The Chinese character for double happiness (Fig. 1) can be obtained with a template that requires several cuts (Fig.2). A question that arises from this template is: is it possible to obtain double happiness with one cut? Conveniently, Erik Demaine (ED), Martin Demaine (MD), and Anna Lubiw had published a theorem for answering this question (Demaine, Demaine & Lubiw, 1998; see also One-cut theorem ; a video lecture is available at: Video lecture ; for additional information, see Demaine and O’Rourke, 2007).
Figure 1. Double happiness (see Ma, 2008, p. 152).
Figure 2. Template for double happiness (see Ma, 2008, p. 153).
ED and MD quickly produced a crease pattern, thus allowing us, in celebration of the Year of the Dragon, to present you with the one-cut double happiness (Fig. 3).
Figure 3. Crease pattern for one-cut double happiness. Click on the picture to download a PDF.
Just as achieving a state of happiness requires a great deal of work, so the one-cut double happiness may pose a challenge. Folding the crease pattern is not difficult, but with regular, 20 lb. paper, the final folded paper, which has 69 layers, is about one cm thick when tightly compressed (Fig. 4) and cannot be cut easily (unless one has access to a laser cutter?).
Figure 4. Folded crease pattern for double happiness.
Folded tracing paper can be cut with scissors, but the tracing paper jams in the copier. Robert Lang (Lang, 2011) suggested taping the leading edge of the tracing paper to regular paper and feeding the paper through the copier manually. One of us (PW-I) tried this method at Staples, and it worked like a charm! But getting ED and MD’s double happiness crease pattern on tracing paper is just the beginning. To obtain a presentable one-cut double happiness requires precise folding and cutting, which PW-I did not achieve, as seen in Figure 5.
Figure 5. One-cut double happiness on tracing paper.
For those without ready access to a copier with manual feed, one can try the following: pre-crease regular paper containing the crease pattern and then re-crease the paper after placing a sheet of tracing paper on top. Remove the regular paper; refold the tracing paper and cut. Thus far, the final tracing paper product leaves much to be desired, but with each attempt, the result has improved.
If you are interested in trying to make the one cut double happiness using other approaches, and with better results, please share your strategies and outcome with your fellow readers of The Fold.
Patsy Wang-Iverson is affiliated with the Gabriella and Paul Rosenbaum Foundation, Erik and Martin Demaine are affiliated with the Massachusetts Institute of Technology, and Liping Ma is a collaborating independent scholar.
Erik D. Demaine, Martin L. Demaine, and Anna Lubiw (1998). “Folding and Cutting Paper”, in Revised Papers from the Japan Conference on Discrete and Computational Geometry (JCDCG'98), Lecture Notes in Computer Science, volume 1763, Tokyo, Japan, December 9–12, 1998, pages 104–117.
Erik D. Demaine and Joseph O'Rourke (2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge, UK: Cambridge University Press.
Erik D. Demaine, Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2010), Fall10 Lectures (retrieved 1/11/12).
Robert J. Lang, Thinnest paper, sharpest scissors email thread, OUSA-members email list, 12/21/11; langorigami (retrieved 1/11/12).
Liping Ma (1999). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum Publishers.
Liping Ma (2008), A Tour of Chinese Culture (10th grade Chinese language textbook). Palo Alto, CA: Liping Ma Press, pp. 152-153.