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Geometric Folding Algorithms: Linkages, Origami, Polyhedra

Erik D. Demaine and Joseph O'Rourke
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Collin Carbno
, on

This book is one of those rare mathematics books that I had a hard time putting down. I wanted to keep reading to find the next insight.

Half the material in the book is less than a decade old. The numerous open problems create the feeling of an unexplored mathematical territory where one is just scratching the surface of a vast new mathematical frontier. The bulk of the open problems, such as “Can every complex polyhedron be cut along edges and be folded flat into one piece without overlapping?” are fairly easy to state and understand. The book is also filled with mind-bending discoveries. For example, there is a planar linkage that can trace out any algebraic curve, even your signature. Thus, there is a linkage that lets you trisect an angle. My daughter and I puzzled over and experimented with the idea that any straight-line drawing can be folded so that the complete drawing can be cut out with one straight scissors cut. It works, except for the practical concern that sometimes the paper is too thick to fold.

The book’s material is organized in three roughly equal sized sections. The first section is about linkages (i.e. one dimensional phenomena like robotic arm movement and protein folding); the second section about paper folding or Origami (i.e. 2D phenomena) and the third section about various polyhedral properties (i.e. 3D phenomena of rigidity and unfolding to planar patterns).

Both authors are computer science professors, and the approach taken on these topics often has a definite computer science feel to it, with numerous discussions of algorithms and explorations of complexity. Still, while this book is not written in a “theorem followed by proof” style, it is a mathematical exploration abounding in deep mathematical results. Throughout the book are wonderful illustrations and solid proofs that bring the results alive.

This is a serious mathematics book whose explorations have significant applications and real mathematical profundity, wonderfully mixed with some fun recreational mathematics. Throughout the book are little tidbits, like the section on tensegrities, that twig the imagination as to possibilities for further research.

The book has a useful index and an extensive bibliography, so when you finish reading it will remain a valuable resource far into the future. There a lot of material in this book and it is really a lot of fun. I highly, highly, recommend this book to anyone with even a passing interest in folding mathematics. 

Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics (relativistic rotating stars) in 1979 at the University of Regina. He has been employed for nearly 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional designation from the Canadian Information Process Society.


Introduction; Part I. Linkages: 1. Problem classification and examples; 2. Upper and lower bounds; 3. Planar linkage mechanisms; 4. Rigid frameworks; 5. Reconfiguration of chains; 6. Locked chains; 7. Interlocked chains; 8. Joint-constrained motion; 9. Protein folding; Part II. Paper: 10. Introduction; 11. Foundations; 12. Simple crease patterns; 13. General crease patterns; 14. Map folding; 15. Silhouettes and gift wrapping; 16. The tree method; 17. One complete straight cut; 18. Flattening polyhedra; 19. Geometric constructibility; 20. Rigid origami and curved creases; Part III. Polyhedra: 21. Introduction and overview; 22. Edge unfolding of polyhedra; 23. Reconstruction of polyhedra; 24. Shortest paths and geodesics; 25. Folding polygons to polyhedra; 26. Higher dimensions.