Appendix A: Margherita Beloch Piazzolla: “Alcune Applicazioni Del Metodo Del Ripiegamento Della Carta Di Sundara Row”

Appendix A: Margherita Beloch Piazzolla: “Alcune applicazioni del metodo del ripiegamento della carta di Sundara Row”

This appendix presents, for the ﬁrst time, a translation from Italian into English of Beloch’s four-page-long paper “Alcune applicazioni del metodo del ripiegamento della carta di Sundara Row,”1 published in 1934 in “Atti dell’Acc. di Scienze Mediche, Naturali e Matematiche di Ferrara Serie II, Vol. XI”; the translation is organized according to the numeration of the original paper (pp. 186–189). The footnotes presented in this section are Beloch’s, while remarks and explanations regarding the content of the paper are given in square brackets in a smaller font. For pp. 186–188, the Italian text is also given in footnotes in square brackets.2 It is also advisable to look from time to time at Fig. 5.25 in order to follow the (implicit) steps in Beloch’s argument, as Beloch herself does not supply any drawing. *** p. 186:

Some Applications of the Method of Paper Folding of Sundara Row

A note by Margherita Beloch Piazzolla, Prof. at the University of Ferrara. In an excerpt from my lessons of a complementary mathematics course taught at the University of Ferrara in the academic year 1933–1934, I proposed the following problem as an application of the paper folding method of Sundara Row,3 a problem

1Beloch (1934a). 2Starting from the end of page 188 until the end of the paper (p. 189), Beloch performs several calculations, and as can be seen, the language she uses is formal; hence, the original text is not given. 3Cf. Sundara Row, Geometric Exercises in Paper Folding (Madras, Addison and Co. 1893; London, Court Company, 1917).

© Springer International Publishing AG, part of Springer Nature 2018 377 M. Friedman, A History of Folding in Mathematics, Science Networks. Historical Studies 59, https://doi.org/10.1007/978-3-319-72487-4 378 Appendix A: Margherita Beloch Piazzolla: “Alcune applicazioni del... that allows for a simple solution to the classic problem of duplicating the cube; the solution has not yet been noticed, to my knowledge. Another application is a method given concerning the graphical resolution of the equations of the third degree, which completes the well-known procedure of Lill. The problem in question is the following: Construct a square, two of whose opposite edges [or their extensions] pass through two given points, respectively, and two of whose adjacent vertices are respectively on two given straight lines. Let A, B be the two given points and r, s be the two given straight lines. Denote by X, Y the vertices of the square to be built, lying respectively on the lines r, s. One edge of the square would then be XY. Suppose that the second edge of the square, which exits from X, passes through A, and the second edge, which exits from Y, passes through B.4

p. 187:

Consider the parabola having its focus as point A and its tangent at its vertex the line r, where we assume that the (unknown) straight line XY is tangent to the parabola. Similarly, consider the parabola having, as its focus, point B, and as tangent at its vertex the line s, where we assume that the same straight line XY is tangent [to the second parabola]. This straight line can then be built as one of the common tangents to the two parabolas, and therefore the points X, Y can be built, and as a result also the required square. Since two parabolas have three common tangents, the problem admits three solutions, including certain real solutions. Those tangents can be found in the simplest way using the aforementioned paper folding method. For this purpose, it is enough to build the directrix of each of the two parabolas, denoted as d1 and d2 respectively, and remembering the property that the [geometric] locus of the symmetrical points regarding the focus [of the parabola] with respect to various tangents to the parabola is the directrix; on this property, the

4[The original text: Stralcio dalle mie lezioni del corso di Matematiche complementari tenuto all’Università di Ferrara nell’anno accademico 1933–1934, il seguente problema, da me proposto come applicazione del metodo del ripiegamento della carta di Sundara Row e che consente una semplice risoluzione mediante il detto metodo del problema classico della duplicazione del cubo, risoluzione che a quanto io sappia non è stata finora notata. Un’altra applicazione è un metodo da me dato per la risoluzione grafica delle equazioni di 3 grado, che completa il noto procedimento di Lill. Il problema di cui si tratta è il seguente: Costruire un quadrato di cui due lati opposti passino rispettivamente per due punti dati, e i due vertici situati sui rimanenti lati stiano rispettivamente su due rette date. Siano A, B i due punti dati: r, s le due rette date. Indichiamo con X, Y i vertici del quadrato da costruirsi, giacenti rispettivamente sulle rette r, s. Un lato del quadrato sarà allora XY. Supponiamo che il secondo lato del quadrato uscente da X passi per A e il secondo lato uscente da Y passi per B.] Appendix A: Margherita Beloch Piazzolla: “Alcune applicazioni del... 379 noticed construction, through the paper folding method, of a parabola with tangents is founded. That is, taking the (straight) edge of a sheet of paper as a directrix of the parabola, and marking the focus at the given distance, it is enough to hold fast the focus and refold the paper on itself so that the folded edge would pass through the focus: the fold of the paper will give a tangent to the parabola, and this way, one can construct all of the tangents. [As Beloch defines it, a symmetric point is the image of the focus P of the parabola after performing the folding of a piece of paper (on which the parabola and P are drawn) along a tangent to the parabola. The collection of all the symmetric points is the directrix (this was also noted by Row; see Sect. 4.2.2.2).] To find a common tangent to the two parabolas (given the directrixes and the foci), it is enough then to fold the paper in such a way that the two given lines (i.e., the directrices) would pass, after folding, through the two given points (i.e., the foci), respectively (which could be achieved by using a transparent piece of paper). The explained operation may be done with the same ease and accuracy with an ordinary drawing, passing a line through two points; this will allow one to find the (real) solutions to the problem.5

p. 188:

The problem concerns, as mentioned above, the graphic solution of equations of the third degree, starting from the data graphic process of Lill,6 and offering a new

