Trisecting an Angle with Origami
By Julie J. Rehmeyer
Many a mathematician has received a long letter from an unknown sender who claims to have found a way to trisect an angle using only a straightedge and compass. The mathematician may read passages of the letter aloud to the guffaws of colleagues at afternoon tea and then toss it out. Without even reading the details, any mathematician would know the writer is incorrect. In the early nineteenth century, the young French mathematician Évariste Galois proved the problem to be impossible.
More recently, mathematicians have found that it is possible to divide an angle into three equal parts by folding paper rather than using a straightedge and compass.
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Robert Lang made this flying walking stick from a single square of untorn paper. To create such complex origami, he designed software that mathematically analyzes shapes to generate the crease patterns for origami sculptures. |
Euclidean geometry is based on a set of axioms that he used to create figures and prove theorems involving those figures. These axioms allow one to create straight lines by connecting two points or by extending an existing line. They also allow one to create circles using a given straight line as the radius. Euclid's lines are mathematical abstractions that can be represented in a drawing or diagram. Generating those diagrams involves the use of two tools: a straightedge and a compass.
Origami artists create lines by folding paper instead of using a straightedge. Several mathematicians, including Humiaki Huzita of the University of Padua in Italy, have explored the range of figures one can create with origami folds. Huzita has codified origami folds into a set of six mathematical axioms for generating abstract lines. His axioms are analogous to Euclid's. Five of them allow one to generate lines that Euclid could have drawn. The sixth axiom allows the creation of lines that Euclid would not have been able to generate.
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Huzita's technique involves folding a paper so that point P lands on line l and point Q lands on line m. |
The result is that origami artists can create any line that Euclid could have drawn, plus some additional lines. In particular, Hisashi Abe of Hokkaido University in Japan discovered how to use the non-Euclidean technique to generate lines that trisect any angle.
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The first step in trisecting an angle is to place one side of the angle along the bottom edge of the paper. |
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The next step is to add two lines that are parallel to the line along the bottom edge, such that the top and bottom parallel lines are equidistant from the middle one. P is the point at the bottom left corner, and Q is the point where the top line meets the side. |
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Fold points P and Q onto lines l and m. |
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When unfolded, the green lines trisect the original angle. The red lines show where the paper was folded. |
Bill Casselman of the University of British Columbia in Vancouver explains the technique in the May 2007 Notices of the American Mathematical Society. His article is based on Project Origami: Activities for Exploring Mathematics (2006, A.K. Peters Ltd.), a recent book by Thomas Hull of Merrimack College in North Andover, Mass.
Nevertheless, Galois's theorem stands unmarred, and the letter-writers are incorrect. It remains impossible to trisect an angle using a straightedge-and-compass method. In fact, Hull also uses origami concepts in his book to illustrate Galois's methods.
References:
Casselman, B. 2007. If Euclid had been Japanese. Notices of the American Mathematical Society 54 (May):626-628. Available at http://www.ams.org/notices/200705/comm-casselman-web.pdf.
Hull, T. 2006. Project Origami: Activities for Exploring Mathematics. Wellesley, Mass.: A.K. Peters Ltd. See http://www.akpeters.com/product.asp?ProdCode=2582 and http://www.akpeters.com/ProjectOrigami/.
Peterson, I. 2006. Folding perfect thirds. Science News Online (June 17). Available at http://www.sciencenews.org/articles/20060617/mathtrek.asp.
______. 2005. Paper bags and tricky folds. Science News Online (July 23). Available at http://www.sciencenews.org/articles/20050723/mathtrek.asp.
______. 2001. Folding maps. Science News Online (Jan. 6). Available at http://www.sciencenews.org/articles/20010106/mathtrek.asp.
______. 1995. Paper folds, creases, and theorems. Science News 147(Jan. 21):44. Available at http://www.sciencenews.org/pages/pdfs/data/
1995/147-03/14703-18.pdf.
Comments
It is interesting that the trisection of an angle (according to Galois's theorem) cannot be accomplished with only compass and ruler IN THE SAME PLANE, but that one can do it if one steps into a third dimension to allow folding.
This is not a particularly new insight, but one is still entitled to wonder how many n-dimensional geometry problems remain "unsolved" because of an arbitrary restriction of dimensional freedom.
A. Schaller
Note Thomas Hull's response. Galois's theorem actually shows that angle trisection is impossible with ruler and compass, even if you allow three-dimensional constructions. --jjr
Posted by: A. Schaller | June 2, 2007 04:43 PM
Actually, origami angle trisecton methods have nothing to do with the 3rd dimension. Each fold is made flat. That is, you could consider each fold to be merely reflecting half the plane about the crease line. The 3D motion of making such folds is not taken into consideration, only the "before" and "after" locations of points and lines in the plane.
One thing that is not mentioned here is that H. Abe discovered this origami angle trisection method in 1980, I believe. Around the same time, and independently, mathematician Jacques Justin of France developed a different method that handles obtuse angles more easily. Both use the "fold two points to two lines" move, however.
Posted by: Thomas Hull | June 3, 2007 12:45 PM
Is the trisectioning of the specific angle(s) n2Pi where n= +/-(0, 1, 2,...) with compass and straight edge considered legitimate, or to be merely dismissed as a trivial nuisance?
The problem is to trisect any given angle. --jjr
Posted by: John Hauber | June 22, 2007 04:14 PM
Actually, to do the origami solution you used the compass to space the horizontal lines equally, and the straight edge to draw them. Same tools were used to transfer the Q and l lines to the back of the paper. Then you used trial and error to get Q onto m and the corner onto l. It would have been much quicker and more accurate to use just the compass to put an arc centered on P from m to the x-axis, then trisect that arc with the compass, correcting each try by removing 1/3 of the prior error. Works as well for fifths, etc.,too. Sorry to be a spoil-sport!
No, the origami solution uses allowed origami moves. It is true, however, that those things could be accomplished with straightedge and compass. The method you outline is an excellent way to get an approximation that is as close as you like to the precise one, but it won't be perfect. As a mathematical abstraction, the origami method will give you the perfect trisection (though, admittedly, if you try it in real life, it will be impossible to carry out the moves with mathematical precision. --jjr
Posted by: Mac Knapp | June 24, 2007 10:41 PM
I am not a math people, I am just playing origami. But I found that "technique to generate lines that trisect any angle" is NOT correct.
I cannot do it with a 180 angle and even a 90 angle.
So that method is really not much useful is origami.
But it is a funny way.
Tell me if I am mis-understanding your method.
And well, it there any method for dividing an angle into one-fifth.?
Posted by: yatching999 | January 18, 2008 03:46 AM