Compiled by Janet Hamilton
A good listing of books and articles on origami and mathematics is Tom Hull’s Origami-Math bibliography at http://www.merrimack.edu/~thull/origamimath.html (homepage http://www.merrimack.edu/~thull/ ). However, Tom himself has been the subject of a number of origami-math articles. The following “Tom Hull” sightings, and various other math-related sightings, were reported by members of the origami-l email list.
MAA's Mathematics Magazine volume 68, no. 3, June 1995, had a review of the article “Paper folds, creases, and theorems” from Science News 147 (21 January 1995, pg 44) by Ivars Peterson. "Origami is regarded as a geometric mathematical recreation, but little has been done to analyze mathematically how the crease patterns are related to the resulting forms. Thomas C. Hull (University of Rhode Island) is pursuing such an analysis by thinking of the crease pattern as a graph but with specific angles between the edges. His findings, and those of Japanese colleagues, has inspired new origami creations, much as mathematical analysis of juggling in recent years has inspired new juggling routines."
In the New England Sampler section of the February 1997 Yankee
Magazine, page 22,
was the article "Origami Man" about Tom Hull and origami-math at the University
of Rhode Island. There's a color photo of Tom with an 810-piece Buckyball-type sphere and the
Hull/Ow Five Intersecting
Tetrahedra.
In the Sept. 14, 1997 Boston Globe New England Section, there was an article about Tom Hull and origami-math. It contained a color photo of Tom sitting next to a 3 ft tall Neale/Simon modular dodecahedron, a big hyperbolic paraboloid, and as 810-unit Buckyball-type sphere.
The 9/27/1997 "Mad or Rad" column on the ABC News web page carried “Origami Unleashed - Know When to Fold ‘Em”. It included an animation of a crane being folded from a square. “Most of Hull's discoveries pertain to a branch of mathematics known as graph theory—the study of networks created by points and connecting lines. Which, according to Hull, is exactly what one gets by unfolding an origami piece.”
The February 6, 1998 issue of Science (Vol. 279, pp 804-805) reported on the American Mathematical Society and the Mathematical Association of Americas meeting. "Proving a Link Between Logic and Origami" (pp 804-805) focuses on the interrelation of math and origami, with quotes from Tom Hull. "Algorigami, Anyone" (page 805) discusses Robert Lang and the algorithm for his TreeMaker program. The March 13, 1998 issue carried another article mentioning origami: "Polyhedra Can Bend But Not Breathe" (page 1637).
In the December 2003 issue of Wired is "Genius on Paper" (page 58) about Tom Hull. Included are pictures of some of his creations.
Advertisements for the computer program Mathematica showcased its power by demonstrating origami folds in motion. The February 1997 (volume 276, #2) of Scientific American carried an ad on page 81 about Mathematica and Origami. Lucy Zamiatina used the program to animate folding some origami models (see http://www.wolfram.com/products/mathematica/experience/its.html).
Crux Mathematicorum, Volume 23, number 2 (March 1997) had an article by Robert Geretschla"ger, on “Folding the Regular Heptagon”. In Volume 23, number 4 (May 1997) Robert Geretschla"ger had another article on “Folding the Regular Nonagon”. The articles showed show how to fold a square piece of paper into a 7-sided and 9-sided polygon. Gerestschla"ger is also the author of a paper on the geometry of origami, “Euclidean Constructions and the Geometry of Origami”, in Mathematics Magazine, volume 68, number 5 (December 1995, pp. 357-371).
The book Mathematical Reflections: In a Room With Many Mirrors by Peter Hilton, Derek Holton, Jean Pedersen (New York: Springer, c1997) has an entire chapter devoted to paperfolding and number theory. It deals mostly with folding strips of paper into polygons. (ISBN 0387947701)
The January 1999 issue of The American Mathematical Monthly had an article called "More on Paperfolding", by Dmitry Fuchs and Serge Tabachnikov, about the mathematics of curved creases. The article included exercises to try.
Ian Stewart's Mathematical Recreations column in the February 1999 issue of Scientific American had an article on origami tessellations. The article discussed the work of engineer Tibor Tarnai of the Technical University of Budapest as reported in his article "Folding of Uniform Plane Tessellations" from the conference proceedings Origami Science & Art, edited by Koryo Miura et al. (Otsu, Shiga, Japan, 1994). It is about tessellating a plane and then using the tessellation to "crumple" a cylinder.
The Winter 1999 issue of Mathematical Intelligenger carried "Molecular Modeling of Fullerenes with Hexastrips" on creating models of various isomers of carbon from strips of paper with 60 degree folds. "Recently [1] Cuccia, Lennox, and Ow showed how origami, the ancient Japanese art of paper folding, can be used for the modeling of fullerenes. They chose modular origami, wherein simple modules are interlocked to form larger and more elaborate structures. In this paper another, and relatively easy, way will be presented to build models of fullerenes and related molecules using hexastrips." This paper was also published in The Chemical Intelligencer (January 1998, pp 40-45). The reference [1] is to The Chemical Intelligencer, 1996, 2(2), 26-31.