5[The original text: Si consideri la parabola avente per fuoco il punto A, e per tangente nel vertice la retta r di cui per proprietà nota la retta (incognita) XY è tangente. Similmente si consideri la parabola avente per fuoco il punto B, e per tangente nel vertice la retta s, di cui per la stessa proprietà la retta XY è tangente. Questa retta si può quindi costruire come una delle tangenti comuni alle due parabole, e determinare quindi i punti X, Y e in conseguenza il quadrato richiesto. Siccome due parabole hanno tre tangenti comuni, il problema ammette tre soluzioni, tra cui certo una reale. Le dette tangenti si possono trovare in modo semplicissimo col suddetto metodo del ripiegamento della carta. Basta all’uopo costruire la direttrice di ognuna delle due parabole, e siano rispettivamente d1 e d2, e ricordare la proprietà che il luogo dei punti simmetrici del fuoco rispetto alle varie tangenti della parabola è la direttrice, proprietà su cui si fonda la nota costruzione d’una parabola per tangenti col metodo del ripiegamento della carta. Prendendo cioè l’orlo (rettilineo) d’un foglio di carta come direttrice della parabola, e segnando il fuoco alla data distanza da questa, basta tener fermo il fuoco e ripiegare la carta su se stessa in modo che l’orlo ripiegato venga a passare per il fuoco: la piega della carta darà una tangente della parabola, e questa quindi si potrà costruire per tangenti. Per trovare una tangente comune alle due parabole basta dunque ripiegare la carta in modo che le due rette date nel ripiegarsi vengano a passare rispettivamente per i due punti dati (ciò che si potrà ottenere usando un foglio di carta trasparente). L’operazione spiegata si potrà eseguire con la stessa facilità e precisione con cui, nel disegno comune, si fa passare una riga per due dati punti, e permetterà di trovare le soluzioni (reali) del problema.] 6Cf. Felix Klein, Elementarmathematik vom höheren Standpunkte aus, II, p. 267, (Berlin, Springer 1925). 380 Appendix A: Margherita Beloch Piazzolla: “Alcune applicazioni del... and simple construction, which certainly follows from that which precedes it, and to which I shall have occasion to return. Another application concerns the classical problem of duplicating the cube, which, as is known, cannot be solved with straightedge and compass. To my knowledge, the construction I give here has not yet been noticed. Take the two lines r, s as orthogonal to each other, and O as their common [intersection] point. Assuming that the point A of the general problem lies on s, and the point B lies on r, and applying the construction mentioned above, one can determine the edge XY of the required square, namely, given the known distances OA ¼ a, OB ¼ b, one can determine the unknown distances OX ¼ x and OY ¼ y (hence solving the problem of the two mean proportionals) with the paper folding method, something that Sundara Row thought to be impossible.7 It is well known that: OA : OX ¼ OX : OY ¼ OY : OB,

Namely, a : x ¼ x : y ¼ y : b. If one takes b ¼ 2a, one obtains

x2 ¼ ay, y2 ¼ 2ax,

and the ﬁrst equation is equivalent to y ¼ x2/a;

p. 189:

Substituting the second, [we obtain]: x4/a2 ¼ 2ax; and since x is different then 0, we obtain: x3 ¼ 2a3. Namely, the segment x that is constructed for b ¼ 2a is the side of the cube of a double volume, like that of a.

7Row, p. 112. [The original text: II problema interessa anche, come sopra accennato, la risoluzione grafica delle equazioni di 3 grado, partendo dai dati del procedimento grafico di Lill, ed offrendo una nuova e semplice costruzione, che segue senz’altro da ciò che precede, e su cui avrò occasione di ritornare. Un’altra applicazione è quella relativa al problema classico della duplicazione del cubo, il quale, come è noto, non si può risolvere con riga e compasso. A quanto io sappia la costruzione che dò non è stata finora notata. Si prendano le due rette r, s tra loro ortogonali e sia O il loro punto comune. Supposto che il punto A del problema generale giaccia su s e il punto B giaccia su r, e applicando la costruzione sopra indicata, si potrà determinare il lato XY del quadrato richiesto, date le distanze (note), OA ¼ a, OB ¼ b, si potranno determinare le distanze (incognite) OX ¼ x e OY ¼ y, (risolvere cioè il problema delle due medie proporzionali) col metodo del ripiegamento della carta ciò che lo stesso Sundara Row riteneva impossibile.] Appendix B: Deleuze, Leibniz and the Unmathematical Fold

The fold, as can be seen in this book, and especially in Chaps. 1 and 6, introduces a double movement: on the one hand, it introduces the “becoming” of an (often marginal) mathematical object inside mathematics, which is, at the same time, marginalized in mathematical discourse; on the other hand, it also represents a certain way in which mathematics is in a process of becoming. What all of this indicates is that the structure of mathematics is constantly in a state of becoming. In the introduction, I followed Derrida and his logic of supplementarity, but there is another philosopher who also thought of the fold as an endless process of becoming—Gilles Deleuze, in his book The Fold: Leibniz and the Baroque. Stating this right from the outset, what I certainly do not intend to thoroughly explore in this appendix is how Deleuze re-conceptualized Leibniz’s thought about the fold, presenting it as having an evident connection to Leibniz’s mathematical concepts and his thoughts on mathematics. Rather, what I aim to briefly survey is Leibniz’s thoughts on the possible connections between the fold and mathematics. Before going into the latter in detail, however, let me begin by briefly surveying the Deleuzian approach and what I see as its essential flaw. Deleuze presents the fold of the Baroque, and along with it, the Leibnizian fold, not only as that which opposes linearity and reduction to basic, finite elements, but also as an event, always in a process of transformation, of unfolding itself; the fold enables differentiation via transformation, but without forcing or forming discontinuity. As Laurence Bouquiaux notes, according to Deleuze, “[t]he [Leibnizian] fold is the metaphor that is appropriate in the phenomenal order—to think of all the degrees of elasticity and fluidity of bodies, to think of these machines of nature, indefinitely folded back on themselves [...]”.8 What is the connection then of the fold to mathematics, or in particular, to the mathematics developed by Leibniz? As Simon Duffy remarks: The reconstruction of Leibniz’s metaphysics that Deleuze provides in The Fold draws upon not only the mathematics developed by Leibniz but also upon developments in mathematics

8Bouquiaux (2005, p. 54).

