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Baroquian Folds: Leibniz on Folded Fabrics and the Disruption of Geometry

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Handbook of the Mathematics of the Arts and Sciences

Abstract

The fold in Leibniz’s philosophy – considered as an image of thought – has received considerable attention during recent decades, mainly because of the work of Gilles Deleuze. For Leibniz the fold often stands for continuous transformation and change, but it is also often mentioned together with references to folded fabrics. But did the folds of fabric prompt new conceptions of geometry in Leibniz’s thought? How does Leibniz’s account on the fall of folds stand in relation to how folded fabrics were drawn in sixteenth- and seventeenth-century Baroque paintings? This chapter will inspect these questions in detail by examining Baroquian painting and specifically what may be termed the Baroquian fold which may be considered as almost un-mathematizable, on the one hand, and Leibniz’ thought on folding, on the other hand. I aim to show that just as the Leibnizian fold resists being reduced to constant, well-defined units, so does the Baroquian fold operate, as it prompts a disruption of geometrization of space.

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Notes

  1. 1.

    For example, Levey (2003) argues that Leibniz’s reflections on the fold are, in fact, a precursor to the mathematical concept of the fractal. See also Friedman (2018, pp. 382–383).

  2. 2.

    See, for example, van Tuinen and McDonnell (2010).

  3. 3.

    For example, Friedman (2018, pp. 386–387): “[Folded] materiality as well as music [see Leibniz (1989 [1714], p. 641)], as enveloping, folded and unfolded, may be taken then as what resists calculation via the human, finite mathematical machinery […] one may claim – as Deleuze does – that Leibniz presents, with his introduction and development of infinitesimal calculus, a metaphysics of the Baroqueian infinite fold [Deleuze (1993, pp. 34–35)] accompanied by the loss of the ‘good form’ [e.g., the circle].”

  4. 4.

    See Serres (1968, pp. 193–200); see also Engelsman (1984, pp. 22–30, and especially pp. 29–30): “The envelope articles [of Leibniz dealing with the mathematical curve called envelope] 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” (cf. Bos (1974, pp. 40–42)).

  5. 5.

    Obviously, the term “envelope curve” had not yet been coined in Leibniz’s lifetime.

  6. 6.

    Cf. Bouquiaux (2005, p. 54): “Le pli est la métaphore qui convient dans l’ordre phénoménal – pour penser tous les degrés d’élasticité et de fluidité des corps, pour penser ces machines de la nature, indéfiniment repliées sur elles-mêmes ou encore pour penser cette toile active et élastique qu’est notre cerveau.”

  7. 7.

    Italics added by M.F.

  8. 8.

    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.”

  9. 9.

    See Hills (2018).

  10. 10.

    See Netz (1999, p. 14): “The lettered diagram is not only a feature of Greek mathematics; it is a predominant feature. Alternatives such as a non-lettered diagram are not hinted at in the manuscripts.”

  11. 11.

    For mathematicians who might have influenced Dürer in developing this method, such as Charles de Bovelles, see Heuer (2011, pp. 262–264). Indeed, although one can already find simple and partial nets in Geometrie en françoys by de Bovelles (1511), Dürer was the first person to systematically introduce polyhedra by folding nets.

  12. 12.

    Regarding Dürer’s nets and his unfolded polyhedra and how he considered the action of folding, see Friedman (2018, pp. 30–82); for an extensive survey of Dürer’s folds in general, see Heuer (2011).

  13. 13.

    As Bruck underlines, at the bottom of the folio one finds a drawing that resembles Dürer’s “knots” from 1507.

  14. 14.

    For how Dürer’s studies on perspective influenced his art and were reflected in it, see the extensive study by Kirsti Andersen (2007, pp. 183–212). Andersen’s study concentrates on the history of the mathematical theory of perspective in the arts but does not deal with how drapery and folds were taken into account.

  15. 15.

    Italics added by M.F.

  16. 16.

    Horst Bredekamp discusses the fall of folds in Schor’s Progetto per il letto di Maria Mancini and its conceptual connection to the “active spaces” in Bredekamp (2020). See also the forthcoming article by Bredekamp (2021).

  17. 17.

    Cf. Strunck (2008) and Walker (2014).

  18. 18.

    Although Mandelbrot often discusses Leibniz and his “Leibniz mania” (Mandelbrot 1983, p. 419), he does state explicitly that Leibniz talked about self-similarity only in the case of the line (ibid., p. 19). On Leibniz’s concept of “self-similarity”, see also de Risi (2007, pp. 238–240).

  19. 19.

    DaCosta Kaufmann (1975) and Bauer (1987).

  20. 20.

    From now on, when referring to Leibniz’s edited works, I will use the letter “A”, followed by one Roman and one Arabic numeral, in order to refer to the edition of Leibniz’s collected works published in the Akademie der Wissenschaften edition of Leibniz’s miscellaneous works (Leibniz 1923).

  21. 21.

    Another reference is from Leibniz’s Protogaea from 1690 to 1691: “[Another] text that Deleuze references (but does not quote) is the Protogaea, Leibniz’s account of the origin of the earth, written around 1690–1, and in particular chapter VIII which is concerned with ‘deposits of metals in the earth and a description of veins’ ” (Lærke 2015, p. 1197).

  22. 22.

    “Totum universum est unum corpus continuum. Neque dividitur, sed instar cerae transfiguratur, instar tunicae varie plicatur” (AVI4, p. 1687). Translation from Lærke (2015, p. 1198).

  23. 23.

    “Nam unitas semper manet quanta maxima potest, salva multitudine, quod fit si corpora plicari potius quam dividi intelligantur” (AVI4, p. 1401). Translation from Lærke (2015, p. 1198).

