Abstract
The end of the fifteenth century signaled a major shift regarding the representation of space: the rise of perspective, a “faithful” representation of three-dimensional objects on a two-dimensional plane, mediated through novel technical instruments and innovations and conceptions such as the “perspective machine” or Alberti’s window. This exact drawing of geometrical forms in a painting—which can be regarded as a doubling of reality consisting of the seen, “outside” reality on the one hand, and the drawn, “flat” reality on the other—was based on mathematical calculations, even if this mathematics, and especially the implied geometry, was sometimes implicit. Folding a piece of paper—so it may seem—did not have a place in this new (doubled) geometrical order, even if this piece of paper were folded into a geometrical form, e.g., a polyhedron.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
See Sect. 1.3 concerning paper—and, as a result, its folding—as a “non-instrument.”
- 3.
For an excellent account on Dürer and his work, see: Panofsky (1955).
- 4.
Erwin Panofsky argues that the Renaissance paintings and treatises, using perspective as the new form of spatial organization, prompted a new conception of space: “the founders of the modern perspectival view of space were the two great painters whose styles, in other ways as well, completed the grand synthesis of Gothic and Byzantine: Giotto [di Bondone (1266/7–1337)] and Duccio [di Buoninsegna (c.1255–1260–c. 1318–1319)]” (Panofsky 1991, p. 54). However, as Vincenzo de Risi remarks: “this new way of representing space was only codified much later, in the late fifteenth century, when a mathematical reflection on those painting techniques began to be available in the works of Alberti, and then Piero della Francesca, Albrecht Dürer, and many others.” (Risi 2015, p. 9). See also: Panofsky (1991, p. 62): “as for Germany, apart from the works of the half Italian Michael Pacher, not a single correctly constructed picture appears to have been produced in the entire fifteenth century: that is, not until the adoption of the exact and mathematically grounded theory of the Italians, in particular through the agency of Albrecht Dürer.”
- 5.
The following discussion will deal mainly with how Dürer mathematically conceptualized a creased folding of the Platonic and Archimedean solids in the framework of Underweysung der Messung. This treatment should by no means imply that Dürer thought of folding and folds only in this sense and in this mathematical framework. Indeed, folding and folded drapery (which do not consist of creases) were essential in Renaissance art, sculpture, sketching and painting, and Dürer gave them great emphasis; see: Heuer (2011) (see also: ibid., p. 251, regarding the usage of the words Falte and fältlen in German-speaking countries at the beginning of the sixteenth century). However, it is important to emphasize that although Dürer attempted to treat folded drapery mathematically (as we will see later in Sect. 2.1.2) from the modern approach to folded cloths (Cf. Cerda et al. 2004), it seems that he did not have at his disposal the mathematical tools that could have enabled such an investigation. In addition, needless to say, Dürer’s mathematical interest was broader than his treatment of the folded nets; on Dürer and geometry in general, see: Peiffer (1995, 1997), Schröder (1980). On Dürer and the German perspectivists, see: Kemp (1990, pp. 53–64).
- 6.
Another subject was the tiling of the plane, in which one of the most famous is the tiling of the plane with regular pentagons and lozenges (see: Dürer 1977 [1525], p. 146). This consequently prompted Kepler—and eventually Roger Penrose—to ask the question as to whether there exists a periodic tiling with fivefold symmetry; the answer is that the only tiling that exists having this property is an aperiodic tiling.
- 7.
For example, he notes, concerning the construction of a regular polygon with 11 and 13 sides, that “[t]his is a mechanical [approximate] not a demonstrative method” (Dürer 1977 [1525], p. 149).
- 8.
Though, when Dürer discusses the construction of the nonagon in one of his preparatory notes (see: Rupprich 1969, pp. 336–337), he does remark that the construction of this polygon came from the praxis of the ateliers, i.e., not from the drawing of a mathematician.
- 9.
Dürer could not have known that, since this was only proved by Pierre Wantzel in 1837. For a modern treatment of Dürer’s constructions, see: Hughes (2012).
- 10.
Pappus, one of the great ancient Greek mathematicians, listed all 13 Archimedean solids, while crediting their discovery to Archimedes. He had not drawn them, but instead wrote the number of different faces each had. His work was mostly forgotten, and the Renaissance artists, mathematicians and artisans (among them Dürer) “rediscovered” these solids. This was completed by Kepler in 1619 with his complete list in his book Harmonices Mundi. For a brief survey on the history of polyhedra, see: Malkevitch (2013).
- 11.
See: Andersen (2007, pp. 183–212), for an extensive discussion of Dürer’s methods of drawing perspective images.
- 12.
- 13.
See: Andrews (2016, p. 416ff).
- 14.
Dürer (1977 [1525], p. 317) (translation modified by M.F.). Dürer writes that these constructions can be made “[…] durch den cirkel vnd richtscheyt” (ibid., p. 316). While Strauss correctly translates the word “cirkel” as “compass” (in today’s German: Zirkel), the word “richtscheyt” (in today’s German: Richtscheit) means “straightedge” and not “ruler.” However, Strauss used the word “ruler” in his translation of this sentence. (Recall that a ruler is a measurement device, whereas a straightedge is a tool for drawing straight lines; i.e., a ruler is a straightedge with some scale on it. Had Dürer used a ruler, he could have constructed, for example, all the regular polygons precisely without any problems as was already known in Antiquity.)
- 15.
The two projections are presented when the solid is already bounded in a sphere.
- 16.
Ibid., p. 316.
- 17.
The German term “Netz” came from Hirschvogel. See Sect. 2.2.2.
- 18.
Or sometimes “offen aufreissen.”
- 19.
Ibid., p. 336.
- 20.
Ibid., p. 334. The German verb “zusammenlegen”, among its various meanings, indicates “to fold together, especially after usage” (see: Grimm and Grimm 1971, vol. 32, col. 755–756).
- 21.
Dürer (1977 [1525], p. 324): “spread out flat on a plane.”
- 22.
See, however, Fig. 2.4, regarding a mistake in the drawing of the net of the truncation of the truncated cube.
- 23.
Ibid., p, 317 (a remark of Walter L. Strauss, referring to: Staigmüller 1891, p. 33).
- 24.
- 25.
Staigmüller (1891, p. 33): “It should be noted that in all horizontal and vertical projections, the circles drawn as the circumcircle of the corresponding projections cannot be regarded as horizontal and vertical projections of the circumscribed spheres, except for the octahedron and icosahedron” [so ist hierzu zu bemerken, dass in allen Horizontal- und Vertikalprojektionen ausser beim Oktaeder und Ikosaeder die als Umkreise der betreffenden Projektionen eingezeichneten Kreise nicht als Horizontal- und Vertikalprojektionen der umschriebenen Kugeln gelten können].
- 26.
This unfolding is impossible mathematically—as an isometry between two surfaces must preserve the Gaussian curvature, which equals 0 for a plane and 1 for a sphere. However, this was only proved during the nineteenth century.
- 27.
Dürer (1977 [1525], p. 329).
- 28.
The solids, whose nets Dürer presents, are the truncated tetrahedron, the cuboctahedron, the truncated octahedron, the truncated cube, the Rhombicuboctahedron, the truncated cuboctahedron and the cubus simus.
- 29.
See: ibid., p. 24 (from Strauss’s introduction): “Dürer experimented as well with perspective renderings of these polyhedra but later abandoned this project […]” and e.g.: Hofmann (1971, pp. 148–149). For a survey of Dürer’s investigations on the Archimedean solids, see e.g.: Weitzel (2007, pp. 139–145). On the solid appearing in the Melencolia I graving, see: Panofsky and Saxl (1923), Schreiber (1999), Weitzel (2004).
