# The material reasoning of folding paper

## Abstract

This paper inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the nineteenth century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails.

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1. 1.

Examples include Mancosu (2008), Manders (2008), Avigad et al. (2009), Leitgeb (2009), Larvor (2012), De Toffoli and Giardino (2014), Larvor (2016).

2. 2.

The use of a compass is prohibited in folding constructions and hence circles cannot be drawn.

3. 3.

Reasoning by folding can thus solve mathematical puzzles that are not solvable by purely Euclidean straightedge and compass constructions. However, to our knowledge folding has never solved a mathematical puzzle that is not solvable by the formal-symbolic means of today.

4. 4.

See Larvor (2017). Explicitly, by the first three postulates we mean the following: (1) a straight-line segment can be drawn joining any two points. (2) Any straight-line segment can be extended indefinitely in a straight line. (3) Given any straight-line segment, a circle can be drawn having the segment as radius and one endpoint as center.

5. 5.

See for example Livingston (2008, esp. chapters 11 and 12).

6. 6.

See for example: Demaine et al (1999).

7. 7.

Three-dimensional paper depicts a two-dimensional plane in an instance of idealization. For more on idealizations in the folding practice, see Sect. 4.1.

8. 8.

Having motivated our philosophical interest in folding, allow us now to point out the joy that comes with paper folding. We invite our readers to read this paper with a sheet of semi-transparent paper (baking paper works nicely) and a pencil to hand to perform some of the folding we will discuss in later sections.

9. 9.

For a survey on Row’s work and on Hermann Wiener’s work, another mathematician who worked with folded models in 1893, see Friedman (2016).

10. 10.

Row (1893, p. i). Note that Eleonore Heerwart, an influential educator who helped spreading the Fröbelian ideas in Great Britain in the second half of the nineteenth century, numbered paper folding as gift num. 8 (Friedman 2018a, p. 239). The Fröbelian ideas came to India in the wake of Great Britain’s colonization activities, which indicates how Row may have encountered them. See Friedman (2018a, pp. 247–250).

11. 11.

12. 12.

By this we mean that by using folding techniques Row was able to construct an infinite number of points that lie on the given curve, hence constructing the curve as the geometrical place of these points. Obviously, the only curve that can be constructed without gaps with paper folding is a line.

13. 13.

It is still not clear how Klein obtained Row’s book.

14. 14.

Klein (1895, p. 33). The translation is taken from Klein (1897, p. 42).

15. 15.

On Hurwitz work on folding, see Oswald (2015). On the connection between Ahrens and Hurwitz conceptions of folding, see also Friedman (2018a, pp. 273–282).

16. 16.

17. 17.

To double the volume of a cube of side-length 1 one needs to construct a cube of side-length third root of 2.

18. 18.

Note that Beloch’s claim, that the operation can always be performed, requires the piece of paper to be indefinitely large. We discuss this requirement in Sect. 4.1.

19. 19.

To see this, we follow the argument in Hull (2011, pp. 310–311): Denote by M the intersection of the fold with the x-axis. Using properties of similar triangles, we obtain:

|OC|/|OA| = |OM|/|OC| = |OB|/|OM|

Substituting |OA| = 1 and |OB| = 2 we obtain:

|OC| = |OM|/|OC| = 2/|OM|

Hence

|OC|3 = |OC|. |OM|/|OC|. 2/|OM| = 2.

20. 20.

The reasons for this ignorance are beyond the scope of this article. For a detailed analysis of these reasons see Friedman (2018a, pp. 336–340).

21. 21.

Examples include Bern and Hayes (1996), Bern et al. (2003), Demaine and O’Rourke (2007) and Alperin and Lang (2009).

22. 22.

See for example Vi Hart’s famous video on YouTube from 2011 on the proof of the Pythagorean theorem via folding (https://www.youtube.com/watch?v=z6lL83wl31E) and Matthew Stone and Una Stojnic analysis of it (Stone and Stojnic 2015).

23. 23.

For the diagrammatic reasoning, an example of this control is, according the Manders, that “practice has to control the production of diagrams and provide for resolution of disagreement among judgments based on their appearances” (ibid., p. 91).

