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Origami 6
Miura K., Kawaskai T., Tachi T., Uehara R., Lang R., Wang-Iverson P., American Mathematical Society, Providence, RI, 2016. 736 pp. Type: Book (978-1-470418-74-8)
Date Reviewed: Aug 8 2016

Cutting and flattening a cardboard box (say, for recycling) results in a planar polygon. It is quite a surprise to learn that some box unfoldings can be refolded to a different box. For example, a 1 x 1 x 5 box can be unfolded and refolded to a 1 x 2 x 3 box (both with surface area 22). R. Uehara describes in these proceedings his discovery of the first example of a polygon that folds to three different boxes. Moreover, he proves there are an infinite number of such 3-box examples. But no one has yet found a 4-box example.

This is just one paper out of the 65 in this delightful 744-page proceedings that emerged from the Origami in Science, Mathematics, and Education conference that is held every few years. Reflecting the growth of this sprawling field, the proceedings is now in two volumes, the first on the mathematics behind origami and the second on its applications to technology and its uses in art and education. I’ll sample mainly from the math volume to give a flavor of the whole.

Z. Abel et al. study the rigid flattening of polyhedra: continuous flattening of a polyhedron to the plane, creased and cut with slits. Here “rigid” means the faces bound by creases cannot bend. It is an open question whether this can always be accomplished without disconnecting the surface, but some slits are necessary. They show that a regular tetrahedron of edge length 1 can be flattened with a single slit of length less than 0.05, and 36 creases. That such a tiny slit suffices is remarkable. A nice contrast to this work on flattening is Y. Miyamoto’s beautiful rotating pop-up dome-like structures, a type of inverse of flattening.

Rigid origami is among the several foci in these volumes, partly because of its application to deployable structures, such as spacecraft solar arrays. T. Evans et al. study rigid origami tessellation twists and show that although no such triangle twist exists, there are many possible quadrilateral twists. Z. Abel, with a different group of colleagues, show that triangulating an origami model with creases does not always suffice to allow it to rigidly fold: the configuration space can be disconnected.

Another focus is curved-crease origami, pioneered by David Huffman (he of Huffman coding fame), who is a posthumous coauthor on one paper. S. Chandra et al. describe a system for designing shapes that can be built from flat aluminum sheets and curve-creased into visually stunning structures.

One of the exciting new directions explored in these proceedings is self-folding structures. For example, M. Ghosh et al. describe designing shapes built from “shape-memory alloys” that reconfigure without collision while remaining gravitationally stable.

The value of origami as an educational vehicle is increasingly recognized. One example is A. Bahmani et al.’s unit for ninth and tenth grade students that introduces fractals (and infinity) through building recursive structures: Palmer’s “flower tower” and Sierpinski’s fractal tetrahedron.

Among my favorite papers in the art section is by T. De Ruysser, a jewelry designer, who uses flexible origami tessellation patterns to manufacture wearable metal origami! I can only summarize these eclectic volumes by affirming the editors’ synopsis: Origami 6 is “a unique collection of papers accessible to a wide audience.”

Reviewer:  Joseph O’Rourke Review #: CR144671 (1610-0742)
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