© Springer International Publishing AG, part of Springer Nature 2018 381 M. Friedman, A History of Folding in Mathematics, Science Networks. Historical Studies 59, https://doi.org/10.1007/978-3-319-72487-4 382 Appendix B: Deleuze, Leibniz and the Unmathematical Fold

made by a number of Leibniz’s contemporaries and a number of subsequent developments in mathematics, the rudiments of which can be more or less located in Leibniz’s own work. Deleuze then retrospectively maps these developments back onto the structure of Leibniz’s metaphysics in order to bring together the different aspects of Leibniz’s metaphysics with the variety of mathematical themes that run throughout his work.9 Although Duffy takes a sympathetic approach to the Deleuzian project, one can already see with respect to the quote cited above that not only is there a “retrospective” mapping, but also that Deleuze “draws upon [...] subsequent developments in mathematics [...]”. This double movement, of a retrospective projection and, at the same time, an “uncovering” of a precursory character to Leibniz’s writing, is an anachronistic move I would like to avoid when discussing Leibniz’s possible mathematical approach to folding. To give an example of this anachronistic approach that transforms Leibniz and his concepts into precursors of the later development of mathematics, Samuel Levey argues that Leibniz’sreflections on the fold within Pacidius to Philalethes (which I will discuss later in this appendix) are, in fact, a precursor to the mathematical concept of the fractal. Levey even indicates that for Leibniz himself, his “folded tunic [...] would fall intermediately between a two-dimensional surface and a solid”.10 This is absurd given the fact that the concept of the fractal (or non-integer) dimension was not defined at the time Leibniz was writing. Moreover, it was originally Deleuze who indicated the same move when discussing the inflection point of a curve, as “the pure Event of the line,”11 that is, as the unfolding of the line, which he then states “moves through virtual transformations.”12 These transformations are then either to be seen, according to Deleuze, in René Thom’s catastrophe theory or with the Koch curve and Mandelbrot’s fractal dimension. Each of these examples, catastrophe theory, continuous curves without any tangent or fractal dimension, was only developed beginning from the end of the nineteenth century and, in particular, during the twentieth century. Thom does discuss the term “unfolding” in his works, appearing, for example, as a technical term (albeit not only as such),13 as the “unfolding of a singularity” or a map, indicating that there is an interplay between unfolding a singularity and the emergence of new singularities, which in turn call for additional unfolding. When Thom discusses morphogenesis and the appropriate models to describe it, starting from the second half of his book Structural Stability and Morphogenesis, unfolding prompts a “creat[ion] [of] successive transitional regimes.”14 But Deleuze’s remarks regarding this are strange: “Rene Thom’s transformations refer in this sense to a morphology of living matter, providing seven

9Duffy (2010, p. 144). 10Levey (2003, p. 400). 11Deleuze (1993, p. 15). 12Ibid., p. 16. 13See. e.g.: Thom (1975, p. 31). 14Ibid., p. 289. Appendix B: Deleuze, Leibniz and the Unmathematical Fold 383 elementary events: the fold; the crease; the dovetail; the butterfly; the hyperbolic, elliptical, and parabolic umbilicus.”15 However, the fact that there are only seven basic types of elementary catastrophe is true only under the assumption that the space we deal with is four-dimensional and the transformations of its processes are calculable.16 Not only does the inscribing of a specific dimension to our world stand in opposition to the Deleuzian project, but also Deleuze’s implicit approval that there is a finite number of “elementary events.” A similar critique can be made regarding the Koch curve and Mandelbrot’s fractals. While, according to Deleuze, the Koch curve presents the way in which “we go from fold to fold and not from point to point,”17 Koch himself did not describe his curve as folded. By contrast, Koch presented the curve geometrically on the background of the purely analytical example of Weierstrass, of a continuous curve with no tangents.18 Mandelbrot’s mathematization of fractal forms not only used the novel digital technologies and mathematical concepts available in the twentieth century, but can also be seen as undermining the Deleuzian project itself: i.e., Mandelbrot attempts to mathematize forms, which during the Baroque were considered, as I will later claim, as inherently resisting this mathematization. Obviously, a more thorough analysis of the Deleuzian conception is needed, and especially the connection Deleuze hinted at between the fold and differential calculus. But as I stated above, what this appendix is concerned with is the image of mathematics—or of mathematical concepts, such as space—that Leibniz presents with his reflections on folding. Hence, I will now briefly survey Leibniz’s thoughts on the possible connections between the fold and mathematics. As I will argue, Leibniz did not try to mathematize the fold or to take it as a mathematical operation, or as a technique that would lead to or prompt operations within mathematics, as Deleuze proposes, for example, with the notion of the enveloping curve.19 Indeed, although Leibniz did deal mathematically with enveloping curves, he never considered them as connected either to folds in general or to folds of drapery in particular. Leibniz’s papers on enveloping curves are few,20 and as Steven Engelsman notes, “the envelope articles [of Leibniz] only constitute an isolated episode in the development of partial differentiation. They failed to have any effect [...]. Even Leibniz himself hardly referred to them again. Thus there was no significant follow up at all.”21 Hence, to emphasize again, the Leibnizian fold is above all a metaphor, and in no way strictly a mathematical concept.