24. 24.

See also Netz (2017) on this theme. See Netz (1999) for a full account.

25. 25.

To stress the point: Row’s argument clarifies that mathematical folding does not require standardized shapes of paper.

26. 26.

Note that this step this is similar to what Giaquinto considers while discussing folding as proof practice; see Giaquinto (2007, p. 51), though he does not mention Row. We will return to the differences between Row’s and Giaquinto’s approach later (see Sect. 5). Notice here that whilst Giaquinto also folds through mid-points, Row’s prior folding operations have constructed these mid-points.

27. 27.

We refer here to Giaquinto’s terms “evidence-providing” and “non-evidential” (Giaquinto 2007, p. 67). We come back to Giaquinto in the next sections.

28. 28.

An envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point.

29. 29.

“Since two parabolas have three common tangents, the problem admits three solutions, including certain real solutions” (Beloch 1934, p. 187).

30. 30.

For ease of exposition, we will assume that the mathematicians fold paper. They could fold any other material that behaves similar to paper when folded, such as papyrus, tin-foil, etc.

31. 31.

Livingston (2008) includes an insightful discussion of the difficulties involved in performing an origami-fold from some given instructions. Paper folding in the sense of this paper faces similar difficulties—it is not easy to fold, say, a point onto a line, especially if the paper is not see-through. The mathematicians practicing paper-folding did not discuss these issues and neither will we do so here. The “you” in the above slogan “if you can fold it” thus refers to a skilled practitioner equipped with adequate material.

32. 32.

Notice the difference to Euclidean practice where not all drawings on paper are permissible constructions. Examples of non-permissible drawing in Euclidean practice include free-hand drawings, moving ruler constructions, Nicomedes chonoid etc. (see Bos 2001, for a discussion of some of these).

33. 33.

With Justin’s and Huzita’s works.

34. 34.

“The point of intersection of the diagonals is called the center of the square” (Row 1893, p. 3).

35. 35.

See for example Sect. 4.2, for Justin’s list of the seven basic operations. One may claim that new points can be marked as a result of basic operations, but these new points result from folding already given points. Note that (new) points can be obtained as a result of folding given points onto a new location (e.g. folding a point on a line, thus obtaining a point on a line); they can be also obtained as a “point of a crease”; for example: halving the segment AB by folding A onto B gives you a point C in the middle of AB.

36. 36.

Note that Huzita, while also trying to find the fundamental operations of folding, gave in Huzita (1989, p. 144), a list of only 6 fundamental operations and noted that there may be more operations to be added to this list.

37. 37.

Justin also gave in his paper a list of restrictions on these operations.

38. 38.

Justin also adds that P not be on D to avoid trivialities (Justin 1989, p. 257).

39. 39.

See e.g. Row (1893, p. 5).

40. 40.

This refers back to the idealization of the embodied aspects of the folding practice, cf. Sect. 4.1.

41. 41.

Note that if one succeeds in folding three different points on three different lines, this does not count as a proof, since the points and lines are not generic, hence this “success” holds only for a special case.

42. 42.

Or perhaps: a folding-version of this proof.

43. 43.

For example, operation 5 discussed in Sect. 4.2.

44. 44.

e.g. “I will argue that those empirical routes to the belief are not ways of discovering it” (Giaquinto 2007, p. 52).

45. 45.

“If [folding] demonstrations do have precise content, it can’t just follow from the physical effects of the demonstrations alone; there must be other cues that determine the content of the speaker’s argument. These cues must help viewers track the features of the demonstration that matter, and help viewers not be distracted by features of the demonstration that don’t.” (Stone and Stojnic 2015, pp. 81–82)

46. 46.

As we have seen, this is not how the mathematicians who fold paper reason, i.e. Giaquinto is losing traction with actual practice here. Arguably Giaquinto was not interested in folding as a practice but rather in folding as an epistemic activity.

47. 47.

“Is it not the same when through visualizing one concludes that the four corner triangles of a square would fit exactly onto he tilted inner square without overlap?” (Giaquinto 2007, p. 53).

48. 48.

Because distances are exact attributes of a diagram; cf. Manders (2008).