- 30.
Staigmüller (1891, p. 36).
- 31.
Dürer (1977 [1525], p. 456).
- 32.
See also: Field (1997, p. 268): “the snub cube cannot in fact be produced by a simple truncation process like those described by Piero Della Francesca, even if we allow subsequent distortion as apparently practiced (but not described) by Pacioli. Thus in the case of the snub cube it seems likely that the construction of a symmetrical net may have been one step in Dürer’s discovery. Moreover, since in his preamble to the discussion of the Archimedean solids he gives instructions for making three-dimensional models from the nets, it seems probable that he tested his candidate solids by making such models. […] the truncated cuboctahedron, can […] be produced by truncation, but we require the truncation to be followed by distortion. For this solid also, investigation starting with the construction of a symmetrical net seems to provide a plausible route to discovery.” See also: Weitzel (2007, p. 141): “It is very unlikely that the model builders produced the models from solid material by sawing off or by a similar process; Instead, they would have built hollow models out of cut-out polygons, a method to which the use of nets of the polygons as a starting point is very close.”
- 33.
For a survey of Joachim Camerarius’s work, see: Baron (1978).
- 34.
Dürer (1532).
- 35.
Peiffer (2000, p. 87).
- 36.
Dürer (1977 [1525], p. 320).
- 37.
Dürer (1532, p. 147).
- 38.
See also: Peiffer (2000), for the reception of Dürer’s language and writings, both in German and in Latin (especially: ibid., p. 89): “It is clear that in the sixteenth century there existed two very distinct audiences for the Underweysung: that of artists and that of mathematicians. But in order to be read, understood and used by each of these two categories of readers, mediation was necessary. Figures, adaptations, selective abstracts have played this role for the former. The Latin translation, done by a humanist philologist, was able to influence the learned world and perhaps even universities.”
- 39.
- 40.
- 41.
See Andrews (2016, pp. 418–419) for Dürer’s sketches of the net of the icosidodecahedron after 1525.
- 42.
Another example of Dürer drawing folded paper can be found around the same period, in 1511, in his woodcarving “Die Messe des heiligen Gregor.” While the woodcarving is abundant with folds of fabric, either of the angles, the pope, or the prayers, at the left corner, one can find the chalice, standing on an unfolded piece of paper, where the creases indicate that the paper was folded to form a three-over-three pattern of rectangles. The carved shades also indicate the order along which the paper was folded. However, comparing other carvings and paintings of the same mass during the same years, one can observe the same paper, on which the chalice stands, with the same pattern (or with a similar one). Hence, it is more reasonable to suggest that this folded paper was a part of the symbolic image of this mass, rather than an attempt to contrast the mathematical, precise folding to the apparently not (yet) mathematizable folds of the fabric.
- 43.
Heuer (2011, pp. 251–259).
- 44.
Heuer mentions that Dürer “likely knew” “the language of folding” from Nicolas of Cusa (1401–1464) (Ibid., p. 254), and while relying on a Deleuzian approach to the fold, he comments: “For Deleuze […] folding simply is the opposite of aesthetic and metaphysical ‘perspectivism’—it just means ontological differentiation. Folding provides a model of the world and its knowing based not upon the ‘appearance’ or ‘interpretation’ of any exterior idea or phenomenon (that is, upon an image) but on the immanence of matter in flux, matter, which cannot (because of this flux) be pinned to any representational code […]. As for Cusanus, ‘enfolding’ for Deleuze is the opposite of image.” (ibid., p. 256. Cursive by M.F.). For a critical discussion on the Deleuzian conception of the fold, see the second appendix in this book.
- 45.
See: Bense (1949, p. 107) (see also Appendix B in this book).
- 46.
- 47.
Bach (1996, pp. 281–282).
- 48.
Ibid., p. 282.
- 49.
Bruck (1905, p. 24).
- 50.
Panofsky (1955, p. 244).
- 51.
Ibid.
- 52.
As we will see in the next section, the claim is not true: a contemporary of Dürer with respect to his investigation of polyhedra with nets was Charles Bovelles.
- 53.
Ibid., p. 259 (cursive by M.F.)
- 54.
Dürer (1977 [1525], p. 24).
- 55.
Peiffer (2004, p. 245).
- 56.
Cf. also Peiffer (1995, p. 55), regarding the treatment by Dürer of the spiral in Underweysung, a description that can also be easily applied here: “The geometric substrate is incorporated into a certain materiality. The treatment of the spiral in the Underweysung […] is very clear in this respect: Dürer starts from the definition of Archimedes of which he realizes a material model.”
- 57.
Panofsky (1955, p. 261).
- 58.
Cf. also Heuer (2011, p. 265). André Scala expresses a different opinion regarding the relations between perspective drawing and the unfolded nets of Dürer. After discussing the essential character of the operation of the unfolding of solids to a two-dimensional net in Dürer’s writing, Scala notes that the same epistemological operation occurs within perspective: “It would not be […] forced to say that perspective […] expresses a double movement of opening of the space of the plane (the Albertian window) and simultaneously of closure (the vanishing point or the horizon line). The double fold of the perspective is the fold [repli] of the third dimension on the second and the unfolding [dépli] of the second into a third of a different nature. And it is by virtue of this eminent duplicity that the perspective duplicates relations of measure, proportion, magnitude, and distance.” (translated from: Scala 2017, p. 52). Peter-Klaus Schuster (in Schuster 1991, pp. 190–191) holds a somewhat similar opinion—i.e., commenting, concerning the “mathematization of nature,” that both the two-dimensional nets and the (projections of the) three-dimensional Platonic and Archimedean solids have, for Dürer, a common aim: both have a similar “Abbildungsfunktion” and both reflect the mathematical structure that lies even in the most irregular phenomena of nature (to avoid confusion, it is to be noted that this mathematization of nature is not expressed by Scala as one of consequences of the nets or the projection).
- 59.
This may also be understood from Panofsky’s remark regarding the “proto-topological” method that such models constitute.
- 60.
In a similar fashion, if one starts from the construction of the folded net, one has to calculate accurately the lengths of the edges and the angles.
- 61.
Peiffer (2004, p. 275).
- 62.
Shephard (1975). Note, though, that Shephard does not mention Dürer in his article.
- 63.
Bern et al. (2003).
- 64.
See: Alpers (1982, p. 187): With Dürer’s perspective machine, “sight or vision is defined geometrically in this art. It concerns our measured relationship to objects in space rather than the glow of light and color.” For a discussion on Dürer’s perspective machine, see: Kemp (1990, pp. 171–173), Dürer (1995, pp. 357–360) (for an appendix on this machine, written by Jeanne Peiffer), Andersen (2007, pp. 207–210).
- 65.
Kemp (1990, p. 167). See also: ibid., pp. 167–189 for a discussion on perspective machines up to the nineteenth century.
- 66.
Doran (2013, p. 86).
- 67.
- 68.
Staigmüller (1891, p. 55) claims that: “Regarding the regular and semi-regular solids, Pacioli always gives a perspective image but never a net; Dürer always a net, and never a perspective picture.” Dürer did, however, draw the polyhedron in Melencolia I from a perspectival view, as we have already seen.
- 69.
- 70.
See: Dürer (1995, p. 91) (in Jeanne Peiffer’s introduction).
- 71.
- 72.
da Vinci (1478–1519), Folio 198r, b.
- 73.
Ibid., Folio 178v, a.
- 74.
- 75.