15Deleuze (1993, p. 16). 16Thom (1975, Chap. 5). As is obvious from Thom’s treatment, once one deals with maps from a ﬁve-dimensional space (or with n-dimensional space, when n > 5), more complex singularities and catastrophe appear. 17Deleuze (1993, p. 17). 18Koch (1906, pp. 145–146). 19Deleuze (1993, pp. 19, 22). 20Leibniz (1692, 1694). For an analysis of Leibniz’s investigations of these curves, see: Serres (1968, pp. 193–200) and Engelsman (1984, pp. 22–30). 21Engelsman (1984, pp. 29–30): Cf. also: Bos (1974, pp. 40–42). 384 Appendix B: Deleuze, Leibniz and the Unmathematical Fold

Before examining how Leibniz did refer to folding, what is initially important to emphasize is the way in which the fold was regarded in the Baroque epoch compared to the Renaissance. Max Bense claims that: The simple, continuous linear arrangement of objects commonly used in Renaissance painting to create clear, simple ratios of proportion and symmetry [...] is replaced in the Baroque by a non-linear, continuous, curved arrangement of objects, and from this results what some describe as the emphatically asymmetrical character of baroque composition.22 Here, Bense pursues a line of thought originally laid down by Heinrich Wölfﬂin, who notes that “the surfaces and folds of the garment [of the sculptures of Bernini] are not only of their very nature restless, but are fundamentally envisaged with an eye to the plastically indeterminate [Platisch-Unbegrenzte]. [...]. The highlights of the folds dash away like lizards [...].”23 In other words, the fold in the Baroque transforms from an object that follows the rules of “proportion and symmetry”,asin the Renaissance, into something that escapes any linear determination and limitation, being the “limitless [das Grenzlose],” and does not come to a standstill.24 While in the Renaissance, as we saw in Chap. 2, Dürer offered a possible geometrical mathematization of the fold—and not only with his nets, but also with folds of drapery—the Baroqueian conception resisted this proposal.25 The folded drapes in the Baroque point towards other, non-linear curves (e.g., parabolas, or non-algebraic ones).26 But these Baroqueian folds, though being a “sign of a rupture with Renais- sance space,”27 have not led—at least in Leibniz’s thought—to the development of new mathematical tools, or new operations within mathematics. This can be clearly seen in Michel Serres’s book Le Système de Leibniz et ses modèles mathématiques,28 where Leibniz’s fold as a mathematical operation or concept is not even mentioned once.29

22Bense (1949, p. 107) (translation taken from: Blümle 2016, p. 84). 23Wölfflin (1950, p. 57) (cursive by M.F.). German original in: Wölfflin (1917, p. 62). 24Wölfflin (1888, p. 21). This aligns with the conception that Deleuze proposes, see, e.g., in: Deleuze (1993, p. 121). See Seppi (2016, p. 57): “Deleuze breaks with the paradigm of linearity in order to replace one kind of line with another—the straight line of the classical age with the curved line or fold of the Baroque—a substitution which also exchanges one kind of philosopher for another, and thus exchanges two types of reason: René Descartes’s with Leibniz’s.” See also: Seppi (2017). 25Taking this into account, it is unclear why Deleuze mentions Dürer’s folding of nets at all (Deleuze 1993, p. 147, footnote 8), since Dürer is certainly considered a representative of the Renaissance. See also the discussion in Sect. 2.1.2 and: Heuer (2011, p. 256): “For Deleuze [...] folding provides a model of the world [...] which cannot (because of this flux) be pinned to any representational code [...].” 26See: Blümle (2016), for an analysis that shows the possible mathematical forms and curves in El Greco’s Annunciations; indeed, one may depict non-linear curves in El Greco’s paintings, and Blümle claims that “the counter-fold, the infinite undulating form and the double fans are rendered visible in the two Annunciations [...].” (ibid., p. 90). See also: ibid., p. 94. 27Deleuze (1993, p. 121). 28See: Serres (1968). 29Given the fact that Deleuze relied heavily on Serres’s study in the development of his own ideas about the fold in Leibniz, this is particularly strange. Appendix B: Deleuze, Leibniz and the Unmathematical Fold 385

Leibniz therefore does not talk about the fold as a mathematical object, or as an object that can be mathematized; on the contrary, the fold, for Leibniz, is what resists the attempt to found everything on presupposed, unchangeable basic units. Hence, as a metaphor, it certainly stands as an opposition to the axiomatic approach or a binary approach. This can be mainly seen in the famous passage from Leibniz’s 1676 dialog Pacidius to Philalethes: [...] the division of the continuum must not be considered to be like the division of sand into grains, but like that of a sheet of paper or tunic into folds. And so although there occur some folds smaller than others infinite in number, a body is never thereby dissolved into points or minima. [...] It is just as if we suppose a tunic to be scored with folds multiplied to infinity in such a way that there is no fold so small that it is not subdivided by a new fold [...] And the tunic cannot be said to be resolved all the way down into points; instead, although some folds are smaller than others to infinity, bodies are always extended and points never become parts, but always remain mere extrema.30 Leibniz also makes other references to the fold (I will return to another one later), though they are few, and while most of them do not refer explicitly to spatial concepts, they nevertheless repeat the understanding of an infinite process.31 What the fold enables, one could say, is the thinking about this infinite process, but one that cannot be described with a finite collection of mathematical laws. Clearly, with Leibniz’s account of continuity, the body does not dissolve itself into a collection of points or minimal elements: there are folds which would always become smaller and smaller, a fold within folds, and one can never claim that a tunic is divided into points. Against the mathematical point, the atom of Gassendi, the Cartesian cogito,32 against all the concepts and terms that exclude the infinite enveloping and the unfolding of indefinite processes—against these, Leibniz posits the folds, which form (themselves) without any rupture, discontinuity or cessation. These folds within folds, stemming seemingly from an ever-shrinking number,33 unfold themselves without any apparent regulative law, an absence that may be seen as the loss of a point of reference. Bouquiaux claims that this loss of the point of reference also reflects Leibniz’s investigations of conic sections, and hence his reflections on the principle of continuity.34 The non-appearance of this law does not mean, however, that there is no mathematical law to describe such folds: the seemingly irregular phenomena, for which only chaos reigns, have a mathematical geometrical