49. 49.

The co-exact attributes of basic folding operation 1 rely solely on the diagram.

50. 50.

This is a similarity to Euclidean geometric reasoning.

51. 51.

However, recall that Giaquinto may not have been interested in the rigor of this reasoning practice.

52. 52.

See Manders (2008) and Larvor (2017).

53. 53.

In footnote 13, Larvor adds: “With the exception of Elements I.4, where a triangle is rigidly displaced. This is well-known to commentators” (p. 7).

54. 54.

Many of these can be found in Bos (2001).

55. 55.

Discussions about what counts as mathematical continue to this day. Consider for example the debate about computer-assisted proofs; e.g. Mackenzie (1999).

56. 56.

Unlike common knots, which are tied in pieces of string with loose ends, mathematical knots are configurations of string whose ends are joined. Note in addition that the “string” in a mathematical knot is one without thickness.

We commented above on the embodied practices and competency of folding. One may similarly remark on the competency of knotting (e.g. give a novice to tie a Winsor knot for a necktie). This gap between diagrammatic instruction and competent enactment of these instructions poses a problem, engagement with which is beyond the scope of this paper.

57. 57.

We are indebted to an anonymous reviewer for pointing out that the performance of an Euclidean proof also requires practical action for which no rules are given at the outset in propositional form; e.g. Livingston (1999).

58. 58.

For an insightful account of conceptual agility in modern mathematics, see Manders’ unpublished article “Euclid or Descartes? Representation and Responsiveness”.

59. 59.

It may be necessary to argue that such moves can be generally performed; see Sect. 4.

60. 60.

Paper as a material on which formulas and mathematical texts are written may also enable mathematical cultures, but this is not the topic of this article.

61. 61.

“The ‘arbitrariness’ of a cultural phenomenon is a function of its particular historical determination. ‘Artificiality’ is related to a different set of problems hinging on the role of culture in human life. Is it a thwarting or fulfilling or both? Is man’s ‘culturalness’ just a thin film, an epiphenomenon, capping his naturalness? Or are cultural features in man’s life so important that culture becomes the capstone to human personality?” (Kroeber and Kluckhohn 1952, p. 52).

62. 62.

Cf. Hatori (2011, p. 10): “In the first years of the Meiji Restoration, in the 1860s and 1870s, the European education system was introduced and adopted in Japan. As a result, European origami was imported to Japan as a part of the kindergarten curriculum.”

63. 63.

This can be seen in Fröbel’s 1850 manuscript: “Anleitung zum Papierfalten” (instruction for paper-folding Fröbel 1874, pp. 371–388; see also Friedman (2018a, pp. 223–227). Fröbel begins the manuscript with the creation of a square from any piece of paper with the help of folding and cutting. He then guides the child to fold the square along various lines and creases. For example, in step number ten (of Fröbel’s instructions), he notes that two different forms may have the same area: folding the square along one of its diagonals results in a triangle whose area is half of the area of the square; and folding one edge on its parallel edge results in a rectangle, whose area is also half of the area of the square. This leads him to declare that “half is the same as a half” (Fröbel 1874, p. 379), introducing, without naming it explicitly, the notion of a fraction, while pointing to its abstraction from a material realization.

64. 64.

For a discussion and further references about the transfer of the Fröbelian methods from Great Britain to India at the second half of the nineteenth century, see Friedman (2018a, pp. 247–250).

65. 65.

We can read works such as De Toffoli and Giardino’s (2014) and (De Toffoli and Giardino 2016) as arguments that cultures of diagrammatic reasoning exist in contemporary mathematics.

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## Acknowledgements

The authors are grateful to event audiences in Nancy, Oxford and Brussels for their patience and helpful questions. We are also thankful to the Novembertagung in the History and Philosophy of Mathematics for providing a point of contact for the first and second author. Valeria Giardino, Yacin Hamami, Brendan Larvor, Lisa Rougetet and two anonymous referees we thank for their insightful remarks. Research for this paper for the first author funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC 2015/1. Research for this paper by the second author has been funded by the Research Foundation—Flanders (FWO), Project G056716N.

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Correspondence to Colin Jakob Rittberg.