John Conway was interviewed on an episode of Nova (10/28/1997) about Andrew Wiles and his proof of Fermat's Last Theorem. Hanging behind Conway's desk was a mobile of various modular origami forms. Conway is the inventor of the Game of Life computer simulation. http://www.pbs.org/wgbh/nova/transcripts/2414proof.html
An undated New York Times clipping states that a patent was awarded to Alan Huan who “developed a new method based on origami, for solving computational problems.”
The December 2000 issue of Mathematics Magazine reviewed the article “A Mathematical Theory of Origami Constructions and Numbers” by Roger C. Alperin, originally published in the New York Journal of Mathematics Volume 6 (2000), pp. 119-133. “Origami, the originally Japanese craft of paper-folding, can be interpreted as an apparatus for mathematical construction. ... this article sets forth axioms for paper-folding and characterizes the resulting origami-constructible points and numbers ...” http://nyjm.albany.edu:8000/j/2000/6-8.pdf
The February 2001 issue of Mathematics Magazine printed the article “Finding Quartic Roots” by B. Carter Edwards and Jerry Shurman. “Recent articles show how certain geometric systems---origami or the "mira"---go beyond straight-edge and compass to solve cubic equations. The key is to construct the common tangents to two parabolas. While this idea is not new, origami and the mira share the novel feature of constructing the tangents from the parabolas' foci and directrices rather than from the parabolas themselves. Thus these methods really use only points and lines. …. This article shows how to solve certain quartic equations similarly, using the common tangents to a parabola and a circle. The geometric construction requires origami-type folds and a compass construction, thus using points, lines, and one circle.”
A front page story in the Boston Globe on 02/17/02 about an young professor who had just been hired at MIT along with his father (Erik Demaine and Martin Demaine) “Road Scholar Finds Home At MIT”:
Origami whiz learned from his
nomad dad
”...It was Martin, with his background in the visual arts, who introduced his
son and his professors to the ancient Japanese art of folding paper.
Long the exclusive terrain of a few researchers, origami math has woken up
slowly in recent years as researchers began to apply it to a lengthening list of
real-world applications: to the folding of proteins in human DNA, or the
unfolding of enormous lenses in orbiting space telescopes, or the folding of air
bags in automobiles.
Erik also became interested in the study of linkage, the dynamics of rigid-sided
polygons in two dimensions. Last year, with the help of mathematicians Robert
Connelly and Gunther Rote, he solved the infamous ''Carpenter's Ruler'' problem,
which had stymied scientists since the 1960s, proving that any such polygon can
be unraveled without breaking - work that would be relevant to the fields of
robotics and genetics...”
The Mathematics Teacher, 1972-10 v LXV n 6, “Some Whimsical Geometry” by Jean J. Pedersen, pp. 513-521. "a method of approximating, by folding a strip of paper, any regular polygon whose number of sides is of the form 2^n + 1, where n is any natural number; or 2^(2n + 1) - 1, where n is any natural number greater than one." And “Collapsible Models of the Regular Octahedron” by Charles Trigg, pp. 530-533, with models that open out and flatten into an "irregular polyhedron", i.e. a "net".
The Mathematics Teacher, (87, 1994) pp 630 – 637, carried the article “Folding n-pointed Stars and Snowflakes” by Steven I. Dutch.
The Mathematics Teacher, (46, 1953) pp 341 – 342, had an article called “The Pentagon and Betsy Ross” by Phillip S. Jones. Included were instructions for folding a five pointed star.
The Jan 18, 2003 issue of New Scientist magazine carried a fairly extensive article about origami-math, with cameo appearances by Tom Hull, Erik Demaine, and Robert Lang, among others.
The Chronicle of Higher Education printed an article on origami mathematics on 7/11/2003 called “Know When to Fold ‘Em” by Lila Guterman. The article mentions Robert Lang, Erik Demaine, Tom Hull, and Joseph Wu. “Although people have been making origami for centuries without the help of mathematical theories, the researchers say they have much to offer.”
From the New York Times 12/2003: "A historian of mathematics at Stanford appears to have solved the mystery of a treatise called Stomachion, written 2,200 years ago by the Greek mathematician Archimedes." "Archimedes asked how many ways (certain 14) pieces can be put together to make a SQUARE." Answer 17,152. In simple terms I would say it is a very complex tangram.
In the Volume 111, No. 1 issue (January 2004) of American Mathematical Monthly is an article by Burkard Polster called "Variations on a Theme in Paper Folding." It's all about the angles you get by folding strips of paper, skinny triangles, and circular paper. The author biography states, "Burkard Polster is well known in and around his home town Melbourne, Australia, as a mathematical juggler...[and] origami master..."
