Abstract
Since Antiquity, and initially in ancient Greece, solutions have been sought for the following three problems, using only compass and straightedge: (1) the squaring of the circle: construct a square equal in area to a circle; (2) the duplication of the cube: given a cube, construct the edge of a second cube whose volume is double that of the first; (3) the trisection of an angle: divide an arbitrary angle into three equal angles.
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- 1.
- 2.
Plutarch (1961, pp. 121–123).
- 3.
Obviously, paper came to Europe only around the eleventh to twelfth centuries AC. Papyrus, and then parchment, were used before that as the writing substrate. I will address the question of materiality later, but the reader should also see Sect. 2.2.1.2 for a more extensive historical survey on the role of paper. Note that I use the name “paper” to refer not only to the substrate of writing, but also to the substrate as such (i.e., also of folding).
- 4.
And same also applies to the other two problems.
- 5.
Wantzel (1837).
- 6.
As I will indicate later, I will not discuss in this book the Japanese traditions (or those stemming from the Far East) of paper folding, later known as “Origami,” which itself is a modern word in Japanese.
- 7.
Messer (1986, p. 284).
- 8.
… and assuming also that one can construct a segment of length \( \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right. \) via folding (the procedure for constructing segments of rational length via folding is called now Haga’s theorem. See: Haga 2002).
- 9.
- 10.
Alexandre-Théophile Vandermonde’s 1771 system of notation for knots arose directly from weaving and the textile industry (see: Epple 1999, p. 52); the origins of computer science had their roots in weaving: Basile Bouchon’s 1725 punch card-controlled loom, further improved on by his assistant Jean-Baptiste Falcon, can be seen as a preliminary version of the highly successful Jacquard loom of 1805 (see: Schneider 2007, esp. Chapter IV, on a system of notation for textiles, and Chapter IX, regarding punch cards). In addition, the mathematician Carl Friedrich Gauss first attempted to find, in a draft dated between 1814 and 1830, an algebraic formalization for the braid: as a collection of bent curves, interwoven within each other, formulized through a sequence of letters (see: Epple 1998).
- 11.
Another epistemological discussion, to be found in the second appendix to the book—which certainly deserves a much more extended and prolonged discussion—is the relation between the Baroque, Leibniz mathematics and the Deleuzian conception of the fold.
- 12.
I refer the reader to Sect. 6.3, to see how the French philosophers Deleuze and Guattari conceptualized these two approaches in mathematics.
- 13.
Rheinberger (1997, p. 28).
- 14.
Ibid., p. 29.
- 15.
Ibid., p. 30.
- 16.
Ibid., p. 226 (cursive by M.F.)
- 17.
I refer here to the notation systems of Akira Yoshizawa, supplemented by Randlett-Harbit, developed during the 1950s and 1960s. This notation, however, is far from being optimal. Demaine and O’Rourke note: “Bern and Hayes’s NP-hardness result [for the Global Flat Foldability problem] shows that this practical difficulty is also a computational difficulty. Effectively, a mountain–valley pattern does not give enough information to specify exactly how to fold. From a theoretical point of view, such origami diagrams should also specify the entire folded state. Unfortunately, we lack a good notation for such a specification […]” (Demaine and O’Rourke 2007, p. 223) (cusrsive by M.F.)
- 18.
Cf. Epple and Krauthausen (2010, pp. 126–127).
- 19.
“Les Mathématiques sont un devenir. Tout ce que nous pouvons faire, c’est essayer d’en comprendre l’histoire, c’est-à-dire, pour situer les Mathématiques parmi d’autres activités intellectuelles, de trouver certaines caractéristiques de ce devenir” (in: Cavaillès and Lautman 1946, p. 7). See also: Benis-Sinaceur (1987).
- 20.
Cavaillès (1938, p. 26): “[Le] travail mathématique [a] deux caractères: devenir imprévisible et valeur absolue: […], devenir imprévisible parce qu’il у a effectivement activité constructrice.”
- 21.
- 22.
Weintraub (1997, pp. 185–186).
- 23.
The usage of future perfect here is, of course, on purpose. See: Rheinberger (1997, p. 113): “As a rule, new developments are at best an irritation at the point where they first appear: they can be approached only in the mode of a future perfect.”
- 24.
The concept of anesthetization is in line with Joseph Vogl’s suggestion that the “becoming” of a medium [Medium-Werden] not only produces new spaces of knowledge and representation but also entails its own potential of an anesthetic effect on its object (see: Vogl 2001).
- 25.
See: Lenoir (1998, p. 18): “every literary [mathematical] form of fact-making is linked to local complexes of technical and social practice […].”
- 26.
I refer here, for example, to Adolf Hurwitz’s private diaries from 1907 (see Sect. 5.1.1); Pacioli’s 1500 book De Viribus Quantitatis, which was discovered only recently (see Sect. 2.3.1); and to the journals of the origami societies in Europe in the twentieth and twenty-first centuries, such as the French “Le Pli,” the German “der Falter,” the English “The British Origami magazine” or the Italian “Quadrato Magico,” which, although playing an essential and crucial role in the dissemination of paper folding mathematics among the origami and paper folding communities, were hardly read by mathematicians.
- 27.
Foucault (1972, pp. 182–183) (cursive by M.F.)
- 28.
- 29.
Kempe (1877, p. 3).
- 30.
Ibid., p. 2.
- 31.
Another solution would be to take a thread and stretch it.
- 32.
Lenoir (1998, p. 6).
- 33.