30Leibniz (2001, pp. 185–187). 31See: Lærke (2015, pp. 1196–1198), for a list of most of the appearances of the fold in Leibniz’s texts. Lærke’s paper is, in a way, essential for seeing how Deleuze re-conceptualized Leibniz’s fold. For example, Lærke presents the Deleuzian-Leibnizian fold as that which can also be seen in the Leibnizian differential calculus (ibid., pp. 1200–1204). As indicated above, Leibniz himself did not use the metaphor of the fold to describe these mathematical concepts. 32See: Leibniz (2001, p. 185). 33See: Bredekamp (2008, p. 15). 34See: Bouquiaux (2005, p. 54). See also Deleuze’s reference to Leibniz’s “ambiguous sign” in: Deleuze (1993, pp. 15, 21), indicating that there is a continuous family of all the (complex) conic sections. See also: Grosholz (2007, pp. 208–213). 386 Appendix B: Deleuze, Leibniz and the Unmathematical Fold explanation, but one that only God can understand, while the reason of the human being is (still) limited in its capability of discovering such regularity.35 Following Horst Bredekamp, one could say that the image that the folds of the tunic suggests (similar to the Leibnizian view regarding irrational numbers)36 is an image of a folded, twisted, even deformed and convoluted, contingent Baroque space, lacking all regularity. This conception obviously contrasts and stands opposed to well- constructed Cartesian space. The folds, as the incalculable, imply that “a trans- mathematical instance comes into force, which enabled an overview of the infinite”.37 Jumping almost forty years from Leibniz’s 1676 Pacidius to Philalethes to 1714, Leibniz’s essay Principes de la nature et de la grâce also shows, in an essential, though different way, how the fold was thought of in an un-mathematical way.38 In this essay, Leibniz refers to the impossibility of a calculation that would allow us to grasp the beauty of music39; several passages before this, he notes that: “One could learn the beauty of the universe in each soul if one could unfold all of its folds, which develop perceptibly only with time [si l’on pouvoit déplier tous ses replis, qui ne se développent sensiblement qu’avec le temps]. But since each distinct perception of the soul includes an infinity of confused perceptions which envelop the entire universe, the soul itself does not know the things which it perceives [...].”40 In addition, Leibniz notes, concerning the death of a Monad, that: “Thus, abandoning their masks or their rags, they merely return, but to a finer stage [...]. Not only souls, therefore, but animals as well, cannot be generated or perish; they are only developed, enveloped, reclothed, stripped, transformed [ils ne sont que développés, enveloppés, revêtus, dépouillés, transformés]”.41 For Leibniz, the folding and unfolding of a cloth and of fabric were taken as a metaphor for a process of change, of a continuous transformation. Materiality as well as music, as enveloping, folded and unfolded, may be taken then as what resists calculation42 via the human, finite mathematical machinery.43 These

35I follow here: Albus (2001, pp. 148–157). 36Bredekamp (2008, p. 100). 37Ibid., p. 105. 38Note, moreover, Lærke (2015, p. 1196): “This stubbornly synchronic view of Leibniz’s philosophical enterprise is, from the historical point of view, an obvious flaw in [...] Deleuze’s [approach].” 39Leibniz (1989, p. 641): “Music charms us, although its beauty consists only in the agreement of numbers and in the counting, which we do not perceive but which the soul nevertheless continues to carry out, of the beats or vibrations of sounding bodies which coincide at certain intervals.” 40Ibid., p. 640 (translation changed by M.F.) 41Ibid., p. 638. 42See also: Deleuze (1993, pp. 77, 79). 43Describing the mathematical methods used by Leibniz to describe the “two levels” of the world— folds of the soul and pleats of matter, Deleuze (ibid., p. 101) indicates that these methods were differential calculus (i.e., calculus of differential relations) and the calculus of minimum and maximum (with the help of the coordinate system). However, Leibniz did not mathematically treat the fold with either of these methods. Appendix B: Deleuze, Leibniz and the Unmathematical Fold 387 transformations, such as the enveloping of the folded drapery, are described by Leibniz as a continuous process of metamorphosis,44 as a continuous process of covering and uncovering, of the convolution of space, which may not be conceptualized mathematically. As a phenomenon of nature, Leibniz does not indicate how the folds that become ever smaller might eventually be described through future laws of geometry and mathematics; it seems that he implicitly suggested that the current mathematical tools he had at his disposal were in no way sufficient for such a task. Indeed, one may claim—as Deleuze does—that Leibniz presents, with his introduction and development of infinitesimal calculus, a metaphysics of the Baroqueian infinite fold,45 accompanied by the loss of the “good form,” i.e., the circle46 and its fixed center, and its replacement by a mathematics that emphasizes a dynamic model of space. Certainly, Leibniz proposed an alternative conception of space to Newton’s “absolute space,” that is, to his “space as container,”47 as Max Jammer puts it, but, as I have claimed, the Leibnizian conception of space hardly considered the fold as a mathematical object, which may possibly be found within such an alternative conception of space.48 The fold may be seen as an image either of nature in a process of metamorphosis, as folded or of space as convoluted, but this image was not thought of or conceptualized as a mathematical concept during Leibniz’s epoch.

