curved origami
History of Curved Origami Sculpture
Erik Demaine and Martin Demaine
There is a surprisingly old history to curved origami sculpture, going back to the 1920s at the Bauhaus. We give here a partial history focusing on the earliest known references.
BauhausThe earliest known reference of curved origami sculpture is from a student’s work at the Bauhaus, from a preliminary course in paper study taught by Josef Albers in 1927–1928. The image on the right is from page 434 of the book Bauhaus: Weimar, Dessau, Berlin, Chicago by Hans M. Wingler [MIT Press, 1969 and 1978 paperback]. It shows a simple yet beautiful model that appears again and again over the years. Take a circular piece of paper and fold it along concentric circles, alternating mountain and valley. You can score circular creases with a compass or a laser cutter. The actual folding is tricky, but once complete, the pleated form will automatically twist into a saddle curve. Albers also taught this model at Black Mountain College in North Carolina, circa 1937–1938, where he was the head of the art department. See page 33 and Figure 29 on page 73 of Esther Dora Adler’s thesis “A New Unity!” The Art and Pedagogy of Josef Albers [University of Maryland, 2004]. 
Of interest to origamists are other foldings (with straight creases) created in 1927–1928 Albers’ class: much earlier models similar to Miuraori and Fujimoto folds, and the earliest known hyperbolic paraboloid. The image on the right is from page 435 of Bauhaus: Weimar, Dessau, Berlin, Chicago.
The hyperbolic paraboloid or hypar (shown in the top left) is similar in spirit to the concentric circle model. Take a square piece of paper and fold concentric squares, alternating mountain and valley, and also fold the diagonals. Again the paper pops automatically into a saddle curve; this time, however, it appears to be a mathematical surface called the hyperbolic paraboloid. Hannes Beckmann writes in his 1970 article “Formative Years” [in Bauhaus and Bauhaus People, edited by Eckhard Neumann, New York: Van Nostrand Reinhold, 1970, page 196]:

Another example of the concentric circle model is by Irene Schawinsky, wife of Alexander “Xanti” Schawinsky who was a Bauhaus student and later taught at Black Mountain College (presumably with Albers). This sculpture appeared at the Museum of Modern Art (MoMA), sometime before 1944.
This sculpture shows a common variation on the concentric circle model, where a center circular hole has been cut out. In this case, the hole is rather large, enabling significant flexibility. The image on the right is from page 42 of the book Paper Sculpture: Its Construction & Uses for Display & Decoration by Paul McPharlin [New York: Marquardt & Company, Incorporated, 1944]. 
Thoki YennIn the origami community, Thoki Yenn popularized the concentric circle model, again with a hole in the center, sometime before 1989. He calls the model “Before the Big Bang”. The images on the right are from Thoki Yenn’s site (archived by Erik Demaine since Thoki’s death) and the British Origami Society. 
David HuffmanDavid Huffman is most famous for his 1952 invention of Huffman codes which are used in almost every digital device and, for example, every JPEG and MP3 file. But since at least 1976 when he wrote his paper “Curvature and Creases: A Primer on Paper”, Huffman has also explored origami with curved creases. The image on the right is one of his many sculptures, Concentric Circular Tower, from The Institute for Figuring. It closely resembles the concentric circle model. We believe, however, that it is from a convex cone of paper, made from a flat disk of paper by cutting out a pie wedge and gluing together the two seams. This small modification causes the circles to remain roughly concentric after folding, instead of curling like the examples above. Many more of David Huffman’s sculptures can be seen on the web. See, for example, Margaret Wertheim’s New York Times article (2004), Grafica Obscura (1996), and Marshall Bern’s Origami Art Show. Most of Huffman’s sculptures are in the possesion of his family. 
Ronald ReschRonald Resch was a contemporary of David Huffman who has also explored curved creases. The two of them apparently also had many discussions about paper folding. The image on the right is from Resch’s page. The sculpture is called “The White Space Curve Fold with 3fold Symmetry”. It was designed and folded by Resch circa 1971–1972, and shown at the 1972 exhibition “Ron Resch and the Computer” at the Museum of Art in Salt Lake City, Utah. 
Erik and Martin DemaineMartin Demaine‘s explorations in curved creases begin in the 1960s. The Demaines’ joint explorations and sculptures began in 1998. When they discovered the hyperbolic paraboloid (hypar) based on concetric squares, they designed and folded a series of “Hyparhedra” sculptures together with Anna Lubiw. The image on the right shows such a sculpture. This work was published in a paper “Polyhedral Sculptures with Hyperbolic Paraboloids” [in Proceedings of the 2nd Annual Conference of BRIDGES: Mathematical Connections in Art, Music, and Science, 1999, pages 91–100]. 
Their experimentation with the concentric circle model began soon thereafter. They have considered several variations, such as concentric ellipses, concentric parabolas, and concentric circles with offset centers. Some of these variations were in joint explorations with MIT students abhi shelat in 2003 and Duks Koschitz in 2007.
Their most recent exploration, titled “Computational Origami”, is a series of three sculptures shown on the right. These sculptures differ from previous models in two ways. First, each sculpture connects together multiple circular pieces of paper (between two and three full circles) to make a large circular ramp of total turning angle much larger than 360° (between 720° and 1080°). Second, each sculpture is also turned a different amount before joining the sliced circles into one big (topological) circle. This approach allows for a wider range of forms that we are just beginning to understand. These sculptures are part of the Museum of Modern Art (MoMA) permanent collection. They were also part of the Design and the Elastic Mind exhibit at MoMA from February 24 to May 12, 2008. 
Last updated July 12, 2008 by Erik Demaine.