- 76.
Bovelles (1510).
- 77.
See the triangles in Fig. 2.12 (which are eventually not part of the faces of the icosahedron) and cf.: Weitzel (2007, p. 142). Weitzel refers to the 1510 manuscript in Latin. Bovelles’s other manuscripts, published in 1511 and 1551 in French (Bovelles 1511, 1551), did not present new ideas or an improvement on the nets he had already presented in Libellus de Matematicis corporibus.
- 78.
- 79.
Sanders (1984, p. 528).
- 80.
See also: Heuer (2011, pp. 263–264), who suggests that along with the Italian influence, the possible influence of Bovelles may, in fact, be a “direct source for Dürer’s polyhedrons [also coming from] Northern handcraft and theology” (ibid., p. 264).
- 81.
For a survey of how print was used in the renaissance, see: Karr Schmidt (2011a, esp. pp. 73–91).
- 82.
“There are extant medieval manuscripts containing scientific diagrams with movable parts” (Murdoch 1984, p. 132; see also: ibid., pp. 266–267, regarding the astrolabe in manuscripts. Cf. also: Braswell-Means 1991). For paper instruments from the end of the fifteenth century and on, see: Karr Schmidt (2006a).
- 83.
Cf. Karr Schmidt (2008).
- 84.
See: Turner (1989, esp. p. 32): “[The] astrolabe and other instruments were most widely known and used. If this be so, and the case of globes where the paper instrument displaced the metal one almost entirely tends to confirm the suggestion, then it entails a considerable revision in our ideas about the distribution and availability of instruments in the late sixteenth and early seventeenth centuries.”
- 85.
In this respect, these paper instruments could be considered as a preliminary version of the physical mathematical models, a tradition that reached its peak at the end of the nineteenth century. See Sect. 4.1.1.
- 86.
Karr Schmidt (2006a, p. 211).
- 87.
Regiomontanus was the pseudonym of Johannes Müller (1436–1476), a German mathematician and astronomer. Erhard Ratdolt reprinted the Calendarium in Venice in 1476.
- 88.
Zinner (1990, p. 169): “Among Dürer’s books was Regiomontanus’s copy of Euclid’s Elements which was later owned by Professor Saxonius of Altdorf, and was lost after the latter’s death in 1625.”
- 89.
Karr Schmidt (2005, p. 12).
- 90.
Eagleton (2010, pp. 159–160). As Eagleton later notes regarding sundials: “Once a woodcut had been prepared, off prints could be sold along with the books describing dials. This potential—to make an instrument by copying or cutting out an illustration, to use a woodcut illustration to make a paper instrument by pasting it to a sheet of wood or metal and attaching a cursor—was built into the illustrations in sixteenth-century dialling books” (ibid., p. 161).
- 91.
Karr Schmidt (2006b, p. 303).
- 92.
Contrary to the modern conception of “hands on mathematics,” which is characterized by problem-solving, here, “hands on” emphasizes the haptic aspect of folding.
- 93.
- 94.
Karr Schmidt (2006a, p. 185).
- 95.
See: Hartmann (2002, p. 198, footnote 22): “One of the words [Hartmann] used for parabola is ‘prenlinien’ which he got from [Dürer’s] Underweysung […].”
- 96.
Citation of Hartmann in a letter to Herzog Albrecht von Preußen, June 8, 1544, in: Klemm (1990, p. 39).
- 97.
Ibid., pp. 38–39: “[Sie] seyn fil leichtlicher zu vorstendigen, ßo man solche mit der handarbeyt anzayget dan mit der schriftt.”
- 98.
For the octagonal sundial, see: Karr Schmidt (2006a, p. 545).
- 99.
Dackerman (2011, p. 294).
- 100.
- 101.
Zinner (1956, p. 80).
- 102.
See: Ibid., pp. 78–84.
- 103.
Gouk (1988, p. 28): “Sundials seem to have been among the first ‘specialty’ ivory products made commercially in Nuremberg. The earliest surviving Nuremberg ivory sundial is probably the small pillar dial now in the Bayerische Nationalmuseum, Munich [from the end of the 15th century]. […] The idea of making a folding instrument, which resembled a writing tablet, book, or small box, obviously gained popularity at a very early date. Most surviving diptychs, […] date from the mid-sixteenth century. Yet scattered among various museums in Europe and America there are a few diptych dials, mostly incomplete, which appear to have been made at the very end of the fifteenth century. This provisional dating is not inconsistent with the first reference to ‘compass-makers’ in Nuremberg city records of the 1480s and 1490s.”
- 104.
Karr Schmidt (2011b, p. 273). See Penelope Gouk’s translation of the 1574 bylaws of Nurnberg, in: Gouk (1988, p. 79): “Compasses should be drawn and divided freehand, but until now compasses have been covered with painted paper which does not endure […] and is a mere deception through which the buyer is cheated […].”
- 105.
Ibid.
- 106.
- 107.
Karr Schmidt (2006a, p. 195).
- 108.
Ibid., p. 209.
- 109.
Although Dürer was not necessarily the only person to have this insight; see: Gouk (1988, p. 82): “The compass-makers were general artisans who presumably had no training in theoretical gnomonics. Yet their instruments incorporated technical features which suggests that they were copied from designs that were initially provided by the collaboration between a mathematician and engraver.”
- 110.
Karr Schmidt (2006a, p. 228). See also ibid., pp. 224–233 for a survey of Globus gores during the sixteenth century.
- 111.
Ibid., p. 225. See also: Dackerman (2011, p. 94).
- 112.
Karr Schmidt (2006a, p. 235).
- 113.
Note that the Globus gores and the gridded terrestrial globe had a direct connection to linear perspective painting: “[Peter] Apian’s and Gemma Frisius’s references effectively suggested that cosmographical maps and globes achieved imitatio, the perfect illusion of visible nature, by applying techniques similar to those of linear perspective painting. In this case, naturalistic representation no longer was a means towards legitimation of cosmography, but offered legitimacy in itself.” (in: Vanden Broecke 2000, p. 138) As was seen before, folding as a way to represent polyhedra was a complementary way to offer this legitimacy.
- 114.
Dürer (1528, fol. Y3v).
- 115.
Dürer (1977 [1525], p. 382, 384). See Heuer, p. 260: “Dürer utilized smaller flaps in his Unterweyssung der Messung to chart distances between a viewer and a cube.”
- 116.
Ibid.
- 117.
Lutz (2010, p. 33): “It seems as if experimentation with practices of opening and closing, as seen in the diptych or triptych, would also lead to a new reflection on the processes of folding in the book and in other media and vice versa.”
- 118.
Cf. ibid., p. 37.
- 119.
Rimmele (2010, pp. 195–196).
- 120.
- 121.
Ibid. See also Heuer (2011, p. 262): “But like the printed triptych, the image of the fold invokes the tension between two opposed states, absent and present, past time and now.”
- 122.
- 123.
- 124.
Hours of Catherine of Cleves (1440, pp. 306–307). See also: Plummer (1964, p. 70), concerning the pages devoted to Saint Agatha. A similar pattern appears in the nineteenth century book “Cassell’s Book of In-Door Amusements, Card Games and Fireside Fun,” where it is presented as a trick within recreational mathematics. See: Cassell (1881, p. 70).
- 125.
- 126.
Ibid., p. 12f.
- 127.
Gumbert (1994).
- 128.
Carey (2003, p. 484) (cursive by M.F.).
- 129.
Other names were also used: plicatif, pieghevole, Faltbuch, Faltband (Gumbert 2016, p. 19). Another class of folded manuscripts is fold out concertinas, see: Ibid., pp. 213–215.