44Leibniz (1989, p. 638): “[...] there is metamorphosis. Animals change, take on, and put off, only parts; in nutrition this takes place little by little and through minute, insensible particles, but continually [...]”. 45Deleuze (1993, pp. 34–35). 46I refer here to Kepler’s discovery of the laws of planetary motion at the beginning of the seventeenth century; Kepler showed that the orbit of a planet is an ellipse and not a circle. 47See: Jammer (1954, pp. xiv–xv) (in the forward by Einstein) and Chap. 5. Obviously, Jammer presents a traditional, “grand” narrative, ascribing the invention of absolute space to the Scientiﬁc Revolution, and advocating the conception that container space was developed only after the Renaissance. One of the aims of this book is, as mentioned in Chap. 6, following Deleuze’s and Guatarri’s distinction between “state sciences” and “minor sciences,” to undermine these “grand narratives” and present, in addition, complementary, “minor” and marginalized accounts of geometry and space. 48Recall that in Vacca’s article on the history of the mathematics of folding, he cites Leibniz’s idea that tailors should have their own geometry (Vacca 1930, p. 45). Vacca, however, did not specify what Leibniz actually meant by this. References

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A Bense, Max, 42, 384 Agazzi, Rosa, 243, 244 Bern, Marshall, 7, 47, 80, 222, 357 Ahrens, Wilhelm, 27, 268, 273–285, 287, 294, Bernstein, Felix, 293, 294, 350–352 297, 301, 302, 337, 375 Berzelius, Jöns Jacob, 202 Alexejeff, W., 345 Bilinear forms, 341–350 Algebraic reasoning, 325, 327 Billingsley, Henry, 25, 67, 73–75 Al-Khayyām, Umar, 21, 93–97 Blintz fold, 60, 220, 221, 224, 227, 307 Analytic geometry, 130, 132, 166, 172, 283, Böhm, Andreas, 21, 93, 96, 97, 290 284, 303, 304 Boole Stott, Alicia, 136–139 Apian, Peter, 54, 55, 58, 75 Brander, Isabelle, 248 Archibald Smith, Nora, 242 Brill, Alexander, 119–121, 123, 204, 205, 365 Archimedean solids, 32, 33, 37, 39, 46, 49, 57, Brill, Ludwig, 133, 165 68–70, 76, 77, 81 Aristotle, 88, 94, 158 Aronhold, Siegfried, 342, 343 C Axiomatization, 10, 18, 19, 27, 28, 153, 173, Canovi, Luisa, 358, 359 203, 273–295, 358–368 Catherine of Cleves, 60 Cavaillès, Jean, 7 Cavalieri, Bonaventura, 308, 309, 313 B Cayley, Arthur, 118, 342, 343 Bach, Friedrich Teja, 42, 43 Cesari, Lamberto, 152 Ball, Katherine, 228, 297–299 Choquet, Gustave, 364 Ball, W.W. Rouse, 228, 298, 299, 321 Clebsch, Alfred, 108, 119, 131, 157, 342, 343 Barbaro, Daniele, 25, 67, 72, 73 Clifford, William Kingdon, 109, 345 Bat books, 59–66 Congruent ﬁgures, 65, 108–110, 253, 254, 261 Beloch, Margherita Piazzolla, 2, 9, 21, 23–25, Convolution, 28, 272, 341, 345, 350–354, 387 27, 28, 259, 261, 271, 272, 282, 285, Cowley, John Lodge, 76–80, 127, 128 290, 295, 318–340, 355, 356, 358–362, Cremona, Luigi, 146–148, 204 364–368, 372, 375, 377–387 Crystallography, 206, 211, 214, 215, 226, 233, Beloch’s fold, 318–340, 358, 364, 365, 367, 247, 265, 266 368 Beltrami, Eugenio, 22, 26, 118, 124–180, 204, 371 D Beman, Wooster Woodruff, 177, 262–264, 272, da Vinci, Leonardo, 49, 50, 83 283 d’Aviso, Urbano, 22, 305, 307–309, 313, 321

© Springer International Publishing AG, part of Springer Nature 2018 415 M. Friedman, A History of Folding in Mathematics, Science Networks. Historical Studies 59, https://doi.org/10.1007/978-3-319-72487-4 416 Index de Arfe, Juan, 76 Francoeur, Louis-Benjamin, 26, 93, 98–103, de Bovelles, Charles, 25, 45, 48–66 107, 140, 141, 180, 185, 203 Dee, John, 25, 67, 73–75 Frigerio, Emma, 339, 340, 362, 370 Deleuze, Gilles, 5, 15, 41, 163, 374, 381–387 Fröbel, Friedrich Demaine, Erik D., 7, 23, 24, 222, 357 Anleitung zum Papierfalten, 217, 221, 223, Demaine, Martin L., 222, 357 234, 239, 240, 244, 245 Derrida, Jacques Fröbel’sinfluence in England, 237–243 supplement, 17, 18, 374 Fröbel’sinfluence in France, 233–237 supplementarity of mathematics, 17, 369, Fröbel’sinfluence in Germany, 229–233 381 Fröbel’sinfluence in Italy, 243–247 Descartes, René, 89, 269 Fushimi, Koji, 258, 330, 336, 356 Descriptive geometry, 115–118, 157, 165, 177, 178, 265, 266, 318 Developable surfaces, 158–164 G Diderot, Denis, 100–103 Gauss, Carl Friedrich, 4, 159 Dieudonné, Jean, 348, 349 Ghersi, Italo, 277, 278 Dodgson, Charles L., 111 Giegher, Matthias, 218–220 Doetsch, Gustav, 351–353 Gleason, Andrew, 258 Dorasawmi Aiyengar, P. V., 248, 249 Globus gores, 36, 50, 55, 58, 59, 108 Douglas Wiggin, Kate, 242 Goldammer, Hermann, 225, 229–231 Dubreuil, Jean, 76 Gordan, Paul A., 342–350, 352 Dudeney, Henry Ernest, 278 Grassmann, Hermann, 130–132 Dupin, Charles, 122 Grunert, Johann August, 82 Dupin, Louis, 26, 125–141, 201, 205, 215 Guattari, Felix, 5, 374 Dürer, Albrecht, 7–9, 11, 16, 20, 24–26, 29–83, Gurney, Mary, 237, 239, 255 88, 90, 91, 97, 115, 117, 124–126, 141, 258, 373, 384 Underweysung der Messung, 20, 25, 29, 32, H 33, 39, 58, 258 Haga, Kazuo, 3, 330 Hantzsch, Arthur, 192, 193, 198 Harsdörffer, Georg Philipp, 218 E Hartmann, Georg, 54–57, 66 Elm, Hugo Andreas, 232 Haüy, René Just, 211, 212, 214 Emsmann, H., 140 Hayes, Barry, 6, 357 Epistemological procedure, 8, 136, 141, 194, Heerwart, Eleonore, 219, 237–242, 255, 269 203, 292, 347 Henrici, Olaus, 26, 28, 93, 104–112, 119, Euclid, 12, 13, 18, 33, 54, 67, 70, 74, 76, 78, 85, 120, 174, 253–255, 261, 262, 264, 88, 89, 94, 96, 104, 108–112, 125, 269, 275 251–256, 287, 295–297, 360, 371 Herbart, Johann Friedrich, 150, 151, 233 Euclid’s Elements, 13, 54, 67, 73, 74, 77–79, Hermann, Felix, 186 104, 110–112, 127, 251, 255, 360, 371 Hermite, Charles, 153, 154, 156, 163 Euler, Leonhard, 21, 71, 72, 82, 116, 142, 158, Heron, 11–14, 88, 89 159 Hessenberg, Gerhard, 348 Hilbert, David, 28, 144, 149, 150, 172, 204, 261, 271, 341, 346–351, 353 F Hinton, Charles Howard, 136, 137 Folded drapery, 32, 39, 41–43, 387 Hirschvogel, Augustin, 8, 25, 34, 67–70, 77, Folding of a gnomon, 83–86 82, 125 Folding of a regular pentagon, 14, 15, 297–318 Hoff, Jacobus Henricus van ‘t Four-dimensional regular polytopes, 126, Die Lagerung der Atome im Raume, 140, 132–134, 137 182, 186 Fourrey, Emile, 22, 313 La chimie dans l’espace, 182, 184, 185, Francesca, Piero della, 31, 32, 37, 45, 49, 72 188, 191 Index 417

