There is a surprisingly old history to curved-crease sculpture, going back to the 1920s at the Bauhaus. We give here a partial history focusing on the earliest known references.
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 Miura-ori 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]:
I remember vividly the first day of the [Preliminary Course]. Josef Albers entered the room, carrying with him a bunch of newspapers. … [and] then addressed us … “Ladies and gentlemen, we are poor, not rich. We can't afford to waste materials or time. … All art starts with a material, and therefore we have first to investigate what our material can do. So, at the beginning we will experiment without aiming at making a product. At the moment we prefer cleverness to beauty. … Our studies should lead to constructive thinking. … I want you now to take the newspapers … and try to make something out of them that is more than you have now. I want you to respect the material and use it in a way that makes sense — preserve its inherent characteristics. If you can do without tools like knives and scissors, and without glue, [all] the better.”
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].
In 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”.
Kunihiko KasaharaKunihiko Kasahara's book Extreme Origami [New York: Sterling Publishing Co., Inc., 2002] has many beautiful photographs of the concentric circle model. Kasahara learned of the model at his first meeting with Thoki Yenn, and he attributes it to both Josef Albers and Thoki Yenn.
The images on the right are the cover and pages 14–15 of the book, and show some of the many forms into which the concentric circle model can be manipulated.
David 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 folding 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 at Xerox PARC. Most of Huffman's sculptures are in the possession of his family.
See our recent paper on Reconstructing David Huffman's Legacy in Curved-Crease Folding.
Ronald 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 3-fold 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 Demaine
Martin 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.
One 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.
The title “Computational Origami” refers to our underlying algorithmic goal of determining the mathematical curved surface that results from different kinds of pleated folding. This kind of “self-folding origami” may have applications to deployable structures that can compress very small by folding tightly and later relax into its natural curved form. To control this process, we must understand what forms result from different pleatings, and how to design pleatings that make desired forms.
With MIT student Jenna Fizel (2004–2006), we have developed software to simulate the physics of pleated paper folding. These simulations accurately reproduce the observed physical behavior. The image on the right shows four examples (first showing the simulation, then a real model of folded paper): the hyperbolic paraboloid, concentric hexagons, concentric octagons, and concentric circles.
We are currently building larger sculptures by joining many smaller elements with various types of curves. See our curved-crease sculpture.