The NY Times carried an article on 2/15/2005 on Eric
Demaine and his work on the mathematical origami, including the single
cutproblem, the carpenter's rule problem, graph theory, and applications in the
study of protein folding. http://www.nytimes.com/2005/02/15/science/15origami.html?pagewanted=1
and http://timesofindia.indiatimes.com/articleshow/1023175.cms
Hasinoff, S. (2000) "Exploring Computational Origami with iFold". http://www.cs.toronto.edu/~hasinoff/origami.doc
Miyazaki, S.Y., Yasuda, T., Yokoi, S. and Toriwaki, J. I. (1996) "An origami playing simulator in the virtual space", Journal of Visualization and Computer Animation, 7 (1): 25-42 Jan-Mar 1996. http://www.om.sccs.chukyo-u.ac.jp/main/research/origami/journal/jvca.html
"An investigation of the usability of software for producing origami instructions" by Tung Ken Lam, 13th Sept, 2005. http://www.angelfire.com/or3/tklorigami0/Dissertation_v10_11.pdf
Burgoon, R., Wood, Z. J. and Grinspun, E. (2006) 'Discrete Shells Origami' in Proceedings of CATA, Seattle, WA, March 2006. http://www.csc.calpoly.edu/~zwood/research/pubs/origamiCATA06.pdf
Terashima, T., Shimanuki, H., Kato, J. and Watanabe, T. (2005) "Method for Representing 3-D Virtual Origami" in Document Analysis and Recognition, 2005. Proceedings. Eighth International Conference on, 1211 - 1215
Ida, T. (n.d.) Computational Origami Project. http://www.score.is.tsukuba.ac.jp/~ida/Ida2004/CompOrigami.htm
Mitani, J. (2006) ORIPA; Origami Pattern Editor. http://mitani.cs.tsukuba.ac.jp/pukiwiki-oripa/index.php?ORIPA%3B%20Origami%20Pattern%20Editor
Zamiatina, L.I. (1994) "On Computer Simulation of Origami", Mathematics in Education, Vol. 3 No. 3, Summer 1994. http://library.wolfram.com/infocenter/Articles/1786/
Miyazaki, S.Y. (2004) "Origami Simulation" http://www.om.sccs.chukyo-u.ac.jp/main/research/origami/index.html
Studio GoGoGo (1999) * How To Make "VRML Origami" http://www1.plala.or.jp/Studio_GoGoGo/vrml/MakingOrigami/MakingOrigami.htm
Nimoy, J. (2002) Making Origami Instructional Symbolics Interactive http://www.jtnimoy.com/itp/origami/
Shimanuki, H., Kato, J. and Watanabe, T. (2003) "Recognition of folding process from origami drill books" in Document Analysis and Recognition, 2003. Proceedings. Seventh International Conference on , (3-6 Aug. 2003) vol.1 p. 550 - 554 http://www.cse.salford.ac.uk/prima/ICDAR2003/Papers/0101_581_shimanuki_h.pdf
"Hybrid Flexahedrons", by Douglas A Engle of Climax, Colorado, published in the Journal of Recreational Mathematics, year unknown.
The Daily Post (New Zealand) printed "Origami's not
just about folding paper" by Kristin Macfarlane. "Who would have
thought folding bits of paper was all about mathematics and science?" The
article talks about the Great Origami Maths and Science Show, featuring Jonathan
Baxter and Hugh Gribben, that was touring New Zealand. http://www.dailypost.co.nz/localnews/storydisplay.cfm?storyid=3696352&thesection=localnews&thesubsection=&thesecondsubsection
April 2005 - Britney Gallivan developed equations to
determine the number of times a piece of paper could be folded in half, given
the thickness and size of the paper. She developed different equations based on
whether the paper was folded in one direction or alternate directions. Britney's
work was mentioned on the prime time CBS television show Numb3rs, and she has a
booklet available on the solution. The photo below shows 11 folds done, she has
completed up to 12. http://pomonahistorical.org:80/12times.htm
Sept 2007 - the American Mathematical Society featured 6 Robert J. Lang models on their Mathematical Imagery page: http://www.ams.org/mathimagery/thumbnails.php?album=16
In the Huntington Library's copy of Billingsley-Dee's English translation and edition of Euclid's Elements, London, 1570, there was a sheet of geometrical figures for the reader to cut out, fold, and paste into the book.
The Daily Gazette, Nove 14, 2007, "O'Rourke on Geometric Reductionism" by Neena Cherayil. "As part of this year’s Math Department lecture series, Philadelphia-native and Smith College professor Joseph O’ Rourke returned this Tuesday to present his lecture entitled “Geometric Folding Algorithms: Linkages, Origami, and Polyhedra.” The talk focused on the computational geometry behind the deconstruction of one, two, and three-dimensional figures." http://www.sccs.swarthmore.edu/org/daily/2007/11/14/orourke-on-geometric-reductionism/ O'Rourke is co-author with Eric Demaine of a new book titled "Geometric Folding Algorithms: Linkages, Origami, Polyhedra".