Hoüel, Jules, 145, 146, 149 Leibniz, Gottfried Wilhelm, 5, 17, 25, 319–323, Hurwitz, Adolf, 9, 27, 233, 273–285, 297, 298, 381–387 302–304, 325, 335, 336, 338, 358, 360, “Le Pli”, 9, 363, 364, 366 364, 366–368, 375 Lill, Eduard, 326, 327, 330–335, 359, 378–380 Hurwitz’s fold, 233, 279, 358, 364, 368 Lill’s method, 326, 330–336, 358, 359, 372 Huzita, Humiaki, 18, 28, 109, 290, 323, 336, Lister, David, 220, 356, 359 339, 340, 356–362, 365–370, 372, 373 Logic of supplementarity, 381 Hyperbolic plane, 125, 143, 144, 150, 204 Lotka, Alfred J., 282–285, 375 Lubiw, Anna, 7, 23, 222, 357 Lucas, Édouard, 274, 277, 305–310, 313, 317, I 321, 369 Ingrami, Giuseppe, 111 Instant geometry, 14, 311 Isomers, 182–184, 187–190, 192, 195–198, M 200, 345 MacLeod, Norma L., 299 Manders, Kenneth, 371 Mannigfaltigkeit, 123, 141, 150, 151, 180, 215 J Marie, François-Charles Michel, 80, 81, Jacobs, Jean-Francois, 233, 234, 244 127–131 Jacoli, Ferdinand, 307–309 Martin, George E., 23, 340, 356, 368 Jorissen, W. P., 187, 192, 193, 198 Mathematical models, 26, 54, 93, 109, 112, Justin, Jacques, 19, 23, 28, 322, 336, 337, 340, 114–126, 142, 157, 165, 170, 173, 175, 356–359, 361–372 180, 267, 353 Mendes, Michel, 358 Messer, Peter, 2, 3, 365 K Meyer, Wilhelm-Franz, 202, 342, 344 Kawasaki, Toshikazu, 321, 363, 370, 373 Mohr, Ernst, 198–201, 205 Kekulé, August, 182, 192, 194, 195, 198, 202, Molteni, Alfred, 129 209, 345 Monge, Gaspard, 115–117, 122, 129, 159, 173 Kempe, Alfred Bay, 11 Morley, Frank, 313, 314, 317, 318, 321, 347, Kepler, Johannes, 16, 32, 33, 66, 78, 79, 82, 371 170, 387 Müller-Wunderlich, Marie, 229, 231, 232 Kergomard, Pauline, 227, 233, 235 Murray, Elsie R., 237, 242, 243 Klein, Felix, 9, 27, 117–124, 126, 162, 165, 166, 173, 177, 180, 201, 202, 233, 254, 262–266, 272, 273, 275, 285, 287, N 293–295, 321, 327, 328, 332, 341, 345, Napkin folding, 60, 90, 217, 218, 220, 221, 227 368, 379 National Indian Association, 248 Knotting of a regular pentagon, 175, 305–318 Nets of polyhedra, 50, 57, 58, 77, 80–83, 126, Koyré, Alexandre, 89 130–136, 246 Kraus-Boelté, Maria, 228, 237 Noether, Emmy, 345–347 Kraus, John, 237 Non-Euclidean geometry, 97, 113, 142, 145

L O Lambert, Heinrich, 21, 96, 97, 290, 329, 330 Olivier, Théodore, 117, 118, 121, 122, 128, 130 Lange, Wichard, 207, 215 O’Rourke, Joseph, 6, 23, 24, 223, 357 Lang, Robert J., 357, 367–370 Ozanam, Jacques, 77, 90 Lardner, Dionysius, 23, 24, 26, 93, 104–112, 140, 141, 321 Lautman, Albert, 7, 349 P Le Bel, Joseph Achille, 181 Pacioli, Luca, 9, 16, 25, 26, 30, 32, 38, 45, Lebesgue, Henri, 157, 164 49–67, 71, 72, 75, 83–87, 90, 93, 105, Legendre, Adrien-Marie, 90, 103, 107, 108, 225, 226, 299, 315, 316, 371 185, 307 De Viribus Quantitatis, 9, 49, 85 418 Index

