origami and mathematics, we need to mention Thomas Hull who has dealt with this topic in both mathematical and artistic ways. Hull is also known for using techniques that combine more than one piece of paper and explore advanced mathematical concepts of origami. Thomas Hull’s ﬁeld 22 N. Budinski

Fig. 38 Prancing pony by David Chang. (Used with author’s permission, source: www.ﬂickr.com/photos/ mitanei)

Fig. 39 Female hair style by David Chang. (Used with author’s permission, source: www.ﬂickr.com/photos/ mitanei)

of research is the mathematics of origami, but he is also known for his models of polyhedrons and geometrical objects. These stunning geometrical objects are an excellent visual example of the bridge among mathematics, origami, and art. In Fig. 44 we can see torus, which is a three-dimensional triangle twist tessellation, while in Fig. 45 we can see 3D grid made from a square grid. Hull also does wet origami techniques, and the result of that work is in Fig. 46, where we can see a hyperbolic paraboloid. The boundary of the paper follows a Hamilton circuit on the cube graph. His model of Five Intersecting Tetrahedra has been selected as one of the ten best origami models of all-time by the British Origami Society. In the Fig. 47, we can see one example of FIT. Hull is also known as an inventor of PHiZZ unit that is used for modular origami (Hull 2006). In Fig. 48 we can see his work called “Phizz Variation 2: rhombicosidodecahedron” made from 120 PHiZZ units. Mathematics and Origami: The Art and Science of Folds 23

Fig. 40 Eurydice 6 by Joel Cooper. (Used with author’s permission, source: www. ﬂickr.com/photos/ origamijoel)

Fig. 41 Hector1byJoel Cooper. (Used with author’s permission, source: www. ﬂickr.com/photos/ origamijoel) 24 N. Budinski

Fig. 42 Queen Box Decoration design by Chris Palmer, folded by Jorge Jaramillo. (CC BY 2.0, source: www.ﬂickr.com/ photos/georigami)

Fig. 43 Five Points Flower Tower design by Chris Palmer, folded by Jorge Jaramillo. (CC BY 2.0, source: www.ﬂickr.com/ photos/georigami)

Beside mathematics, origami can be combined with robotics in artistic expression. Matthew Gardiner works in the ﬁelds of art and science, connecting origami and robotics. He coined the term that describes his work as orirobotics. He bases his research of nature, origami, and robots and connects aesthetics and biomechanics. In Fig. 49 we can see Matthew Gardiner’s interactive gardens, rich in color, form, and material. Jun Mitani, a Japanese computer scientist, is referred as an origami magician due to his creations of extraordinary complex and so-called organic origami forms. The fusion of his two passions, computer science and origami, resulted in developing software for origami called Ori-revo and Ori-revo-morth. The ﬁrst one generates 3D origami shapes with rotational symmetry that can be folded from a sheet of paper. The second one is for animation of origami folding and unfolding, so users can observe the process of generating 3D origami objects a from a single sheet of paper. Mathematics and Origami: The Art and Science of Folds 25

Fig. 44 Torus 1 by Thomas Hull. (Used with author’s permission, source: www. ﬂickr.com/photos/ 33761183@N00)

Fig. 45 3D Grid by Thomas Hull. (Used with author’s permission, source: www. ﬂickr.com/photos/ 33761183@N00)

In Figs. 50 and 51, we can see models of Mitani’s “Triangle of whipped cream” and “A column embossed with circles” on the left sides, while on the right sides are the crease patterns (Mitani 2016). Mitani recognizes origami technology as a tool for designing various products, for example, in fashion industry. The dynamic collaboration with fashion designer Issey Miyake resulted in creating garments from a single sheet of fabric without cutting or sewing, only by permanent pleats folding and using imperceptible snaps. Each of these origamist-scientists-mathematicians-artists has their own speciﬁc approach, own folding skills and techniques, and area of expertise. Each one is pushing the limits of the disciplines of their interest. From simple paper forms, like cranes and frogs, this traditional craft has become modern, global, mathematical, scientiﬁc and artistic, suitable for creating unexpected results. Prominent art museums worldwide celebrate work of masters of origami folds and origami art can be found as part of the collection in the Museum of Modern Art in New 26 N. Budinski

Fig. 46 Cube 1 by Thomas Hull. (Used with author’s permission, source: www. ﬂickr.com/photos/ 33761183@N00)

Fig. 47 Five Intersecting Tetrahedra, design by Thomas Hull. (CC BY 2.0, source: www.ﬂickr.com/ photos/fdecomite/)

York, Hangar 7 in Salzburg, Tikotin Museum of Japanese art in Haifa, and in the Pendulum Museum in Vancouver. Exploration and experimentation with traditional and modern media gives one of the oldest materials such as paper many abilities for creative expression. Fusion of paper and computer, past and present, new and old, opens endless possibilities for mathematics and art as a uniﬁed concept. The dream of Akahira Yoshizawa has become truth, and origami has become a boundless creative art. Even more, it has become a mathematical and scientiﬁc discipline and also found its place in mental therapies and education, media and design. Origami’s Mathematics and Origami: The Art and Science of Folds 27

Fig. 48 Phizz Variation 2: rhombicosidodecahedron by Thomas Hull. (Used with author’s permission, source: www.ﬂickr.com/photos/ 33761183@N00)

Fig. 49 Interactive Gardens by Matthew Gardiner. (Source: www.ﬂickr.com/photos/ arselectronica, CC-BY-NC-ND-2.0) richness comes from its purity. The requirements are small: a piece of paper, hands, and little bit of imagination. The outcomes are astonishing: proofs of theorems, wings of spaceships, medical devices, clothes, furniture, sculptures, and many more that the future will bring. 28 N. Budinski

Fig. 50 “Triangle of Whipped Cream” and its crease pattern. (Source: www.mitani.cs.tsukuba. ac.jp,CC4.0)

Fig. 51 “A Column Embossed with Circles” and its crease pattern. (Source: www.mitani.cs. tsukuba.ac.jp,CC4.0)

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