Mathematics and Origami: The Art and Science of Folds

  • Natalija BudinskiEmail author
Living reference work entry


There is no consensus where origami originated, but it is assumed that its roots are in China associated with the discovery of paper. However, the craft of origami blossomed in Japan, where it is treated as national art. Origami is usually connected to fun and games, and the most common association with origami is a crane which has a special place in Japanese culture. The popularity of modern origami has grown in many aspects, mathematical, scientific, artistic, or even an enjoyable craft.


Origami Mathematical Artistic Crane Scientific 


  1. Abel Z, Cantarella J, Demaine E, Eppstein D, Hull T, Jason K, Lang R, Tomohiro T (2016) Rigid origami vertices: conditions and forcing sets. J Comput Geom 7(1):171–184MathSciNetzbMATHGoogle Scholar
  2. Adler D (2004) A new unity, the art and pedagogy of Joseph Albers. PhD thesis University of MarylandGoogle Scholar
  3. Alperin R, Lang R (2006) One-, two-, and multi-fold origami axioms. Origami4 fourth international meeting of origami science, mathematics and education. A K Peters Ltd, Natick, MAGoogle Scholar
  4. Balkcom D (2002) Robotic origami folding. PhD thesis, Carnegie Mellon UniversityGoogle Scholar
  5. Burton D (2006) The history of mathematics: an introduction. McGraw−Hill, New YorkGoogle Scholar
  6. Davis E, Demaine ED, Demaine ML, Ramseyer J, Tessellations O (2013) Reconstructing David Huffman’s origami tessellations 1. J Mech Des 135(11):111010, ISSN 1050–0472. Scholar
  7. Demaine E, Demaine M (2009) Mathematics is art. In: Proceedings of 12th annual conference of BRIDGES: mathematics, music, art, architecture, culture, Banff, Alberta, Canada, pp 1–10Google Scholar
  8. Demaine E, Demaine M, Lubiw A (1999) Polyhedral sculptures with hyperbolic paraboloids. In: Proceedings of the 2nd annual conference of BRIDGES: mathematical connections in art, music, and science (BRIDGES’99), pp 91–100Google Scholar
  9. Demaine M, Hart V, Price G, Tachi T (2011) (Non)existence of pleated folds: how paper folds between creases. Graphs Combin 27(3):377–397MathSciNetCrossRefGoogle Scholar
  10. Dureisseix D (2012) An overview of mechanisms and patterns with origami. Int J Solid Struct 27(1):1–14MathSciNetCrossRefGoogle Scholar
  11. Felton S, Tolley M, Demaine E, Rus D, Wood R (2014) A method for building self-folding machines. Science 345(6197):644–646CrossRefGoogle Scholar
  12. Fenyvesi K, Budinski N, Lavicza Z (2014) Two solutions to an unsolvable problem: connecting origami and GeoGebra in a Serbian high school. In: Greenfield G, Hart G, Sarhangi R (eds) Proceedings of bridges 2014: mathematics, music, art, architecture, culture. Tessellations Publishing, Phoenix, pp 95–102Google Scholar
  13. Harbin R (1956) Paper Magic. Oldbourne Press, LondonGoogle Scholar
  14. Fujimoto S (1976) Twist origamiGoogle Scholar
  15. Hatori K (2001) K’s Origami, Fractional LibraryGoogle Scholar
  16. Hull T (1994) On the mathematics of flat origamis. Congr Numer 100:215–224MathSciNetzbMATHGoogle Scholar
  17. Hull T (2002) The combinatorics of flat foldis survey, The proceedings of the third international meeting of origami science, mathematics, and education, pp 29–38Google Scholar
  18. Hull T (2003) Counting mountain-valley assignments for flat folds. Ars Combin 67:175–188Google Scholar
  19. Hull T (2006) Project origami: activities for exploring mathematics. AK Peters. WellesleyGoogle Scholar
  20. Huzita H (1989) Axiomatic development of origami geometry. In: Proceedings of the 1st international meeting of origami science and technology, pp 143–158Google Scholar
  21. Huzita H (1992) Understanding geometry through origami axioms. COET91: proceedings of the first international conference on origami in education and therapy, British Origami Society, pp 37–70Google Scholar
  22. Jackson P (2011) Folding techniques for designers, from sheet to form. Laurence King Publishing, London, 118Google Scholar
  23. Justin J (1986a) Mathematics of origami, part 9. Br Origami Soc 118:28–30Google Scholar
  24. Justin J (1986b) Résolution par le pliage de l’équation du troisième degré et applications géométriques. L’Ouvert - Journal de l’APMEP d’Alsace et de l’IREM de Strasbourg (in French) 42:9–19Google Scholar