Palmyre, Martin, 236 S Panofsky, Erwin, 30, 31, 37, 43–49, 71, 81 Sachse, Hermann, 26, 126, 180–203, 205, 228, Paper instruments, 25, 45, 49–66, 74, 87, 89, 345, 371 90, 117, 125 Sainte-Laguë, André, 307 Parallel postulate, 21, 94–98, 143, 290, 361 Sallas, John, 60, 90, 218–220 Pasquali, Pietro, 243–247, 297–298, 301, 309, Salmon, George, 118 310, 320, 321 Savineau, C., 233, 236, 237, 310 Pasteur, Louis, 181, 182 Schilling, Martin, 120, 121, 133, 135 Paternò, Emmanuele, 182 Schläﬂi, Ludwig, 82, 131, 132 Patrizi, Francesco, 89, 90, 96 Schlegel, Victor, 26, 82, 125–141, 176, 204, Peano, Giuseppe, 125, 126, 152–155, 157, 163, 268 319–322 Schmid, Wolfgang, 25, 53–59, 67–70 Pellegrino, Consolato, 358, 359 Schouten, Jan Arnoldus, 348 Pepe, Luigi, 319, 340, 359 Schwenter, Daniel, 70, 91 Perspective, 8, 9, 16, 19–29, 31–33, 37, 41, Scimemi, Benedetto, 340, 359, 361, 362, 373 43–49, 54, 58, 59, 69, 70, 72, 73, 75–77, Serret, Joseph Alfred, 152, 153, 155, 156 81–83, 89, 123, 124, 127, 141, 171, 172, Smith, David Eugene, 177, 262, 272 174, 175, 201, 244, 251, 257, 265, 273, Stereochemistry, 126, 181, 187, 192, 194 277 Stifel, Michael, 76 Perspective machine, 9, 29, 47, 48, 75, 76, 81, Stringham, Washington Irving, 132, 135, 136, 89 139 Pestalozzi, Johann Heinrich, 207, 211, 228, Structural chemistry, 186 233, 251 Struik, Dirk Jan, 77, 348 Picard, Émile, 164, 165 Suzanne, Pierre-Henri, 93, 98–103 Platonic solids, 33–35, 37, 45, 50, 52, 53, 67, Sylvester, James Joseph, 342, 343, 345 68, 70–72, 74, 76, 77, 79, 131, 166, 167, Symmetry, 10, 26, 30, 32, 35, 77, 90, 91, 93, 170, 175, 252 98, 100–112, 138, 140, 141, 170, 175, Polyhedral sundials, 55, 56 176, 184, 185, 211, 214, 227, 231, 243, Proto-topological, 45, 47, 72, 82 251, 253, 259–261, 268, 299, 335, 359, 364, 374, 384

R Ramus, Peter, 76 T Rao, B. Hanumantha, 260–262, 264–269 Technical object, 5, 30, 58, 76, 194, 203, 275 Recreational mathematics, 9, 17, 22, 27, 60, 86, Technical procedure, 26, 81, 87, 91, 139, 141, 87, 90, 91, 256, 268, 273–275, 277, 278, 152, 246, 266, 268, 277, 344, 347, 374 287, 307, 311, 316–318, 328, 356, 372, Tit, Tom, 310–316 375 Treutlein, Peter, 121, 171, 174 Reﬂection, 8, 9, 14, 25, 31, 58, 65, 95, 96, 140, Trisection of an angle, 1, 245, 257, 356, 362, 171, 173–176, 274, 314, 320, 322, 339, 363 361, 364, 372, 382, 383, 385 Trisectrix, 1, 11, 14 Regiomontanus, 54 Riemann, Bernhard, 130, 141, 142, 148, 150–152, 204, 341 U Rivelli, Alfonso, 81, 246, 247 Umkehrung, 140, 141 Ronge, Bertha, 228, 237, 238, 247 Umstülpung, 139–141, 176, 204 Ronge, Johannes, 239 Row, Sundara Tandalam, 2, 19, 22–24, 27, 28, 107, 108, 111, 113, 118, 124, 126, 131, V 206, 216, 226, 233, 247, 249, 250, Vacca, Giovanni, 22, 23, 104, 321–325, 328, 253–269, 271–321, 323, 326–328, 330, 330, 337, 376, 389 334, 340, 359, 364, 372, 374, 375, van Calcar, Elise, 229 377–380 van Esveldt, Steven, 220 Rupp, C. A., 272–285, 334, 338, 375 Virgilio, Luisa, 294 Index 419 von Baeyer, Adolf, 195 Werner, Alfred, 192, 193 von Dyck, Walter., 120, 133–135, 157, Wiener, Christian, 26, 108, 118, 119, 126, 152, 163–166, 172, 176, 177, 180, 263, 267, 157, 204, 261, 266, 371, 372 274, 275 Wiener, Hermann, 22, 26, 119, 126, 152, 158, von Kügelgen, Wilhelm, 221 161–180, 261, 262, 272, 273 von Marenholtz-Bülow, Bertha, 217, 223, half turn [Umwendung], 173–175 228–232, 234 Wiener, Norbert, 352 Wright, Richard P., 26, 93, 110

W Wantzel, Pierre, 2, 3, 33 Y Weierstrass, Karl, 108, 124, 161–163, 204, Young, Grace Chisholm 383 The First Book of Geometry, 285–293, 295, Weiss, Christian Samuel, 208, 211, 212, 214, 328 215 ﬂat pattern, 291, 292