- 130.
Ibid., p. 17.
- 131.
Ibid., p. 23.
- 132.
Ibid.
- 133.
Ibid., p. 19.
- 134.
Ibid., p. 20.
- 135.
Kunze (1975, p. 151).
- 136.
Imposition is a printing term and refers to one of the steps in the printing process. It consists of the arrangement of the printed product’s pages on the printer’s sheet, folding a sheet of a printed paper (which already consists of several printed pages) in order to obtain a faster printing technique, and to simplify binding.
- 137.
Febvre and Martin (1976, p. 30).
- 138.
Ibid., p. 32.
- 139.
Ibid.
- 140.
Ibid., p. 16: “[…] the introduction of paper […] did not replace parchment, but supplemented it and made possible the production of less expensive books alongside the traditional luxury manuscripts.”
- 141.
On printing as a mass media production, see: Stewart (2013).
- 142.
Febvre and Martin (1976, p. 41).
- 143.
See: ibid., pp. 180–186. In the first 50 years of the existence of the printing industry, until the end of the fifteenth century, “[f]irst in Germany, then in Italy and finally in France, the industry had developed centers of large-scale production.” (ibid., p. 186.)
- 144.
Ibid., p. 70. Note that from the fourth century onwards, the codex, made of parchment, gradually replaced the scroll, usually made of papyrus. The codex was a book constructed of a number of sheets of paper, vellum, papyrus, or similar materials, which were bound together. See: Funke (1999, pp. 67–71).
- 145.
Febvre and Martin (1976, p. 70).
- 146.
Clemens and Graham (2007, p. 14).
- 147.
Ibid., pp. 15–16.
- 148.
Ibid., p. 16.
- 149.
See: Febvre and Martin (1976, pp. 68–70).
- 150.
Note, however, that this may be a historically idealized picture, see: ibid., p. 36: “So as long as rag remained the essential raw material of papermaking—from the fourteenth to the nineteenth centuries—the expansion of the papermaking industry appeared forever threatened by lack of raw materials. At Troyes and perhaps Venice in the sixteenth century, in the Auvergne and Angoumois in the seventeenth and eighteenth centuries, papermakers were forced to sacrifice quality to quantity in face of the increasing demand.”
- 151.
In contrast to what can be seen, for example, in the forms of the diptych, the triptych and the polyptych (from which the codex stemmed; Funke 1999, p. 70). The folding of wood tablets was a form of folding along an already fixed and pre-determined crease.
- 152.
In contrast to vellum, which could not be mass-produced.
- 153.
Krauthausen (2016, p. 42).
- 154.
Ibid., p. 38.
- 155.
It is essential to remember that Dürer lived in Nuremberg, preparing numerous copper engraving and woodcuts to be printed. By Dürer’s time, Nuremberg was already a well-established center for printing.
- 156.
- 157.
With the drawing of the dodecahedron (Schmid 1539, p. 22).
- 158.
See: Richter (1994, pp. 60–61). Richter notes that the different methods used to draw the three-dimensional projections of the Platonic solids designate Schmid as a “person eager to try out new things” (ibid., p. 61).
- 159.
Schmid (1539, p. 22).
- 160.
Ibid., pp. 22–23.
- 161.
Ibid., p. 23.
- 162.
- 163.
- 164.
See: Hirschvogel (1543a), at the end of the first part.
- 165.
Cf. Andrews (2016, p. 420): “Hirschvogel’s two-volume Geometria was exceptionally careful to include self-similar information on the same page and to accurately cross-reference image and text, perhaps in response to the natural confusion that might have arisen from Underweysung.” For a survey of Hirschvogel’s whole book, see: Kühne (2002, pp. 244–250). Cf. also Weitzel (2007, pp. 145–146).
- 166.
Hirschvogel carved several other solids, yet he did so without carving their corresponding nets. See: Schwarz (1917, pp. 198–199).
- 167.
Hirschvogel (1543b, p. AVIII) and on the cover (p. AI).
- 168.
- 169.
Hirschvogel (1543a, section 4).
- 170.
The text can be found in: Addabbo (2015).
- 171.
See: Lagarias and Zong (2012).
- 172.
The text was edited and translated from Latin to Italian in: Addabbo (2015). See: ibid., pp. 134–142, for a critical analysis, and pp. 286–296 for the translation.
- 173.
Ibid., p. 289.
- 174.
Ibid., p. 290.
- 175.
Ibid., p. 291.
- 176.
Another possible explanation is that Maurolico may have thought that the number of the different nets for a given polyhedron has a certain meaning, except being greater than one (for example, for two polyhedron dual to each other, this number is the same for both). However, Maurolico does not explain why, for several polyhedra, he presents that there are more nets than with the others.
- 177.
Ibid., p. 140.
- 178.
Cf. also: Lakatos (1976) for an analysis of the proof of Euler’s formula, a survey which, of course, does not take into account Maurolico’s unpublished manuscript.
- 179.
Barbaro mentions him on the first page of the introduction to his book (Barbaro 1569, at the “Proemio”).
- 180.
See: Field (1997, pp. 269–271) concerning the acquaintance of Barbaro with Pacioli’s and Piero della Francesca’s writings.
- 181.
See: Field (1997, p. 271): “Thus in Chapter 16 Barbaro describes a solid (allegedly formed by cutting away solid angles of the cuboctahedron) whose net shows that it has vertices of two different kinds, one of which is surrounded by two equilateral triangles and two hexagons. Such a vertex would be flat, but neither author nor illustrator seems to have noticed, and a perspective view of the solid duly appears in the text. It would be possible to construct such a solid if some of its triangular and hexagonal faces were not regular, but the text does not mention such a possibility, and the illustrations appear to show all the polygonal faces as regular. A similar solecism occurs in Chapter 19, where we have a solid whose net shows that some of its vertices are surrounded by three hexagons, again making flat Vertices. Again, the silence in the accompanying text suggests that Barbaro had not noticed this fact, though the revisions to his earlier draft had eliminated other solids beset with the same kind of problem.”
- 182.
Field (1997, p. 272).
- 183.
Ibid., pp. 271–272.
- 184.
The title of the second chapter is “spiegatura, dritto et adombratione della Piramide” (Barbaro 1569, third part, p. 45).
- 185.
Ibid.
- 186.
- 187.
Concerning the translations of Euclid’s Elements into English, see: Barrow-Green (2006, esp. pp. 4–7), concerning Dee and Billingsley.
- 188.
To be exact, 38 of the figures found are pop-up constructions. The pop-up diagrams are cut to lie flat. Hence, the book can be closed without gaining thickness. Cf. also Evenden-Kenyon (2008, p. 150): “The illustrations required for Billingsley’s translation included in Book II a series of onlays intended to be pasted at one edge over the illustration to create individual three-dimensional diagrams.”
- 189.
Ibid.
- 190.
Whereas nets of other solids appear in: Billingsley (1570, folios 322–322v).
- 191.
Ibid., 320v.
- 192.
Ibid., preface, sig. ciiii recto, where Dee mentions Dürer’s Vier Bücher von Menschlicher Proportion; see also: Yates (1969, pp. 23–24, 191).
- 193.
Cf. also: Archibald (1950, p. 450).
- 194.
According to Elizabeth Evenden-Kenyon (private communication, 16.1.2017), John Day may have been inspired by the 1564 edition of Peter Apian’s Cosmographia, an astrological book that also included pop-up diagrams and movable parts.
- 195.