  24. 24.

    (AVI5, manuscript number 2655, pp. 265502–265503) (from the Vorausedition, Nov 2018): “Il faut dire encor suivant l’exacte correspondance de l’âme et du corps; que le corps organique subsiste tousjours, et ne sçauroit jamais estre détruit, de sorte que non seulement l’âme, mais même l’animal doit demeurer. Cela vient de ce que la moindre partie du corps organique est encor organique; les machines de la nature estant repliées en elles mêmes à l’infini. Ainsi ny le feu, ny les autres forces exterieures n’en sçauroient jamais deranger que l’écorce. […] Et tout corps organique de la nature, estant infiniment replié, est indestructible. Et la preuve qu’il est infiniment replié, est, qu’il exprime tout. De plus le corps doit exprimer l’estat futur de l’âme ou de l’Entelechie qu’il a et cela en exprimant son propre estat futur.”

  25. 25.

    See also Albus (2001, p. 147).

  26. 26.

    See also Albus (2001, pp. 148–157).

  27. 27.

    On the relations between Jungius’s thought and Leibniz, see Kangro (1969). See also De Risi (2007, pp. 34–35).

  28. 28.

    Fogel (1634–1675) with Johann Vagetius (1633–1691) published several works of Jungius. Leibniz had correspondence with both of them, and after Fogel’s death Vagetius was able to provide Leibniz with a few other of Jungius’s manuscripts. Leibniz attempted to prompt the publication of an edition of the manuscripts in Jungius’s Nachlaß, but most of it was destroyed by a fire in Vagetius’s house several years later.

  29. 29.

    See also Bredekamp (2005). The discussion of Leibniz on the garter is in AVI4, part B, p. 1230.

  30. 30.

    “Strumpfbandel binden mit 3 falten ohn die zwey zipfel gibt ein schohn exempel confusae cognitionis et distinctae.”

  31. 31.

    “Wenn dies der scopus ist dass zwey falten oben und eine samt den zipfeln unten kommen sollen, vulgo wird es leichtlich tentando secundum granditatem rectarum gefunden, und also confuse behalten und gewohnt, wie ein knabe die lettern im munde formiren lernet.”

  32. 32.

    “If he now knew clearly, he could tell you what, according to his hand, had to be the longest [band] and then, like an apron or a sling, in what order it had to be superimposed.” (“Wenn ers nun distincte wuste kondte er einem sagen, was nach seiner hand das langste seyn mus, und denn wie eine shurz oder schlinge und mit was ordnung uber einander gehn.”)

  33. 33.

    A similar claim is expressed in Nouveaux essais sur L’entendement humain (Leibniz 1996, Chap. IV.i, p. 360): “Geometers do not derive their proofs from diagrams, although the expository approach makes it seem so. The cogency of the demonstration is independent of the diagram, whose only role is to make it easier to understand what is meant and to fix one’s attention.”

  34. 34.

    “Ob vicinitatem, implectere flechten texere; intricare verwirren, implicare bäugen, plica valte, fere implicare dicunt Latini pro implectere quia brachio et digitis plicamus, id est flectimus hoc est in angulum figuramus. Intricatio est quasi excessus implexionis, bendel machen ist ein weben, texere et plectere differunt quia texere fit compendio quodam, plectere ist wie 〈schnühr〉 machen und Knüppelen” (AVI4, part B, p. 1227).

  35. 35.

    “Nomina ambigua plerumque non tam inopia dictionum aut taedio onomathetisandi, sed quod censerent res aliquas falso notionem aliquam communem habere. […] Notionum confusio a sensus observatione imperfecta” (ibid., pp. 1227–1228).

  36. 36.

    For a critical analysis of Vacca’s paper, see Friedman (2018, pp. 319–323). It is important to note that Vacca was also very much interested in the Chinese culture. Cassina (1953, p. 186) remarks that in Vacca’s work, one can clearly distinguish two main strands: one related to China and the Far East, the other related to mathematics and the history of science. One can therefore consider Vacca’s article on paper folding as an attempt to integrate these two strands together.

  37. 37.

    “Non possono in alcun modo dare quelle sensazioni tattili che dà soltanto la carta piegata, per acuire e sviluppare il senso geometrico.”

  38. 38.

    “Queste pieghettature egiziane e greche dovrebbero piuttosto far parte di un’altra ricerca, lo studio della geometria dei sarti, geometria sartorum, di cui Leibniz sembra per il primo aver avuto l’idea, e la quale, mentre ha un certo sviluppo nelle scuole professionali, non sembra finora aver preso contatto con la geometria pura.”

  39. 39.

    How Leibniz considered a future, possible geometry prompted from weaving, is outside the scope of this chapter and will be dealt in a forthcoming article (Friedman 2021).

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Acknowledgements

The author wishes to thank warmly Ellen Harlizius-Klück and Angelika Seppi for inspiring conversations on Leibniz and folding, and the referee for helpful comments and remarks. The author acknowledges the support of the Cluster of Excellence Matters of Activity. Image Space Material funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2025.

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  1. Excellence Cluster Matters of Activity, Humboldt University, Berlin, Germany

    Michael Friedman

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  1. Michael Friedman
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  1. Department of Mathematical Sciences, University of Montana, Missoula, MT, USA

    Bharath Sriraman

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  1. Department of Mathematical Sciences, The University of Montana, Missoula, MT, USA

    Bharath Sriraman

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Friedman, M. (2020). Baroquian Folds: Leibniz on Folded Fabrics and the Disruption of Geometry. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-70658-0_93-1

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