  25. Kasahara K (1973) Origami made easy. Japan Publications Inc, JapanGoogle Scholar
  26. Kasahara K (2003) Extreme origami. Sterling, New YorkGoogle Scholar
  27. Konjevod G (2008) Origami science, mathematics and technology, poetry in paper, the first origami exhibition in Croatia with international origami masters. Open University Krapina, Krapina City GalleryGoogle Scholar
  28. Lakes RS, Witt R (2002) Making and characterizing negative Poisson’s ration materials. Int J Mech Eng Educ 30(1):50–58CrossRefGoogle Scholar
  29. Lalloo M (2014) Applied origami. Ingenia 61:33–37Google Scholar
  30. Lang R (1995) Origami insects and their kin. General publishing co., Toronto, OntarioGoogle Scholar
  31. Lang R (2003) Origami and geometric constructionsGoogle Scholar
  32. Lang R (2009) Mathematical methods in origami design. Bridges 2009: mathematics, music, art, architecture, culture, pp 13–20Google Scholar
  33. Lebee A (2015) From folds to structures, a review. Int J Space Struct 30(2):55–74. Multi Science PublishingCrossRefGoogle Scholar
  34. Lister D (1997) Introduction to the third edition. In: Harbin R (ed) Secrets of origami: the Japanese art of paper folding. Dover Publications, Mineola, pp 1–3. (Originally published, 1964)Google Scholar
  35. Liu K (2017) The magic and mathematics of paper-folding. Tor.comGoogle Scholar
  36. Lv C, Krishnaraju D, Yu H, Jiang H (2014) Origami based mechanical metamaterials. Sci Report 4:5979CrossRefGoogle Scholar
  37. Maehara H (2010) Reversing a polyhedral surface by origami-deformation. Eur J Comb 31(4):1171–1180MathSciNetCrossRefGoogle Scholar
  38. Magrone P (2015) Form and art of closed crease origami. In: Proceedings of 14th conference of applied mathematics, Slovak University of Technology in BratislavaGoogle Scholar
  39. Mahadevan L, Rica S (2005) Self organized origami. Science 307:1740CrossRefGoogle Scholar
  40. Messer P (1986) Problem No. 1054. Crux Math 12:284–285Google Scholar
  41. Mitani J (2016) 3D origami art. A K Peters/CRC Press, Natick, MAGoogle Scholar
  42. Mukerji M (2007) Marvelous modular origami. A K Peters Ltd, Natick, MAGoogle Scholar
  43. Nishiyama Y (2012) Miura folding: applying origami to space exploration. Int J Pure Appl Math 79(2):269–279zbMATHGoogle Scholar
  44. Orlean S (2017) Origami lab: why a physicist dropped everything for paper folding. The New York TimesGoogle Scholar
  45. Peraza-Hernandez E, Hartl D, Malak D Jr, Lagoudas D (2014) Origami-inspired active structures: a synthesis and review. Smart Mater Struct 23(9):1–50CrossRefGoogle Scholar
  46. Row S (1893) Geometrical exercises in paper folding. Addison & Co, Mountain RoadGoogle Scholar
  47. Schneider J (2004) Flat-foldability of origami crease patterns.
  48. Smith J (1993) Some thoughts of minimal folding. British Origami.
  49. Smith J (2011) Simplicity and realism in origamiGoogle Scholar
  50. Sorguç A, Hagiwara I, Selçuk S (2009) Origamics in architecture: a medium of inquiry or design in architecture. METU J Fac Archit 26(2):235–247CrossRefGoogle Scholar
  51. Tachi T (2011) Rigid-foldable thick origami. Origami5: fifth Int. meeting of origami science, mathematics, and education, pp 253–263CrossRefGoogle Scholar
  52. Turner N, Goodwine B, Sen M (2016) A review of origami application in mechanical engineering. Proc Inst Mech Eng C J Mech Eng Sci 230(14):2345–2362CrossRefGoogle Scholar
  53. Verrill H (1998) Origami tessellation. Bridges: mathematical connections in art, music, and science, Winfield, Kansas, pp 55–68Google Scholar
  54. Versnic P (2004) Folding for fun. Orihouse, p. 36Google Scholar
  55. Wertheim M (2004) Cones, curves, shells, towers: he made paper Jump to life. NYork Times.
  56. Zhmud L (2006) The origin of the history of science in classical antiquity. Walter de Gruyter, BerlinzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Rusinska 63Petro Kuzmjak SchoolRuski KrsturSerbia

Section editors and affiliations

  • Bharath Sriraman
    • 1
  • Kyeong-Hwa Lee
    • 2
  1. 1.Department of Mathematical SciencesThe University of MontanaMissoulaUSA
  2. 2.Department of Mathematics Education, College of EducationSeoul National UniversitySeoulSouth Korea

Personalised recommendations