Although there is an abundance of diagrams in the manuscript of Billingsley. For the role of the diagrams in Billingsley’s translation, see: Barany (2010).
- 196.
Euclid (1908a, p. 110) (cursive by M.F.)
- 197.
Kemp (1990, p. 184).
- 198.
Stifel (1544, p. 5) in the “De Erratis.”
- 199.
Ramus (1569, pp. 163–168).
- 200.
See: Arfe (1795 [1585], pp. 31–42).
- 201.
Dubreuil (1647, pp. 88–123).
- 202.
Ibid, p. 88, 89, 119.
- 203.
- 204.
See the translation of Stevin’s work in Struik (1958, p. 223): “in the said description of Albert [Dürer] there are also two other solids which are composed by the folding of planes, one of which cannot be folded; the reason is that for the construction of one solid angle, three equal plane angles equal to four right angles have been put together, which do not constitute a solid angle […]. And the other solid is not included between the boundaries which are set […], for which reasons we have omitted those two solids.”
- 205.
Ozanam (1778, p. 318 and plates 6 and 7).
- 206.
Andersen (2007, pp. 570–571).
- 207.
Cowley (1752), “Advertisement,” between plate 1 and plate 2.
- 208.
Ibid.
- 209.
See: Hart (2000, p. 188): “The only parallelogram-faced zonohedra [a zonohedron is a convex polyhedron in which every face is a polygon with point symmetry, or alternatively, the set of points in 3-dimensional space constructed from vectors v i by taking the sum of a i v i , where each a i is a scalar between 0 and 1] known in the classical literature, including Euclid’s Elements, are parallelepipeds, which includes the special cases of rhombohedra and the cube. The next examples date to the German renaissance, when Johannes Kepler discovered the rhombic dodecahedron (RD) and the rhombic triacontahedron (RT) […].
“Kepler’s RD is geometrically distinct from (but topologically identical to) the form […] called ‘the rhombic dodecahedron of the second kind’ (RD2). […] The RD2 is often attributed to Bilinski, who described it in 1960, but the form appears two centuries earlier in John Lodge Cowley’s 1751 book Geometry made Easy. It contains a series of paper pop up models of polyhedra and conic sections, one of which […] has the form of the RD2.” (in the above citation, Hart refers to: Bilinski 1960).
- 210.
Grünbaum (2010, p. 14): “Since the angles of the rhombi are, as close as can be measured, 60° and 120°, the obtuse angles of the shaded rhombus would be incident with two other 120° angles—which is impossible.”
- 211.
ibid.
- 212.
The preface calls the attention of the reader to “moveable diagrams,” referring perhaps to glued flips or pop-up models (Cowley, 1758, p. A).
- 213.
Ibid, p. 7.
- 214.
Ibid., plates VI–X..
- 215.
In vol. XX of the 1759 “The Monthly Review” (London: Griffiths), p. 563.
- 216.
Holtzapffel (1846, pp. 380–385 and esp. p. 384, footnote †).
- 217.
Marie (1835, p. iii): “Lorsqu’on veut passer du rabattement d’un corps a sa représentation réelle, il faut relever la figure et la plier […].”
- 218.
See: Aleksandrov (1950).
- 219.
In: Bern et al. (2003). Note that the abstract of this paper is enlightening with respect to the question of the limits of foldability: “Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ‘open’ polyhedra with triangular faces may not be unfoldable no matter how they are cut.” (ibid., p. 51)
- 220.
This stagnation may be connected to another stagnation, this time between the practical and the theoretical work of the plans of Dürer, as Lefèvre hinted: “one has to state that the art of technical drawing in the sixteenth century did not keep pace with the theoretical foundations which Albrecht Dürer had developed for constructing interrelated plans at the beginning of the century and which were not continued and developed further earlier than in the seventeenth century by, for example, Samuel Maralois […] and Gerard Desargues” (Lefèvre 2003, p. 71).
- 221.
For example, these techniques also appeared in books from the end of the nineteenth century. In 1897, Alfonso Rivelli published his book Stereometria applicata allo sviluppo dei solidi ed alla loro costruzione in carta, containing numerous nets of polyhedra (see: Rivelli 1897).
- 222.
- 223.
See: Richter (1994, pp. 69–82). Especially important is Jaminitzer’s Perspectiva. Corporum Regularium, published in 1568, containing, among others, “variations” of the Platonic and Archimedean solids.
- 224.
Kepler mentions an error in Dürer’s construction of the heptagon in Dürer’s second book of the Underweysung (See: Kepler 1997 [1619], pp. 75–76).
- 225.
See, for example: Grunert (1827).
- 226.
Cf. also Lakatos (1976, p. 95, footnote 146).
- 227.
Schläfli (1901, p. 43).
- 228.
Hoff (1877, p. 46).
- 229.
Schlegel (1892, p. 67).
- 230.
For an extensive survey of Pacioli’s life and work, see: Mackinnon (1993).
- 231.
Agostini (1924).
- 232.
Pacioli (1997).
- 233.
- 234.
Pacioli (1997, p. 180).
- 235.
Ibid., p. 182.
- 236.
Ibid., pp. 182–183.
- 237.
Ibid., p. 248: “saper fare una squadra giusta subito materiale senza sexton.”
- 238.
Ibid.: “Et perché molte volte in facto non si trova havere dicto strumento, né riga e ne sexto, el modo ligiadro et giusto qui te insegno avendo tu adosso, che raro sia a chi sa legere un poco dc carta scripta netta.”
- 239.
Ibid., p. 249.
- 240.
Ibid., pp. 300–302.
- 241.
See: Bossi (2008, p. 124).
- 242.
Risi (2015, p. 3).
- 243.
Cf. also the discussion on the Derridean economy of excess and lack of the fold presented in Sect. 1.4.
- 244.
As I have shown, this has changed dramatically with Dürer’s nets.
- 245.
Russo (1998, p. 195).
- 246.
Rosenfeld (1988, p. 110). (cursive by M.F.)
- 247.
As Vincenzo de Risi remarks, Patrizi “was among the first to succeed in formulating that new concept of space as a three-dimensional extension independent of the bodies occupying it which was to become the core of much metaphysics (and of much science) once the modern age had reached its maturity.” (Risi 2016, p. 60).
- 248.
Risi (2015, p. 6): “The algebraization of geometry, in fact, required geometrical figures to be conceived as embedded in a larger (indefinite) extension, that can be captured in a system of coordinates.”
- 249.
Cf. Henry (2001).
- 250.
Koyré (1957, p. viii).
- 251.
Risi (2016, p. 66).
- 252.
This already shows the difference between this conception and Heron’s (see Chap. 1): Heron’s rotation pre-assumed the existence of an infinite three-dimensional space.
- 253.
Regier and Vermeir (2016, p. 12).
- 254.
See: Hon and Goldstein (2008) and the next chapter.
- 255.
See, among others: Schwenter (1636, pp. 521–522, 568), Ozanam (1723, plate 9), Alberti (1747, p. 200–202) and Halle (1787, p. 311). The folding exemplified in these manuscripts has an implicit geometry, and although these books were written by mathematicians and scientists, only few mathematical constructions were developed with these geometric folded forms. This also includes the art of napkin-folding (see: Sallas 2010), starting from the fifteenth century, which I will discuss briefly in Sect. 4.2.1.2, and which was also based on implicit geometrical principles.
- 256.
The encounter between recreational mathematics and folding and the ways in which, on the one hand, this encounter used folding to spread scientific knowledge practically and, on the other, conceived of it mainly as a practical and not as a theoretical procedure is beyond the scope of the present book. See however: Friedman and Rougetet (2017).