This book might, in fact, be the work of Diophantus. See: Knorr (1993).
- 34.
Heron (1912, p. 16, 18) (in German); ibid., p. 17, 19 (in Greek).
- 35.
Ibid., pp. 54–55.
- 36.
Ibid., p. 61.
- 37.
Russo (1998).
- 38.
Ibid., pp. 214–216.
- 39.
Lenoir (1998, p. 12).
- 40.
Cf. Epple and Krauthausen (2010, pp. 129–132), where “paper techniques” [Papiertechniken] are equated with “operation on paper” (ibid., p. 131) and seen eventually as a training for the imagination.
- 41.
Rheinberger (1997, p. 104): “Upon closer inspection, any representation ‘of’ turns out to be always already a representation ‘as.’”
- 42.
Derrida (1997, p. 149).
- 43.
The term “instant geometry” is taken from Tom Tit’s reflections on folding a regular pentagon (see Sect. 5.1.3.3).
- 44.
Seppi (2016, p. 50).
- 45.
Derrida (2005, p. 14) (cursive by M.F.)
- 46.
Indeed, Dürer considers for his solids both the perspectival approach (a method based on a point of reference, which may be located at infinity) as well as unfolding his polyhedra (a method not based on any reference point) to present them (see Sect. 2.1). The famous example of this loss of the center and the fixed point, starting from the seventeenth century, is seen with Galileo, supporting the Copernican system, in which the Earth orbits the sun; and with Kepler, proving that the Earth revolves in an elliptical orbit around the sun, i.e., there is an “empty center” (as an ellipse has two foci) that determines the orbit of the earth. For the loss of the reference point and the fixed point, see: Serres (1968, pp. 647–810).
- 47.
Derrida (1997, p. 154).
- 48.
- 49.
Rheinberger (1997, p. 4).
- 50.
See: Cabanne (1971, p. 61). I am grateful to Angelika Seppi for this observation.
- 51.
For example, when folding was seen as an activity to be associated with recreational mathematics, then it was considered not yet mathematical: “[m]athematical recreations […] provided mathematicians with an opportunity to explore mathematical dimensions of activities that are clearly not entirely mathematical, yet related to mathematics.” (in: Chemla 2014, p. 370).
- 52.
See: Volkert (1986).
- 53.
See: Derrida (1997, pp. 75–81).
- 54.
Ibid., p. 27.
- 55.
See: Husserl (1989, p. 160): “The Pythagorean theorem, [and indeed] all of geometry, exists only once, no matter how often or even in what language it may be expressed. It is identically the same in the ‘original language’ of Euclid and in all ‘translations’; and within each language it is again the same, no matter how many times it has been sensibly uttered, from the original expression and writing down to the innumerable oral utterances or written and other documentations.” Martin Heidegger followed this line of thought in 1966: “The sciences, i.e., even for us today the natural sciences (with mathematical physics as the fundamental science) are translatable into all the languages of the world—or, to be exact, they are not translated but the same mathematical language is spoken.” (in: Heidegger 1981 [1966], p. 63)
- 56.
Derrida (1997, p. 9): “I have already alluded to theoretical mathematics; [when] its writing […] understood as a sensible graphie […] [this] already presupposes an identity, therefore an ideality, of its form […].”
- 57.
Ibid., p. 144.
- 58.
Ibid., p. 145.
- 59.
Row (1893, p. ii).
- 60.
Ibid., pp. ii–iii.
- 61.
Hatori (2011) surveys the Japanese tradition of origami, mainly during the eighteenth and nineteenth centuries. Hatori notes: “Not only did the repertoires [of eastern and western paper folding] have little overlap, the folding styles also differed completely between the East and West. The Japanese origami models before the middle of the nineteenth century were made of sheets in various shapes: squares, rectangles, hexagons, octagons, and even many eccentric shapes. They were also folded with many cuts as well as with sophisticated folding techniques, and often were painted. Their European counterparts were made mainly from squares, sometimes from rectangles, and had few cuts. In addition, their crease lines were mostly limited to square grids and diagonals.” (Ibid., p. 10)
- 62.
See, for example, Sect. 2.2.1.2.
- 63.
See, e.g., the discussion in Sect. 3.1.1 on Al-Khayyām’s attempt to prove the parallel postulate.
- 64.
- 65.
Fourrey (1924, p. 113) (“La question semble être restée dans l’oubli jusqu’en 1893”). Fourrey also mentions that he was not able to obtain Wiener’s paper.
- 66.
Vacca (1930).
- 67.
I refer here to volumes 14–24 of this journal. See Sect. 6.2 for a more extensive survey.
- 68.
Justin (1984a, p. 2).
- 69.
Other examples include the survey of Koshiro Hatori on the history of origami in the east and the west before the twentieth century (Hatori 2011), concentrating especially on Japan. Yates (1941, p. 54), gives a list of “bibliography and further reading,” mentioning several of the mathematicians that I will discuss. However, Yates does not attempt to sketch any history of folding in mathematics. In addition, he notes that the book by Row was “translated,” despite the fact that the book was originally written in English. Martin (1998, p. 145), mentions Row’s book and then directly discusses two manuscripts that deal with paper folding, both from 1949.
- 70.
Demaine and O’Rourke (2007), back cover.
- 71.
This is the title of Section 10.2, in: ibid.
- 72.
Ibid., pp. 168–169.
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Friedman, M. (2018). Introduction. In: A History of Folding in Mathematics. Science Networks. Historical Studies, vol 59. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72487-4_1
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