Reflection about a vertical axis.
Commissioned by high school mathematics teacher David Masunaga for a talk about origami, and published in Peter Engel's book Origami from Angelfish to Zen.
In April 1988 I attended the 25th anniversary reunion of the Design Science program at Harvard University's Carpenter Center for the Visual Arts. Although I hadn't actually attended the program, let alone Harvard, I felt such kinship with this merry band of artist-mathematicians that I showed up anyway.
The program, founded by Harvard professor Arthur Loeb, is a uniquely interdisciplinary of 3d polyhedral geometry and sculpture. Equal parts art, mathematics and engineering, the program has produced one of the most interesting collection of alumnae I've ever seen. Talks at the event included:
-- Peggy Weil talking about her vision of the Weatherium: a multi-story inverted polyhedral globe that you view from a platform floating on the inside, with a live projected image of earth's surface as seen from weather satellites.
-- Amy Edmundson, who wrote a most illuminating book about Buckminster Fuller's geometry called A Fuller Explanation, giving a hands-on workshop on building structures out of toothpicks and marshmallows. You quickly learn that in order to build anything stable you have to use triangulated structures like tetrahedra.
-- A talk about radomes, which are geodesic domes built to protect radar dishes from high winds. Geodesic domes are strong, light, easy to assemble and require only a few distinct shapes. Unfortunately, the nearly parallel bands of creases across the surface of a geomesic dome tend to interfere with the waves being received. A more randomly constructed dome would not have that problem, but would be much harder to build and require manufacturing many more different shapes.
To solve this problem, the various pentagonal nodes spaced around the surface were all twisted slightly in whorls, an operation that Buckminster Fuller dubbed "jitterbugging." The complexity of assembly and number of distinct shapes stayed the same, but the added irregularity cleared up the reception problem. As a lover of polyhedra, I've always been fond of the jitterbug transformation, but never suspected that it had a practical application. To learn more about Buckminster Fuller's Geometry, check out the Design Science Consortium, which was started by the same people that for many years ran the Buckminster Fuller Institute.
-- High school mathematics teacher David Masunaga and mathematician-architect-writer Peter Engel talking about the mathematics of origami. I already knew David because he had brought me to Hawaii to give talks about inversions to mathematics students throughout the islands. Peter, whose mathematically virtuosic constructions put him at the forefront of origami, I met for the first time at this event.
David Masunaga teaches at Iolani School in Honolulu, Hawaii. He continues the design science tradition by filling his classrooms with stunning polyhedral models built by students. Ever the resourceful teacher, David challenged me to create an inversion on "origami". Since origami is the art of paper folding, I created a design that illustrates the symmetry of folding. I also created a bilingual version that imbeds "origami" in Chinese characters inside "origami" in English (this inversion will be posted next week!). David showed these inversions in his talk, taking advantage of the way an overhead transparency can be manipulated.
After the talk, Peter Engel asked if he could use the designs in his forthcoming book "Folding the Universe", which was re-released by Dover Publications as Origami from Angelfish to Zen" I cleaned up the designs a bit and created the version you see here. The Folding Universe is a unique combination of origami instruction and mind-expanding philosophy...sort of the Goedel, Escher, Bach of paper folding.
For a wonderful portrait of Peter and his art, see Discover magazine, June 1988. For photographs of origami folds by Peter Engel and others, see John Paulsen's origami menagerie. For more information about Peter Engel's book and other origami supplies, visit Fasinating Folds: Origami and the Paper Arts.
I generally prefer rotational symmetry to reflective symmetry, but in this case reflection was clearly appropriate. The hardest part of this design was figuring out how to turn the two letters I, with their dots, into parts of oher letters. The G/A combination in the middle is a slam-dunk, except that the two letters are of different cases.