- 257.
Regier and Vermeir (2016, p. 28).
References
Addabbo C (2015) Il “Libellusaurolico e la tassellazione dello spazio, PhD. Dissertation, University of Pisa
Agostini A (1924) De Viribus Quantitatis di Luca Pacioli. Period Mat 4:165–192
Alberti GA (1747) I giochi numerici. B. Borghi, Bologna
Aleksandrov AD (1950) Vypuklye mnogogranniki [Russian: Convex Polyhedra]. Gosudarstv Izdat Tekhn-Teor Lit, Moskow-Leningrad
Alpers S (1982) Art history and its exclusions: the example of Dutch art. In: Broude N, Garrard MD (eds) Feminism and art history: questioning the litany. Harper & Row, Boulder, pp 183–199
Andersen K (2007) The geometry of an art. The history of the mathematical theory of perspective from Alberti to Monge. Springer, New York
Andrews N (2016) Albrecht Dürer’s personal Underweysung der Messung. Word Image 32(4):409–429
Archibald RC (1950) The first translation of Euclid’s elements into English and its source. Am Math Mon 57(7):443–452
Bach FT (1996) Struktur und Erscheinung: Untersuchungen zu Dürers graphischer Kunst. Gebr. Mann, Berlin
Barany MJ (2010) Translating Euclid’s diagrams into English, 1551–1571. In: Heeffer A, Van Dyck M (eds) Philosophical aspects of symbolic reasoning in early modern mathematics. College Publications, London, pp 125–163
Barbaro D (1569) La pratica della perspettiva. Borgominieri, Venice
Barbin É (2007) Les Récréations: des mathématiques à la marge. Science 30:22–25
Baron F (ed) (1978) Joachim Camerarius: (1500–1574). Beiträge zur Geschichte des Humanismus im Zeitalter der Reformation. Fink, Munich
Barrow-Green J (2006) ‘Much necessary for all sortes of men’: 450 years of Euclid’s elements in English. BSHM Bull 21(1):2–25
Bense M (1949) Konturen einer Geistesgeschichte der Mathematik. Die Mathematik in der Kunst, vol 2. Claassen & Goverts, Hamburg
Bern M, Demaine ED, Eppstein D, Kuo E, Mantler A, Snoeyink J (2003) Ununfoldable polyhedra with convex faces. Comput Geom 24(2):51–62
Bilinski S (1960) Über die Rhombenisoeder. Glasnik Mat Fiz Astr 15:251–263
Billingsley H (1570) The elements of Geometrie. John Day, London
Bossi V (2008) Magic and card tricks in Luca Paciolo’s De Viribus Quantitatis. In: Demaine E, Demaine M, Tom R (eds) A lifetime of puzzles. A K Peters, Wellesley, pp 123–129
Braswell-Means L (1991) The vulnerability of Volvelles in manuscript codices. Manuscripta 35(1):43–54
Bressanini D, Toniato S (2011) I giochi matematici di fra’ Luca Pacioli. Dedalo, Bari
Broecke SV (2000) The use of visual media in renaissance cosmography: the cosmography of Peter Apian and Gemma Frisius. Paedagog Hist 36(1):130–150
Bruck R (ed) (1905) Das Skizzenbuch von Albrecht Dürer. Heitz, Straßburg
Carey HM (2003) What is the folded Almanac?: the form and function of a key manuscript source for astro-medical practice in later medieval England. Soc Hist Med 16(3):481–509
Cassell (ed) (1881) Cassell’s book of in-door amusements, card games and Fireside Fun, 3rd edn. Cassell, Petter, Galpin, London
Cerda EA, Mahadevan L, Pasini JM (2004) The elements of draping. Proc Natl Acad Sci 101(7):1806–1810
Clemens R, Graham T (2007) Introduction to manuscript studies. Cornell University Press, Ithaca
Cowley JL (1752) Geometry made easy. Mechel, London
Cowley JL (1758) An appendix to Euclid’s Elements. Watkins, London
da Vinci L (1478–1519) Codice Atlantico (Hoepli ed). http://www.leonardodigitale.com/index.php?lang=ENG
Dackerman S (ed) (2011) Prints and the Pursuit of knowledge in early modern Europe. Yale University Press, New Haven
de Arfe, J (1795 [1585]) De Varia commensuración para la escultura y arquitectura, 7th edn. Plácido Barco Lopez, Madrid
de Bovelles C (1510) Libellus de Mathematicis corporibus. Henri Estienne, Paris
de Bovelles C (1511) Geometrie en Françoys. Henri Estienne, Paris
de Bovelles C (1551) Geometrie Practique. Regnaud Chaudiere, Paris
de Risi V (2015) Introduction. In: de Risi V (ed) Mathematizing space. The objects of geometry from antiquity to the early modern age. Birkhäuser, Cham, pp 1–13
de Risi V (2016) Francesco Patrizi’s conceptions of space and geometry. In: Vermeir K, Regier J (eds) Boundaries, extents and circulations. Space and spatiality in early modern natural philosophy. Springer, Basel, pp 55–106
Dee J (1975) The mathematical preface to the elements of geometrie of Euclid of Megara (1570). Science History Publications, New York
Doran S (2013) The culture of yellow- or, the visual politics of late modernity. Bloomsbury, New York
Dubreuil J (1647) Seconde partie de la perspective pratique qui donne une grande facilité à trouver les apparences de tous les corps solides, tant reguliers qu’irreguliers … veuve de FranÓois L’Anglois dit Chartres, Paris
Dürer A (1528) Vier Bücher von menschlicher Proportion. Hieronymus Formschneider, Nuremberg
Dürer A (1532) Institutionum Geometricarum. Christian Wechel, Paris
Dürer A (1977 [1525]) The Painter’s manual (trans with comments: Strauss WL). Abaris Books, New York
Dürer A (1995) Géométrie (presentation, trans and comments: Peiffer J). Seuil, Paris
Eagleton C (2010) Monks, manuscripts and sundials: the Navicula in medieval England. Brill, Leiden
Euclid (1908a) The thirteen books of the elements, vol 1: Books 1–2 (trans: Heath TL). The University Press, Cambridge
Evenden-Kenyon E (2008) Patents, pictures and patronage: John Day and the Tudor Book Trade. Ashgate, Hampshire
Eves HW (1969) In mathematical circles: Quadrants I and II. Prindle, Weber & Schmidt, Boston
Febvre L, Martin H-J (1976) The coming of the book. The impact of printing 1450–1800 (trans: Gerard D). NLB, London
Felfe R (2015) Naturform und bildnerische Prozesse. Walter de Gruyter GmbH, Berlin
Field JV (1997) Rediscovering the Archimedean Polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler. Arch Hist Exact Sci 50(3):241–289
Field JV (2005) The invention of infinity: mathematics and art in the renaissance, 2nd edn. Oxford University Press, Oxford
Friedman M, Rougetet L (2017) Folding in Recreational Mathematics during the 17th–18th Centuries: Between Geometry and Entertainment. Acta Baltica Historiae et Philosophiae Scientiarum 5(2):5–34.
Funke F (1999) Buchkunde: Ein Überblick über die Geschichte des Buches. K.G. Saur, Munich
Gouk P (1988) The ivory sundials of Nuremberg 1500–1700. Whipple Museum of the History of Science, Cambridge
Grimm J, Grimm W (1971) Deutsches Wörterbuch von Jacob und Wilhelm Grimm, 16 vol in 32 sub-volumes, Lepizig. http://woerterbuchnetz.de/DWB/
Grünbaum B (2010) The Bilinski dodecahedron and assorted parallelohedra, zonohedra, monohedra, isozonohedra, and otherhedra. Math Intell 32(4):5–15
Grunert J (1827) Einfacher Beweis der von Cauchy und Euler Gefundenen Sätze von Figurennetzen und Polyedern. J Reine Angew Math 2:367
Gumbert JP (1994) Über Faltbücher, vornehmlich Almanache. In: Rück P, Boghardt M (eds) Rationalisierung der Buchherstellung im Mittelalter und in der frühen Neuzeit. Institut für Historische Hilfswissenschaften, Marburg an der Lahn, pp 111–121
Gumbert JP (2016) Bat books. A catalogue of folded manuscripts containing almanacs or other texts. Brepols, Turnhout
Hafner I (2008) Space filling with a rhombic dodecahedron of the second kind. http://demonstrations.wolfram.com/SpaceFillingWithARhombicDodecahedronOfTheSecondKind/. Last accessed 15 Mar 2017
Halle JS (1787) Magie, oder die Zauberkräfte der Natur. Johann Thomas Edlen, Trattnern
Hart GW (2000) A color-matching dissection of the rhombic enneacontahedron. Symmetry: Cult Sci 11(1–4):183–199
Hartmann G (2002) Hartmann’s Practika: a manual for making sundials and astrolabes with the compass and rule (trans, ed: Lamprey J). Lamprey, Bellvue
Henry J (2001) Void space, mathematical realism and Francesco Patrizi da Cherso’s use of atomistic arguments. In: Lüthy C, Murdoch JE, Newman WR (eds) Late medieval and early modern corpuscular matter theories. Brill, Leiden, pp 133–161
Heuer CP (2011) Dürer’s folds. RES: Anthropol Aesthet 59/60:249–265
Hirschvogel A (1543a) Eigentliche und gründliche Anweisung in die Geometria. Self-published, Nuremberg
Hirschvogel A (1543b) Geometria (the accompanying plates book to Hirschvogel, 1543a). Self-published, Nuremberg
Hofmann JE (1971) Dürers Verhältnis zur Mathematik. In: Albrecht Dürers Umwelt, Festschrift zum 500. Geburtstag Albrecht Dürers. Verein für Geschichte der Stadt Nürnberg, Nürnberg, pp 132–151
Holtzapffel C (1846) Turning and mechanical manipulation. Bradbury and Evans, London
Hon G, Goldstein BR (2008) From summetria to symmetry: the making of a revolutionary scientific concept. Springer, New York
Hours of Catherine of Cleves (ca. 1440) Utrecht. http://www.themorgan.org/collection/hours-of-catherine-of-cleves/346. Last accessed 23 July 2017
Hughes GH (2012) The polygons of Albrecht Dürer – 1525. https://arxiv.org/abs/1205.0080. Last accessed 15 Mar 2017
Johnston S (1996) The identity of the mathematical practitioner in 16th-century England. In: Hantsche I (ed) Der “mathematicus”: Zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators. Brockmeyer, Bochum, pp 93–120
Karr Schmidt S (2005) Adult entertainment: collecting movable books and prints in the renaissance. Movable Book Soc Q 13(3):1, 12–13
Karr Schmidt S (2006a) Art—a user’s guide: interactive and sculptural printmaking in the renaissance, Ph.D. dissertation, Yale University
Karr Schmidt S (2006b) Interactive and sculptural prints: introducing an overlooked early modern genre. Print Q 23:(3):302–304
Karr Schmidt S (2008) Johannes Krabbes Papierastrolabium. In: Heitzmann C (ed) Die Sterne lügen nicht. Harrassowitz Verlag, Wiesbaden, pp 117–121
Karr Schmidt S (2011a) Altered and adorned. In: Nichols K (ed) Using renaissance prints in daily life. Art Institute of Chicago, Chicago
Karr Schmidt S (2011b) Georg Hartmann and the development of printed instruments in Nuremberg. In: Dackerman S (ed) Prints and the pursuit of knowledge in early modern Europe. Yale University Press, New Haven, pp 268–279
Kemp M (1990) The science of art: optical themes in western art from Brunelleschi to Seurat. Yale University Press, London
Kepler J (1997 [1619]) The harmony of the world (trans: Aiton EJ, Duncan AM, Field JV). Memoirs of American Philosophical Society, vol 209
Klemm HG (1990) Georg Hartmann aus Eggolsheim (1489–1564): Leben und Werk eines fränkischen Mathematikers und Ingenieurs. Gürtler-Druck, Forchheim
Knoespel KJ (1987) The narrative matter of mathematics: John Dee’s preface to the elements of Euclid of Megara (1570). Philol Q 66:35–54
Koyré A (1957) From the closed world to the infinite universe. The Johns Hopkins Press, Baltimore
Krauthausen K (2016) Folding the narrative: the dimensionality of writing in French structuralism (1966–1972). In: Friedman M, Schäffner W (eds) On folding. Towards a new field of interdisciplinary research. transcript, Bielefeld, pp 31–48
Krutzsch M (2008) Falttechniken an altägyptischen Handschriften. In: Graf J, Krutzsch M (eds) Ägypten lesbar machen – die klassische Konservierung. Walter de Gruyter, Berlin, pp 71–83
Krutzsch M (2009) Kniffen, Knicken, Falten. Was Objekte erzählen. In: Peltz U, Zorn O (eds) kulturGUTerhaltenRestaurierung archäologischer Schätze an den Staatlichen Museen zu Berlin. Philipp von Zabern, Mainz, pp 111–114
Kühne A (2002) Augustin Hirschvogel und sein Beitrag zur praktischen Mathematik. In: Gebhardt R (ed) Verfasser und Herausgeber mathematischer Texte der frühen Neuzeit. Adam-Ries-Bund, Annaberg-Buchholz, pp 237–252
Kunze H (1975) Geschichte der Buchillustration in Deutschland. Das 15. Jahrhundert, vol 1. Insel, Leipzig
Lagarias JC, Zong C (2012) Mysteries in packing regular tetrahedral. Notices AMS 59:1540–1549
Lakatos I (1976) Proofs and refutations. The logic of the mathematical discovery. Cambridge University Press, Cambridge
Le Goff J-P (1991) Aux confins de l’art et de la science: De prospectiva pingendi de Piero della Francesca. In: Bessot D, Hellegouarch Y, Le Goff J-P (eds) Destin de l’art. Desseins de la science. Actes du colloque A.D.E.R.H.E.M., Université de Caen, Caen, pp 185–254
Lefèvre W (2003) The limits of pictures cognitive functions of images in practical mechanics – 1400 to 1600. In: Lefèvre W, Renn J, Schoepflin U (eds) The power of images in early modern science. Birkhäuser, Basel, pp 69–88
Lutz H (2010) Medien des Entbergens. Falt- und Klappoperationen in der altniederländischen Kunst des späten 14. und frühen 15. Jahrhunderts. In: Engell L, Siegert B, Vogl J (eds) Archiv für Mediengeschichte – Renaissancen. Wilhelm Fink, Munich, pp 27–46
Mackinnon N (1993) The portrait of Fra Luca Pacioli. Math Gaz 77(479):130–219
Malkevitch J (2013) Milestones in the history of polyhedra. In: Senechal M (ed) Shaping space. Exploring polyhedra in nature, art, and the geometrical imagination. Springer, New York, pp 53–63
Marie F-CM (1835) Géométrie stéréographique, ou Reliefs des polyèdres. Bachalier, Paris
Münkner J (2008) Eingreifen und Begreifen. Handhabungen und Visualisierungen in Flugblättern der Frühen Neuzeit. Erich Schmidt, Berlin
Murdoch JE (1984) Album of science. Antiquity and the middle ages. Scribner’s and Sons, New York
Ozanam J (1723) Récréations mathématiques et physiques, vol 4. Jombert, Paris
Ozanam J (1778) Récréations mathématiques et physiques. Jombert, Paris
Pacioli L (1997) De viribus quantitatis (trans: Garlaschi M, ed: Marinoni A). Ente Raccolta Viniciana, Milano
Panofsky E (1955) The life and art of Albrecht Dürer. Princeton University Press, Princeton
Panofsky E (1991) Perspective as symbolic form (trans: Wood CS). Zone Books, New York
Panofsky E, Saxl F (1923) Dürers Melencolia I. Teubner, Leipzig
Peiffer J (1995) La style mathématique de Dürer et sa conception de la géométrie. In: Dauben J, Folkerts M, Knobloch E, Wußing H (eds) History of mathematics: state of the art. Academic, San Diego, pp 49–61
Peiffer J (1997) Dürers Geometrie als Propädeutik zur Kunst. In: Knobloch E (ed) Wissenschaft—Technik—Kunst, Interpretationen, Strukturen, Wechselwirkungen. Harrasowitz, Wiesbaden, pp 89–103
Peiffer J (2000) La creation d’une Langue mathematique allemande par Albrecht Dürer. In: Chartier R, Corsi P (eds) Sciences et langues en Europe. Éditions EHESS, Paris, pp 77–90
Peiffer J (2004) Projections embodied in technical drawings: Dürer and his followers. In: Lefèvre W (ed) Picturing machines. MIT Press, Cambridge, pp 245–275
Plummer J (1964) The book of hours of Catherine of Cleves. Pierpont Morgan Library, New York
Ramus P (1569) Arithmeticae libri. Episcopius, Basileae
Regier J, Vermeir K (2016) Boundaries, extents and circulations: an introduction to spatiality and the early modern concept of space. In: Vermeir K, Regier J (eds) Boundaries, extents and circulations. Space and spatiality in early modern natural philosophy. Springer, Basel, pp 1–32
Richter F (1994) Die Ästhetik geometrischer Körper in der Renaissance. Gerd Hatje, Stuttgart
Rimmele M (2010) Das Triptychon als Metapher, Körper und Ort. Semantisierungen eines Bildträgers. Wilhelm Fink, München
Rivelli A (1897) Stereometria applicata allo sviluppo dei solidi ed alla loro costruzione in carta. Hoepli, Milano
Rosenfeld BA (1988) A history of non-Euclidean geometry. Evolution of the concept of a geometric space (trans: Shenitzer A). Springer, New York
Rupprich H (ed) (1969) Dürer: schriftlicher Nachlass, vol 3. Deutsche Verein für Kunstwissenschaft, Berlin
Russo L (1998) The definitions of fundamental geometric entities contained in book I of Euclid’s elements. Arch Hist Exact Sci 52(3):195–219
Sallas J (2010) Gefaltete Schönheit. Die Kunst des Serviettenbrechens. Freiburg i. Breisgau, Wien
Sanders PM (1984) Charles de Bovelles’s treatise on the regular polyhedra (Paris, 1511). Ann Sci 41(6):513–566
Scala A (2017) Anmerkungen zur Genese der Zwiefalt bei Heidegger. In: Friedman M, Seppi A (eds) Martin Heidegger: Die Falte der Sprache. Turia + Kant, Vienna, pp 39–52
Schläfli L (1901) Theorie der vielfachen Kontinuität. Springer, Basel
Schlegel V (1892) Ueber Projectionen der mehrdimensionalen regelmässigen Körper. Jahresber Dtsch Math-Vereinigung 2:66–69
Schmid W (1539) Das erste Buch der Geometria. Petrejus, Nürnberg
Schreiber P (1999) A new hypothesis on Dürer’s enigmatic polyhedron in his copper engraving ‘Melencolia I’. Hist Math 26:369–377
Schröder E (1980) Dürer – Kunst und Geometrie. Akademie-Verlag/Birkhäuser, Berlin/Basel
Schuster P-K (1991) Melancolia I. Dürers Denkbild, vol 1. Gebr. Mann, Berlin
Schwarz K (1917) Augustin Hirschvogel: ein deutscher Meister der Renaissance. Bard, Berlin
Schwenter D (1636) Deliciae physico-mathematicae, oder mathematische und philosophische Erquickstunden. Dümler, Nürnberg
Shephard GC (1975) Convex polytopes with convex nets. Math Proc Camb Philos Soc 78(3):389–403
Sheppard E (2003) Marketing mathematics: Georg Hartmann and Albrecht Dürer, a comparison, Master dissertation in: History of Science, University of Oxford
Singmaster D (2008) De Viribus Quantitatis by Luca Pacioli: the first recreational mathematics book. In: Demaine E, Demaine M, Rodgers T (eds) A lifetime of puzzles. A K Peters, Wellesley, pp 77–122
Staigmüller HCO (1891) Dürer als Mathematiker. K. Hofbuchdrukker, Stuttgart
Steck M (1948) Dürers Gestaltlehre der Mathematik und der bildenden Künste. Max Niemeyer, Halle
Stewart AG (2013) The birth of mass media. In: Bohn B, Saslow JM (eds) A companion to renaissance and baroque art. Wiley, Chichester, pp 253–273
Stifel M (1544) Arithmetica integra. Nürnberg
Struik DJ (1958) The principal works of Simon Stevin, Mathematics, vol 2. C. V. Swets & Zeitlinger, Amsterdam
Turner AJ (1989) Paper, print, and mathematics: Philippe Danfrie and the making of mathematical instruments in late 16th century Paris. In: Blondel C et al (eds) Studies in the history of scientific instruments. Rogers Turner, London, pp 22–42
Hoff JH van’t (1877) Die Lagerung der Atome im Raume (trans: Hermann F). Vieweg, Braunschweig
Weitzel H (2004) A further hypothesis on the polyhedron of A. Dürer’s Engraving Melencolia I. Hist Math 31(1):11–14
Weitzel H (2007) Zum Polyeder auf A. Dürers Stich Melencolia I—ein Nürnberger Skizzenblatt mit Darstellungen archimedischer Körper. Sudhoffs Arch 91(2):129–173
Wölfflin H (1917) Kunstgeschichtliche Grundbegriffe: Das Problem der Stilentwicklung in der neueren Kunst, 2nd edn. Bruckmann, Munich
Wölfflin H (1950) Principles of art history (trans: Hottinger MD). Dover, New York
Yates FA (1969) Theatre of the world. Routledge/Kegan Paul, London
Young GC, Young WH (1905) The first book of geometry. Chelsea Publishing Company, New York
Zinner E (1956) Deutsche und Niederländische Astronomische Instrumente des 11.–18. Jahrhunderts. C.H. Beck, Munchen
Zinner E (1990) Regiomontanus, his life and work (trans: Brown E). North-Holland, Amsterdam
Zorach R (2009) Meditation, idolatry, mathematics: the printed image in Europe around 1500. In: Zorach R, Wayne Cole M (eds) The idol in the age of art. Ashgate, Aldershot, pp 317–342
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Friedman, M. (2018). From the Sixteenth Century Onwards: Folding Polyhedra—New Epistemological Horizons?. 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_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-72487-4_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-72486-7
Online ISBN: 978-3-319-72487-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)